On the Motion of a Charged Colloid with a Harmonic Trap

Abstract

In this study, we derive the Fokker–Planck equation for a colloidal particle subject to a harmonic trap and viscous forces under the influence of a magnetic field. We then extend the analysis to a charged colloid driven by both thermal and active noises in the same magnetic environment. Finally, the case of a charged colloid experiencing a harmonic trap together with thermal and active noises is investigated. Analytical solutions for the joint probability density are obtained in the limits of $t\ll \tau$, $t\gg \tau$, and $\tau =0$. For a colloid under a harmonic trap and magnetic field, the mean squared displacement exhibits a superdiffusive scaling proportional to $t^3$ in the short-time regime ($t\ll \tau$), while the mean squared velocity scales as $\sim t$ when $\tau =0$. For a charged colloid with thermal noise, the mean-squared displacement follows a superdiffusive form $\sim t^{2h+1}$ for $t\ll \tau$, and the mean squared velocity again scales linearly with time for $\tau =0$. When the active noise is included together with a harmonic trap, the characteristic time scale grows as $\sim t^4$ in the short-time regime, while the mean squared velocity becomes normally diffusive at $\tau =0$. In the long-time limit ($t\gg \tau$) and for $\tau =0$, the moments of the joint probability density under combined thermal and active noises scale as $\sim t^{4h+2}$, consistent with our analytical results. Notably, as $h\to 1/2$, the entropy of the joint probability density with thermal noise ${\zeta }_{\mathrm{th}}(t)$ coincides with that obtained for active noise ${\zeta }_{\mathrm{ac}}(t)$ in both $t\gg \tau$ and $\tau =0$ limits.

1. Introduction

Recently, many researchers have focused on the dynamics of particle systems classified into passive and active groups [1,2,3]. Passive particles, such as Brownian particles, undergo random motion due to thermal fluctuations and spatial interactions with their surroundings. In contrast, active particles—such as micro-swimmers—exhibit self-propelled motion with a sense of direction or purpose. The dynamical behaviors of macro-scale passive and micro-scale active particles have been extensively explored through theoretical, numerical, and experimental studies, and they continue to attract attention in ongoing research.

Lee and Kwon [4] investigated the motion of a colloidal particle confined in a harmonic trap under an external nonconservative force that produces a torque in the presence of a uniform magnetic field. They demonstrated that a steady state exists within a certain range of parameters, including the viscosity coefficient, the stiffness of the harmonic potential, and the magnetic field strength in the overdamped limit. Diffusion without a confining potential is inherently a nonequilibrium process, yet the presence of a magnetic field introduces additional complexity, as observed in many plasma systems. Diffusion under magnetic fields has been extensively studied [5,6,7,8], and nonequilibrium systems [9,10] in time-varying potentials have also been analyzed in the presence of constant magnetic fields [11,12,13].

Over the past few decades, anomalous diffusion dynamics [14,15,16,17,18] have been widely discussed across natural and complex systems [19,20,21,22]. For a typical anomalous diffusion process, the mean squared displacement scales as $⟨x^2(t)⟩\sim t^{\alpha }$, where the scaling exponent $\alpha$ characterizes the type of motion: subdiffusive ($0<\alpha <1$) if the mean squared displacement grows sublinearly with time, and superdiffusive ($\alpha >1$) if it grows superlinearly. Subdiffusion has been observed in endogenous submicron tracers within biological cells [21,23,24,25,26], artificially crowded biological media [27,28,29,30,31], protein dynamics in supercomputing studies [32,33], and lipid bilayer membranes with varying crowding conditions [34,35,36,37,38]. Superdiffusion, on the other hand, has been reported in several cellular systems [39,40,41].

As one example, the restoring force exerted by optical tweezers within biological cells [42,43] follows a fractional Ornstein–Uhlenbeck process [44]. In the underdamped limit, the active Ornstein–Uhlenbeck process exhibits ballistic motion, while more complex underdamped dynamics lead to hyperdiffusion for $t\ll \tau$. Furthermore, the active fractional Langevin equation (AFLE) has been developed to describe transport phenomena in active viscoelastic media, revealing weak ergodicity breaking—a phenomenon not observed in equilibrium systems. For Gaussian processes, statistical properties can be fully determined from the mean and covariance functions. The ergodic behavior of such processes is often inferred by comparing the mean-squared displacement and the time-averaged mean-squared displacement [45,46,47].

The diffusive motion of active particles has been analyzed primarily by solving generalized and fractional generalized Langevin equations and by deriving the corresponding fractional Fokker–Planck equations [48]. The Fokker–Planck framework remains a fundamental tool across various fields, including statistical physics, solid-state physics, quantum optics, applied statistics, chemical physics, theoretical biology, and circuit theory.

In this study, we present a new analytical approach to obtain the joint probability density of a charged colloid subjected to a harmonic and viscous force under the combined effects of thermal and active noises. We first derive the Fokker–Planck equation for a charged colloid and then apply a double Fourier transform to the joint probability density of displacement and velocity. The organization of this paper is as follows. In Section 2, we treat the motion of a charged colloid with a harmonic trap in the presence of a magnetic field. We derive the Fokker–Planck equation in our model. For a charged colloid with the thermal equilibrium noise ${\zeta }_{th}(t)$ and the active noise (exponentially decaying correlation) ${\zeta }_{ac}(t)$, we obtain the analytical solutions for the joint probability density and the numerical values of mean squared displacement and mean squared velocity from the joint probability density, which is observed in the limits of $t<<\tau$, $t>>\tau$ and for $\tau =0$, where $\tau$ is the correlation time. In Section 3, we solve the fractional Langevin equation for a charged colloid with ${\zeta }_{th}(t)$ and ${\zeta }_{ac}(t)$. The x- and y- component joint probability density for charged colloid with a harmonic force, a viscous force, and thermal and active noises is obtained in three-time domains. In Section 5, we present the calculated results of the non-Gaussian parameter, the correlation coefficient, and the moment related to the displacement and the velocity. Finally, we provide and account for a conclusion summarizing our key findings in Section 6.

2. Motion with a Harmonic Trap in the Presence of a Magnetic Field

In this section, we derive the Fokker–Planck equation and obtain the joint probability density for a charged colloid. From the joint probability density for a charged colloid, we get the mean squared displacement and the mean squared velocity in the limits of $t<<\tau$, $t>>\tau$ and for $\tau =0$, where $\tau$ is the correlation time.

2.1. The Joint Probability Density

The Hamiltonian for a charged colloid with a harmonic trap in the presence of a magnetic field is given by

$$H={\displaystyle \frac{1}{2m}}[(p_x+{\displaystyle \frac{q{B}_{x}y}{2}}{)}^2+(p_y-{\displaystyle \frac{q{B}_{y}x}{2}}{)}^2]+{\displaystyle \frac{1}{2}}(k_1{x}^2+k_2{y}^2),$$

(1)

where $\overrightarrow{B}=B_x(t)\overrightarrow{x}+B_y(t)\overrightarrow{y}+B_z(t)\overrightarrow{z}$. The two equations of motion for a charged colloid with $\overrightarrow{B}=B_z(t)\overrightarrow{z}$ are given by

$$m{\dot{v}}_x={\displaystyle \frac{1}{2}}(y{\dot{B}}_z+2v_y{B}_z)-k_{1}x-{\gamma }_1{v}_x+{\eta }_x(t),$$

(2)

$$m{\dot{v}}_y={\displaystyle \frac{q}{2}}(x{\dot{B}}_z+2v_x{B}_z)-k_{2}y-{\gamma }_2{v}_y+{\eta }_y(t).$$

(3)

Here, ${\dot{v}}_x={\displaystyle \frac{d}{dt}}{v}_x$ and ${\dot{B}}_z={\displaystyle \frac{d}{dt}}{B}_z$. The correlated Gaussian force ${\eta }_x(t)$ depends exponentially on the time difference: $<{\eta }_i\left(t\right){\eta }_i\left(t^{\prime }\right)>={\displaystyle \frac{{\eta }_{0i}^2}{2\tau }}\mathit{exp}(-{\displaystyle \frac{|t-t\prime |}{\tau }})$ and ${\eta }_{0i}^2=2{\gamma }_i{k}_B{T}_i$ for $i=x,y$. The viscous forces denote $-{\gamma }_1{v}_x$ and $-{\gamma }_2{v}_y$, the harmonic forces $-k_{1}x$, and $-k_{2}y$, and the dimensionless mass $m=1$.

For our model, taking the z-component magnetic field $\overrightarrow{B}=B\overrightarrow{z}$, the modified equations of motion from Equations (2) and (3) reduce to

$${\displaystyle \frac{d}{dt}}{v}_x\left(t\right)=-q^2{B}^2{v}_x\left(t\right)-k_{1}x\left(t\right)-{\gamma }_1{v}_x\left(t\right)+{\eta }_x\left(t\right)+qB{\eta }_y\left(t\right),$$

(4)

$${\displaystyle \frac{d}{dt}}{v}_y\left(t\right)=-q^2{B}^2{v}_y\left(t\right)-k_{1}y\left(t\right)-{\gamma }_2{v}_y\left(t\right)+{\eta }_y\left(t\right)-qB{\eta }_x\left(t\right).$$

(5)

In this paper, we derive the Fokker–Planck equations for $p(x,v_x,t)$ and $p(y,v_y,t)$, and find the solutions of $p(x,v_x,t)$ and $p(y,v_y,t)$ from two equations of motion, Equations (4) and (5). Firstly, the joint probability densities $p(x,v_x,t)$ and $p(y,v_y,t)$ for the displacement $x,y$ and the velocity $v_x,v_y$ are defined by $p(x,v_x,t)=<\delta (x-x(t))\delta (v_x-v_x(t))>$ and $p(y,v_y,t)=<\delta (y-y(t))\delta (v_y-v_y(t))>$. By taking time derivatives of the joint probability density, we have the differential equations as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p(x,v_x,t)=-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}}<{\displaystyle \frac{\mathit{\partial }x}{\mathit{\partial }t}}\delta (x-x(t))\delta (v_x-v_x(t))>-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_x}}<{\displaystyle \frac{\mathit{\partial }{v}_x}{\mathit{\partial }t}}\delta (x-x(t))\delta (v_x-v_x(t))>$$

(6)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p(y,v_y,t)=-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }y}}<{\displaystyle \frac{\mathit{\partial }y}{\mathit{\partial }t}}\delta (y-y(t))\delta (v_y-v_y(t))>-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_y}}<{\displaystyle \frac{\mathit{\partial }{v}_y}{\mathit{\partial }t}}\delta (y-y(t))\delta (v_y-v_y(t))>.$$

(7)

Inserting Equations (4) and (5) into Equations (6) and (7) and manipulating the method of Ref. [34], the Fokker–Planck equations for $p(x,v_x,t)$ and $p(y,v_y,t)$ with an exponentially Gaussian force are, respectively, derived as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p(x,v_x,t)=[-v_x{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}}+(q^2{B}^2+{\gamma }_1){\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_x}}{v}_x+k_{1}x{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_x}}]p(x,v_x,t)+({\alpha }_1+qB{\alpha }_2)[-b_1(t){\displaystyle \frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }{v}_x}}+a_1(t){\displaystyle \frac{{\mathit{\partial }}^2}{\mathit{\partial }{v}_x^2}}]p(x,v_x,t).$$

(8)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p\left(y,v_y,t\right)=\left[-v_y{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }y}}+\left(q^2{B}^2+{\gamma }_2\right){\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_y}}{v}_y+k_{2}y{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_y}}\right]p\left(y,v_y,t\right)+\left({\alpha }_2-qB{\alpha }_1\right)\left[-b_2\left(t\right){\displaystyle \frac{{\mathit{\partial }}^2}{\mathit{\partial }x\mathit{\partial }{v}_x}}+a_2\left(t\right){\displaystyle \frac{{\mathit{\partial }}^2}{\mathit{\partial }{v}_x^2}}\right]p\left(x,v_x,t\right).$$

(9)

Here, the parameters ${\alpha }_1$, ${\alpha }_2$, $a_1(t)$, $a_2(t)$, $b_1(t)$, and $b_2(t)$ are, respectively, given by ${\alpha }_1={\eta }_{0x}^2/2$, ${\alpha }_2={\eta }_{0y}^2/2$, $a_1(t)=a_2(t)=1-\mathit{exp}(-t/\tau )$, and $b_1(t)=b_2(t)=(t+\tau )\mathit{exp}(-t/\tau )-\tau$. As our result for one charged colloid, we will derive the Fokker–Planck equations, Equations (8) and (9).

Now, the double Fourier transforms of the joint probability density are defined by

$$p({\zeta }_1,{\nu }_1,t)={\int }_{-\infty }^{+\infty }dx{\int }_{-\infty }^{+\infty }d{v}_x\mathit{exp}(-i{\zeta }_{1}x-i{\nu }_1{v}_x)P(x,v_x,t), p({\zeta }_2,{\nu }_2,t)={\int }_{-\infty }^{+\infty }dy{\int }_{-\infty }^{+\infty }d{v}_y\mathit{exp}(-i{\zeta }_{2}y-i{\nu }_2{v}_y)P(y,v_y,t).$$

(10)

By using the double Fourier transform of Equation (10), we derive the Fourier transforms of the Fokker–Planck equation for a charged colloid with correlated random force as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p({\zeta }_1,{\nu }_1,t)=[-k_1{\nu }_1{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\zeta }_1}}+[{\zeta }_1-(q^2{B}^2+{\gamma }_1){\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}]p({\zeta }_1,{\nu }_1,t)+({\alpha }_1+qB{\alpha }_2)[b_1(t){\zeta }_1{\nu }_1-a_1(t){\nu }_1^2]p({\zeta }_1,{\nu }_1,t),$$

(11)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p({\zeta }_2,{\nu }_2,t)=[-k_2{\nu }_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\zeta }_2}}+[{\zeta }_2-(q^2{B}^2+{\gamma }_2){\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}]p({\zeta }_2,{\nu }_2,t)+\left({\alpha }_2-qB{\alpha }_1\right)\left[b_2\left(t\right){\zeta }_2{\nu }_2-a_2\left(t\right){\nu }_2^2\right]p\left({\zeta }_2,{\nu }_2,t\right).$$

(12)

Now, Equations (11) and (12) are the Fourier transforms of the Fokker–Planck equations. Later, we will derive the time evolutions of the joint probability densities $p(x,v_x,t)$ and $p(y,v_y,t)$, Equations (8) and (9), in Appendix B.

2.2. Probability Densities $p(x,t)$ and $p(v_x,t)$ in Three-Time Domains

2.2.1. p(x,t) and $p(v_x,t)$ in the Time Domain $t<<\tau$

By separating the variables ${\zeta }_1$ and ${\nu }_1$ as $p({\zeta }_1,{\nu }_1,t)=p({\zeta }_1,t)p({\nu }_1,t)$, Equation (11) becomes

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p({\zeta }_1,t)=-k_1{\nu }_1{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\zeta }_1}}p({\zeta }_1,t)+{\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2)[b_1(t){\zeta }_1{\nu }_1-a_1(t){\nu }_1^2]p({\zeta }_1,t)+A_{1}p({\zeta }_1,t),$$

(13)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p\left({\nu }_1,t\right)=\left[{\zeta }_1-\left(q^2{B}^2+{\gamma }_1\right){\nu }_1\right]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}p\left({\nu }_1,t\right)+{\displaystyle \frac{1}{2}}\left({\alpha }_1+qB{\alpha }_2\right)\left[b_1\left(t\right){\zeta }_1{\nu }_1-a_1\left(t\right){\nu }_1^2\right]p\left({\nu }_1,t\right)-A_{1}p\left({\nu }_1,t\right).$$

(14)

Here, $A_1$ is the separation constant. Taking ${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p({\zeta }_1,t)=0$ for ${\zeta }_1$ in the steady state, $p({\zeta }_1,t)$ becomes $p^{st}({\zeta }_1,t)$. The Fourier transformed equations for the steady joint probability density separate as

$$-k_1{\nu }_1{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\zeta }_1}}{p}^{st}({\zeta }_1,t)+{\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2)[b_1(t){\zeta }_1{\nu }_1-a_1(t){\nu }_1^2]p^{st}({\zeta }_1,t)+A_1{p}^{st}({\zeta }_1,t)=0,$$

(15)

$$[{\zeta }_1-(q^2{B}^2+{\gamma }_1){\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}{p}^{st}({\nu }_1,t)+{\displaystyle \frac{{\alpha }_1}{2}}[b_1(t){\zeta }_1{\nu }_1-a_1(t){\nu }_1^2]p^{st}({\nu }_1,t)-A_1{p}^{st}(\nu ,t)=0.$$

(16)

The stationary solution for ${\zeta }_1$ is

$$p^{st}({\zeta }_1,t)=\mathit{exp}[{\displaystyle \frac{1}{2k_1{\nu }_1}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{b_1(t)}{2}}{\nu }_1{\zeta }_1^2-a_1(t){\zeta }_1{\nu }_1^2]]$$

(17)

Introduce a factorization $p({\zeta }_1,t)\equiv q({\zeta }_1,t)p^{st}({\zeta }_1,t)$. Then $p({\zeta }_1,t)$ is calculated as

$$p({\zeta }_1,t)=q({\zeta }_1,t)\mathit{exp}[{\displaystyle \frac{1}{2k_1{\nu }_1}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{b_1\left(t\right)}{2}}{\nu }_1{\zeta }_1^2-a_1(t){\zeta }_1{\nu }_1^2]].$$

(18)

To obtain $p({\zeta }_1,t)$, the truncation of higher-order terms is justified under the assumption that the contributions of these terms are negligible in the long-time or weak-coupling regime, as supported by previous studies on Fokker–Planck dynamics for similar systems. Physically, these transformations allow separation of the stationary component from time-dependent deviations, facilitating analytical tractability while preserving essential dynamical features. In addition, supplementary paragraphs and formulas of the joint probability density are summarized in Appendix A. By repeating this transformation successively, we obtain

$$q({\zeta }_1,t)=r({\zeta }_1,t)\mathit{exp}[-{\displaystyle \frac{1}{2(k_1{\nu }_1{)}^2}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{b_1^{\prime }(t)}{6}}{\nu }_1{{\zeta }_1}^3-{\displaystyle \frac{a_1^{\prime }(t)}{2}}{{\zeta }_1}^2{{\nu }_1}^2]],$$

(19)

$$r\left({\zeta }_1,t\right)=s({\zeta }_1,t)\mathit{exp}[{\displaystyle \frac{1}{2(k_1{\nu }_1{)}^3}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{b_1^{″}(t)}{24}}{\nu }_1{{\zeta }_1}^4-{\displaystyle \frac{a_1^{″}(t)}{6}}{{\zeta }_1}^3{{\nu }_1}^2]],$$

(20)

$$s\left({\zeta }_1,t\right)=t\left({\zeta }_1,t\right)\mathit{exp}[-{\displaystyle \frac{1}{2(k_1{\nu }_1{)}^4}}\left({\alpha }_1+qB{\alpha }_2\right){\displaystyle \frac{b_1^{‴}\left(t\right)}{120}}{{\zeta }_1}^5{\nu }_1].$$

(21)

Here, $a\prime (t)=da(t)/dt$ and $a″(t)=d^{2}a(t)/d{t}^2$. Neglecting the terms proportional to $1/{\tau }^3$, $t({\zeta }_1,t)$ is given by ${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}t({\zeta }_1,t)=-k_1{\nu }_1{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\zeta }_1}}t({\zeta }_1,t)$. Hence an arbitrary function of the combination $t-{\zeta }_1/k_1{\nu }_1$ becomes $t({\zeta }_1,t)=\mathit{\Theta }[t-{\zeta }_1/k_1{\nu }_1]$. Expanding derivatives to second order in powers of $t/\tau$ and performing cancellations yields

$$\begin{gathered} p({\zeta }_1,t)=t({\zeta }_1,t)s^{st}({\zeta }_1,t)r^{st}({\zeta }_1,t)q^{st}{\zeta }_1,t)p^{st}({\zeta }_1,t)=\mathit{\Theta }[t-{\zeta }_1/k_1{\nu }_1]s^{st}({\zeta }_1,t)r^{st}({\zeta }_1,t)q^{st}{\zeta }_1,t)p^{st}({\zeta }_1,t) \\ =\mathit{exp}[-{\displaystyle \frac{({\alpha }_1+qB{\alpha }_2)t^3}{12k_1\tau }}[1-{\displaystyle \frac{3}{2}}{\displaystyle \frac{1}{\tau t}}]{{\zeta }_1}^2-{\displaystyle \frac{({\alpha }_1+qB{\alpha }_2)t^3}{12\tau }}[1-{\displaystyle \frac{\tau }{6t}}]{\nu }_1{\zeta }_1-{\displaystyle \frac{({\alpha }_1+qB{\alpha }_2)t^3}{4\tau }}[1+2{\displaystyle \frac{k_1}{t}}]{{\nu }_1}^2]. \end{gathered}$$

(22)

Using the inverse Fourier transform, the spatial probability density $p(x,t)$ is

$$p(x,t)=[\pi {\displaystyle \frac{({\alpha }_1+qB{\alpha }_2)t^3}{3k_1\tau }}[1-{\displaystyle \frac{3}{2\tau t}}]{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3k_1\tau x^2}{({\alpha }_1+qB{\alpha }_2)t^3}}[1-{\displaystyle \frac{3}{2\tau t}}{]}^{-1}].$$

(23)

The mean squared displacement for $p(x,t)$ follows:

$$<x^2(t)>={\displaystyle \frac{1}{6k_1\tau }}({\alpha }_1+qB{\alpha }_2)t^3[1-{\displaystyle \frac{3}{2}}{\displaystyle \frac{1}{\tau t}}].$$

(24)

For the short-time domain $t<<\tau$, to obtain a special solution for ${\nu }_1$ we expand $[{\zeta }_1-(q^2{B}^2+{\gamma }_1){\nu }_1{]}^{-1}\cong {{\zeta }_1}^{-1}[1+(q^2{B}^2+{\gamma }_1){\nu }_1/{\zeta }_1]$. The Fourier transform of the steady probability density $p^{st}({\nu }_1,t)$ becomes

$$p^{st}({\nu }_1,t)=\mathit{exp}[{\displaystyle \frac{1}{2{\zeta }_1}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{a_1(t)}{3}}{{\nu }_1}^3-{\displaystyle \frac{b_1(t)}{2}}{\zeta }_1{{\nu }_1}^2]+{\displaystyle \frac{{\epsilon }_1}{2{{\zeta }_1}^2}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{a_1(t)}{4}}{{\nu }_1}^4-{\displaystyle \frac{b_1(t)}{3}}{\zeta }_1]{{\nu }_1}^3+{\displaystyle \frac{A_1}{{\zeta }_1}}{\nu }_1+{\displaystyle \frac{A_1{\epsilon }_1}{{{\zeta }_1}^2}}{\displaystyle \frac{{{\nu }_1}^2}{2}}].$$

(25)

Applying the same successive-transformation procedure:

$$p({\nu }_1,t)=q({\nu }_1,t)\mathit{exp}[{\displaystyle \frac{1}{2{\zeta }_1}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{a_1(t)}{3}}{{\nu }_1}^3-{\displaystyle \frac{b_1(t)}{2}}{\zeta }_1{{\nu }_1}^2]+{\displaystyle \frac{{\epsilon }_1}{2{{\zeta }_1}^2}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{a_1(t)}{4}}{{\nu }_1}^4-{\displaystyle \frac{b_1(t)}{3}}{\zeta }_1{{\nu }_1}^3]],$$

(26)

$$q({\nu }_1,t)=r({\nu }_1,t)\mathit{exp}[{\displaystyle \frac{1}{2{{\zeta }_1}^2}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{a_1^{\prime }(t)}{12}}{{\nu }_1}^4-{\displaystyle \frac{b_1^{\prime }(t)}{6}}{\zeta }_1{{\nu }_1}^3] +{\displaystyle \frac{{\epsilon }_1}{2{{\zeta }_1}^3}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{a_1^{\prime }\left(t\right)}{20}}{{\nu }_1}^5-{\displaystyle \frac{b_1^{\prime }\left(t\right)}{12}}{\zeta }_1{{\nu }_1}^4]],$$

(27)

$$r\left(v_1,t\right)=s(v_1,t)\mathit{exp}[{\displaystyle \frac{1}{2{{\zeta }_1}^3}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{a_1^{″}(t)}{60}}{{\nu }_1}^5-{\displaystyle \frac{b_1^{″}(t)}{24}}{\zeta }_1{{\nu }_1}^4]+{\displaystyle \frac{{\epsilon }_1}{2{{\zeta }_1}^4}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{a_1^{″}(t)}{120}}{{\nu }_1}^6-{\displaystyle \frac{b_1^{″}(t)}{60}}{\zeta }_1{{\nu }_1}^5]],$$

(28)

$$s(\nu ,t)=t({\nu }_1,t)\mathit{exp}[-{\displaystyle \frac{1}{{{\zeta }_1}^3}}({\alpha }_1+qB{\alpha }_2){\displaystyle \frac{b_1^{‴}(t)}{240}}{{\nu }_1}^5-{\displaystyle \frac{{\epsilon }_1}{{{\zeta }_1}^4}}({\alpha }_1+qB{\alpha }_2){\displaystyle \frac{b_1^{‴}(t)}{720}}{{\nu }_1}^6].$$

(29)

Discarding $1/{\tau }^3$-terms, $t({\nu }_1,t)$ satisfies

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}t({\nu }_1,t)=[{\zeta }_1-(q^2{B}^2+{\gamma }_1){\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}t({\nu }_1,t).$$

(30)

Thus $t\left({\nu }_1,t\right)$ is a function of $t+{\nu }_1/[{\zeta }_1-(q^2{B}^2+{\gamma }_1){\nu }_1]$, i.e., $t({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[{\zeta }_1-(q^2{B}^2+{\gamma }_1){\nu }_1]]$. Expanding derivatives to second order in $t/\tau$ yields after cancellations

$$\begin{array}{cc}\hfill p\left({\nu }_1,t\right)& =\mathit{\Theta }\left[t+{\displaystyle \frac{{\nu }_1}{{\zeta }_1-\left(q^2{B}^2+{\gamma }_1\right){\nu }_1}}\right]s^{st}\left({\nu }_1,t\right)r^{st}\left({\nu }_1,t\right)q^{st}\left({\nu }_1,t\right)p^{st}\left({\nu }_1,t\right)\hfill \\ & =\mathit{exp}[-{\displaystyle \frac{3}{4}}(q^2{B}^2+{\gamma }_1{)}^2({\alpha }_1+qB{\alpha }_2)t^3{{\nu }_1}^2-{\displaystyle \frac{1}{4\tau }}({\alpha }_1+qB{\alpha }_2)t^4[1-{\displaystyle \frac{\tau }{t}}]{{\zeta }_1}^2].\end{array}$$

(31)

The inverse Fourier transform of $p(v_1,t)$ yields

$$p(v_x,t)=[\pi 3(q^2{B}^2+{\gamma }_1{)}^2({\alpha }_1+qB{\alpha }_2)t^3/2{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{2v_x^2}{3(q^2{B}^2+{\gamma }_1{)}^2({\alpha }_1+qB{\alpha }_2)t^3}}].$$

(32)

Hence, its mean squared velocity for $p(v_x,t)$

$$<v_x^2(t)>={\displaystyle \frac{3}{2}}(q^2{B}^2+{\gamma }_1{)}^2\left({\alpha }_1+qB{\alpha }_2\right)t^3\left[1+{\displaystyle \frac{2}{3\left(q^2{B}^2+{\gamma }_1\right)t}}\right].$$

(33)

2.2.2. p(x,t) and $p(v_x,t)$ in the Time Domain $t>>\tau$

In the long-time domain, we approximate Equation (13) for ${\zeta }_1$ as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{{\zeta }_1}\left({\zeta }_1,t\right)\cong {\displaystyle \frac{1}{2}}\left({\alpha }_1+qB{\alpha }_2\right)\left[b_1\left(t\right){\zeta }_1{\nu }_1-a_1\left(t\right){{\nu }_1}^2\right]p_{{\zeta }_1}\left({\zeta }_1,t\right).$$

(34)

Thus,

$$p_{{\zeta }_1}({\zeta }_1,t)=\mathit{exp}[{\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2)\int [b_1(t){\zeta }_1{\nu }_1-a_1(t){{\nu }_1}^2]dt].$$

(35)

Defining $q_{{\zeta }_1}^{st}({\zeta }_1,t)$ via $q_{{\zeta }_1}({\zeta }_1,t)p_{{\zeta }_1}({\zeta }_1,t)$, we get

$$q_{{\zeta }_1}^{st}({\zeta }_1,t)=\mathit{exp}[-{\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2)\int [b_1(t){\zeta }_1{\nu }_1-a_1(t){{\nu }_1}^2]dt].$$

(36)

From Equation (13), the stationary factor $p^{st}({\zeta }_1,t)$ is

$$p^{st}({\zeta }_1,t)=\mathit{exp}[{\displaystyle \frac{1}{2k_1{\nu }_1}}({\alpha }_1+qB{\alpha }_2)[{\displaystyle \frac{b_1(t)}{2}}{\nu }_1{{\zeta }_1}^2-a_1(t){{\nu }_1}^2{\zeta }_1]].$$

(37)

Taking arbitrary functions of $t-{\zeta }_1/k_1{\nu }_1$, i.e., $r({\zeta }_1,t)=\mathit{\Theta }[t-{\zeta }_1/k_1{\nu }_1]$ and expanding in powers of $t/\tau$ yield after cancellations

$$\begin{array}{cc}\hfill p({\zeta }_1,t)=& r({\zeta }_1,t){qst}_{{\zeta }_1}({\zeta }_1,t)p^{st}({\zeta }_1,t)=\mathit{\Theta }[t-[{\zeta }_1/k_1{\nu }_1]]){qst}_{{\zeta }_1}({\zeta }_1,t)p^{st}({\zeta }_1,t)\hfill \\ & =\mathit{exp}[-{\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2)t[1+\tau ]{{\zeta }_1}^2+{\displaystyle \frac{3}{4}}{k}_1({\alpha }_1+qB{\alpha }_2)t^2{\nu }_1{\zeta }_1-{\displaystyle \frac{{\alpha }_1}{6}}{k}_1^2({\alpha }_1+qB{\alpha }_2)t^3[1-{\displaystyle \frac{3\tau }{k_1^{2}t}}]{{\nu }_1}^2].\hfill \end{array}$$

(38)

Here

$$\mathit{\Theta }(u)=\mathit{exp}[{\displaystyle \frac{1}{2}}{k}_1\tau ({\alpha }_1+qB{\alpha }_2)t{{\nu }_1}^{2}u-{\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2)(t-\tau ){{\nu }_1}^2+{\displaystyle \frac{1}{4}}{k}_1\tau ({\alpha }_1+qB{\alpha }_2){{\nu }_1}^2{u}^2-{\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2){{\nu }_1}^{2}u].$$

(39)

Similarly, for ${\nu }_1$ in the long-time domain, we approximate

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{{\nu }_1}({\nu }_1,t)\cong {\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2)[b_1(t){\zeta }_1{\nu }_1-a_1(t){{\nu }_1}^2]p_{{\nu }_1}({\nu }_1,t),$$

(40)

so that

$$p_{{\nu }_1}({\nu }_1,t)=\mathit{exp}[{\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2)\int [b_1(t){\zeta }_1{\nu }_1-a_1(t){{\nu }_1}^2]dt.$$

(41)

Using $\int a(t)dt=t-\tau$ (with $a(t)=1$) and $\int b(t)dt=-\tau t,(\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} b(t)=-\tau$) in the long-time domain and setting $p({\nu }_1,t)=q_{{\nu }_1}({\nu }_1,t)p_{{\nu }_1}^{st}({\nu }_1,t)$, we obtain

$$q_{{\nu }_1}^{st}({\nu }_1,t)=\mathit{exp}[{\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2)\int [a_1(t){{\nu }_1}^2-b_1(t){\zeta }_1{\nu }_1]dt.$$

(42)

Hence, in the long-time domain,

$$p^{st}({\nu }_1,t)=\mathit{exp}[{\displaystyle \frac{1}{2{\zeta }_1}}({\alpha }_1+qB{\alpha }_2)\int [1+{\displaystyle \frac{{\nu }_1}{{\zeta }_1}}(q^2{B}^2+{\gamma }_1)][a_1(t){{\nu }_1}^2-{\displaystyle \frac{b_1(t)}{2}}{\zeta }_1{\nu }_1+A_1]d{\nu }_1].$$

(43)

Choosing $r({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[{\zeta }_1-(q^2{B}^2+{\gamma }_1){\nu }_1]]$ and expanding in $t/\tau$, we obtain the expression for $p({\nu }_1,t)$ after some cancellations as

$$p({\nu }_1,t)=r({\nu }_1,t)q_{{\nu }_1}^{st}({\nu }_1,t)p^{st}({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[{\zeta }_1-(q^2{B}^2+{\gamma }_1){\nu }_1]]q_{{\nu }_1}^{st}({\nu }_1,t)p^{st}({\nu }_1,t).$$

(44)

Therefore, combining Equations (38) and (44), we calculate that

$$\begin{array}{cc}\hfill p({\zeta }_1,{\nu }_1,t)=& p({\zeta }_1,t)p({\nu }_1,t)=\mathit{exp}[-(q^2{B}^2+{\gamma }_1{)}^2({\alpha }_1+qB{\alpha }_2)t^3[1+{\displaystyle \frac{\tau }{2(q^2{B}^2+{\gamma }_1{)}^{2}t}}]{{\nu }_1}^2\hfill \\ & -{\displaystyle \frac{1}{2}}(q^2{B}^2+{\gamma }_1{)}^2({\alpha }_1+qB{\alpha }_2)t^4[1+{\displaystyle \frac{\tau }{(q^2{B}^2+{\gamma }_1{)}^{2}t}}]{\nu }_1{\zeta }_1-{\displaystyle \frac{1}{6}}({\alpha }_1+qB{\alpha }_2)t^4{{\zeta }_1}^2]\hfill \end{array}$$

(45)

Using the inverse Fourier transform, the long-time densities are

$$p(x,t)=[2\pi ({\alpha }_1+qB{\alpha }_2)t^4/3{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3x^2}{2({\alpha }_1+qB{\alpha }_2)t^4}}],$$

(46)

$$p\left(v_x,t\right)={[4\pi {\left(q^2{B}^2+{\gamma }_1\right)}^2\left({\alpha }_1+\mathit{qB}{\alpha }_2\right)t^3[1+{\displaystyle \frac{\tau }{2{(q^2{B}^2+{\gamma }_1)}^{2}t}}]]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{x^2}{4(q^2{B}^2+{\gamma }_1{)}^2({\alpha }_1+qB{\alpha }_2)t^3}}[1+{\displaystyle \frac{\tau }{2(q^2{B}^2+{\gamma }_1)t}}{]}^{-1}].$$

(47)

Thus, the mean squared quantities for $P(x,t)$ and $P(v_x,t)$ from Equations (46) and (47) are, respectively, given by

$$<x^2(t)>={\displaystyle \frac{1}{3}}({\alpha }_1+qB{\alpha }_2)t^4,<v_x^2(t)>=2(q^2{B}^2+{\gamma }_1{)}^2({\alpha }_1+qB{\alpha }_2)t^3[1+{\displaystyle \frac{\tau }{2(q^2{B}^2+{\gamma }_1{)}^{2}t}}].$$

(48)

2.2.3. p(x,t) and $p(v_x,t)$ for $\tau =0$

For $\tau =0$ (so $a_1(t)=1$, $b_1(t)=0$), Equations (13) and (14) for ${\zeta }_1$ and ${\nu }_1$ reduce to

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p({\zeta }_1,t)=-k_1{\nu }_1{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\zeta }_1}}p({\zeta }_1,t)-{\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2){{\nu }_1}^{2}p({\zeta }_1,t),$$

(49)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p({\nu }_1,t)=[{\zeta }_1-(q^2{B}^2+{\gamma }_1){\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}p({\nu }_1,t)-{\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2){{\nu }_1}^{2}p({\nu }_1,t).$$

(50)

In the steady state, we calculate $p^{st}({\zeta }_1,t)$ and $p^{st}({\nu }_1,t)$ as

$$p^{st}({\zeta }_1,t)=\mathit{exp}[-{\displaystyle \frac{1}{2k_1{\nu }_1}}({\alpha }_1+qB{\alpha }_2){{\nu }_1}^2{\zeta }_1],$$

(51)

$$p^{st}({\nu }_1,t)=\mathit{exp}[{\displaystyle \frac{1}{2{\zeta }_1}}({\alpha }_1+qB{\alpha }_2){\displaystyle \frac{{{\nu }_1}^3}{3}}+{\displaystyle \frac{1}{2{{\zeta }_1}^2}}(q^2{B}^2+{\gamma }_1)({\alpha }_1+qB{\alpha }_2){\displaystyle \frac{{{\nu }_1}^4}{4}}].$$

(52)

Using the approximation $[{\zeta }_1-(q^2{B}^2+{\gamma }_1){\nu }_1{]}^{-1}\cong {{\zeta }_1}^{-1}[1+(q^2{B}^2+{\gamma }_1){\nu }_1{{\zeta }_1}^{-1}]$ in Equation (50), dependent forms are

$$p({\zeta }_1,t)=\mathit{\Theta }[t-{\displaystyle \frac{{\zeta }_1}{k_1{\nu }_1}}]p^{st}({\zeta }_1,t),$$

(53)

$$p({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[{\zeta }_1-(q^2{B}^2+{\gamma }_1){\nu }_1]]p^{st}({\nu }_1,t).$$

(54)

Therefore,

$$\begin{array}{cc}\hfill p({\zeta }_1,{\nu }_1,t)& =p({\zeta }_1,t)p({\nu }_1,t)=\mathit{exp}[-{\displaystyle \frac{1}{8}}(q^2{B}^2+{\gamma }_1)({\alpha }_1+qB{\alpha }_2)t^4[1+{\displaystyle \frac{4}{3}}[(q^2{B}^2+{\gamma }_1)t{]}^{-1}]]{{\zeta }_1}^2\hfill \\ & -{\displaystyle \frac{1}{2}}(q^2{B}^2+{\gamma }_1)({\alpha }_1+qB{\alpha }_2)t^3[1+[(q^2{B}^2+{\gamma }_1)t{]}^{-1}]{\xi }_1{\nu }_1-{\displaystyle \frac{1}{2}}({\alpha }_1+qB{\alpha }_2)t[1-{\displaystyle \frac{1}{4}}[(q^2{B}^2+{\gamma }_1)t]]{{\nu }_1}^2].\hfill \end{array}$$

(55)

Inverse Fourier transforms give

$$p(x,t)=[\pi (q^2{B}^2+{\gamma }_1)({\alpha }_1+qB{\alpha }_2)t^4/2{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{2x^2}{(q^2{B}^2+{\gamma }_1)({\alpha }_1+qB{\alpha }_2)t^4}}],$$

(56)

$$p(v_x,t)=[2\pi ({\alpha }_1+qB{\alpha }_2)t{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{v_x^2}{2({\alpha }_1+qB{\alpha }_2)t}}].$$

(57)

Thus, the mean-squared deviations for $p(x,t)$ and $p(v_x,t)$ are as

$$<x^2(t)>={\displaystyle \frac{1}{4}}(q^2{B}^2+{\gamma }_1)({\alpha }_1+qB{\alpha }_2)t^4[1+{\displaystyle \frac{4}{3}}[(q^2{B}^2+{\gamma }_1)t{]}^{-1}], <v_x^2(t)>=({\alpha }_1+qB{\alpha }_2)t[1-{\displaystyle \frac{1}{4}}(q^2{B}^2+{\gamma }_1)t]$$

(58)

2.3. Probability Densities $p(y,t)$ and $p(v_y,t)$ in Three-Time Domains

From the Fourier transforms of the Fokker–Planck equation for the y-component of a charged colloid,

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p({\zeta }_2,{\nu }_2,t)=[-k_2{\nu }_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\zeta }_2}}+[{\zeta }_2-(q^2{B}^2+{\gamma }_2){\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}]p({\zeta }_2,{\nu }_2,t)+({\alpha }_2-qB{\alpha }_1)[b_2(t){\zeta }_2{\nu }_2-a_2(t){\nu }_2^2]p({\zeta }_2,{\nu }_2,t)$$

We proceed analogously to Section 2.2 by separating $p({\zeta }_2,{\nu }_2,t)=p({\zeta }_2,t)p({\nu }_2,t)$. The time derivatives of $p\left({\zeta }_2,t\right)$ and $p({\nu }_2,t)$ yield

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p({\zeta }_2,t)=-k_2{\nu }_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\zeta }_2}}p({\zeta }_2,t)+{\displaystyle \frac{1}{2}}({\alpha }_2-qB{\alpha }_1)[b_2(t){\zeta }_2{\nu }_2-a_2(t){\nu }_2^2]p({\zeta }_2,t)+A_{2}p({\zeta }_2,t),$$

(59)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}p({\nu }_2,t)=[{\zeta }_2-(q^2{B}^2+{\gamma }_2){\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}]p({\nu }_2,t)+{\displaystyle \frac{1}{2}}({\alpha }_2-qB{\alpha }_1)[b_2(t){\zeta }_2{\nu }_2-a_2(t){\nu }_2^2]p({\nu }_2,t)-A_{2}p({\nu }_2,t).$$

(60)

Here, $A_2$ is the separation constant. By the same method as in Section 2.2 we obtain the three-time-domain results for the $y$-component of a charged colloid.

2.3.1. p(y,t) and $p(v_y,t)$ in the Time Domain $t<<\tau$

The probability densities $p(y,t)$ and $p(v_y,t)$ are presented by

$$p(y,t)=[\pi {\displaystyle \frac{({\alpha }_2-qB{\alpha }_1)t^3}{3k_2\tau }}[1-{\displaystyle \frac{3}{2\tau t}}]{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3k_2\tau y^2}{({\alpha }_2-qB{\alpha }_1)t^3}}[1-{\displaystyle \frac{3}{2\tau t}}{]}^{-1}],$$

(61)

$$p(v_y,t)=[\pi 3(q^2{B}^2+{\gamma }_2{)}^2({\alpha }_2-qB{\alpha }_1)t^3/2{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{2v_y^2}{3(q^2{B}^2+{\gamma }_2{)}^2({\alpha }_2-qB{\alpha }_1)t^3}}].$$

(62)

Thus, the probability densities $p(y,t)$ and $p(v_y,t)$ yield

$$<y^2(t)>={\displaystyle \frac{1}{6k_2\tau }}({\alpha }_2-qB{\alpha }_1)t^3[1-{\displaystyle \frac{3}{2}}{\displaystyle \frac{1}{\tau t}}], <v_y^2(t)>={\displaystyle \frac{3}{2}}(q^2{B}^2+{\gamma }_2{)}^2({\alpha }_2-qB{\alpha }_1)t^3[1+{\displaystyle \frac{2}{3(q^2{B}^2+{\gamma }_2)t}}].$$

(63)

2.3.2. p(y,t) and $p(v_y,t)$ in the Time Domain $t>>\tau$

The probability densities $p(y,t)$ and $p(v_y,t)$ are presented by

$$p(y,t)=[2\pi ({\alpha }_2-qB{\alpha }_1)t^4/3{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3y^2}{2({\alpha }_2-qB{\alpha }_1)t^4}}],$$

(64)

$$p\left(v_y,t\right)={[4\pi {\left(q^2{B}^2+{\gamma }_2\right)}^2\left({\alpha }_2+\mathit{qB}{\alpha }_1\right)t^3[1+{\displaystyle \frac{\tau }{2{(q^2{B}^2+{\gamma }_2)}^{2}t}}]]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{y^2}{4(q^2{B}^2+{\gamma }_2{)}^2({\alpha }_2-qB{\alpha }_1)t^3}}[1+{\displaystyle \frac{\tau }{2(q^2{B}^2+{\gamma }_2)t}}{]}^{-1}].$$

(65)

Hence, the mean squared displacement and the mean squared velocity for $p(y,t)$ and $p(v_y,t)$ are, respectively, given by

$$<y^2(t)>={\displaystyle \frac{1}{3}}({\alpha }_2-qB{\alpha }_1)t^4,<v_y^2(t)>=2(q^2{B}^2+{\gamma }_2{)}^2({\alpha }_2-qB{\alpha }_1{)}^3[1+{\displaystyle \frac{\tau }{2(q^2{B}^2+{\gamma }_2{)}^{2}t}}].$$

(66)

2.3.3. p(y,t) and $p(v_y,t)$ in the Time Domain $\tau =0$

$p(y,t)$ and $p(v_y,t)$ are, respectively, presented by

$$p(y,t)=[\pi (q^2{B}^2+{\gamma }_2)({\alpha }_2-qB{\alpha }_1)t^4/2{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{2y^2}{(q^2{B}^2+{\gamma }_2)({\alpha }_2-qB{\alpha }_1)t^4}}],$$

(67)

$$p(v_y,t)=[2\pi ({\alpha }_2-qB{\alpha }_1)t{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{v_y^2}{2({\alpha }_2-qB{\alpha }_1)t}}].$$

(68)

Therefore, the mean-squared deviations for $p(x,t)$ and $p(v_x,t)$ are

$$<y^2(t)>={\displaystyle \frac{1}{4}}(q^2{B}^2+{\gamma }_2)({\alpha }_2-qB{\alpha }_1)t^4[1+{\displaystyle \frac{4}{3}}[(q^2{B}^2+{\gamma }_2)t{]}^{-1}], <v_y^2(t)>=({\alpha }_2-qB{\alpha }_1)t[1-{\displaystyle \frac{1}{4}}(q^2{B}^2+{\gamma }_2)t].$$

(69)

3. Fractional Generalized Langevin Equation

In this section, we introduce a class of nonequilibrium dynamic models referred to as the fractional Langevin equation, which arises from a viscoelastic memory effect characterized by a power-law kernel $K(t-t\prime )=|(t-t\prime )/\tau {|}^{2h-2}$. The active fractional Langevin equation, as a subclass of nonequilibrium dynamic models, is presented, and its unique viscoelastic memory effect is characterized by a power-law decay of correlations over time. We derive the Fokker–Planck equation for the joint probability density from the fractional generalized Langevin equation with thermal equilibrium noise.

The thermal fractional generalized Langevin equations with a viscous force, derived from Equations (4) and (5), are modified and expressed as

$${\displaystyle \frac{d}{dt}}{v}_{x_{th}}(t)=-q^2{B}^2{v}_{x_{th}}(t)-{\gamma }_{1,th}{\int }_0^{t}dt\prime |{\displaystyle \frac{t-t\prime }{{\tau }_{th}}}{|}^{2h-2}{v}_{x_{th}}(t\prime )+{\zeta }_{x_{th}}(t)+qB{\zeta }_{y_{th}}(t),$$

(70)

$${\displaystyle \frac{d}{dt}}{v}_{y_{th}}(t)=-q^2{B}^2{v}_{y_{th}}(t)-{\gamma }_{2,th}{\int }_0^{t}dt\prime |{\displaystyle \frac{t-t\prime }{{\tau }_{th}}}{|}^{2h-2}{v}_{y_{th}}(t\prime )+{\zeta }_{y_{th}}(t)-qB{\zeta }_{x_{th}}(t).$$

(71)

The active fractional generalized Langevin equations in our model are given by

$${\displaystyle \frac{d}{dt}}{v}_{x_{ac}}(t)=-q^2{B}^2{v}_{x_{ac}}(t)-{\gamma }_{1,ac}{\int }_0^{t}dt\prime |{\displaystyle \frac{t-t^{\prime }}{{\tau }_{ac}}}{|}^{2h-2}{v}_{x_{ac}}(t\prime )+{\zeta }_{x_{ac}}(t)+qB{\zeta }_{y_{ac}}(t),$$

(72)

$${\displaystyle \frac{d}{dt}}{v}_{y_{ac}}(t)=-q^2{B}^2{v}_{y_{ac}}(t)-{\gamma }_{2,ac}{\int }_0^{t}dt\prime |{\displaystyle \frac{t-t^{\prime }}{{\tau }_{ac}}}{|}^{2h-2}{v}_{y_{ac}}(t\prime )+{\zeta }_{y_{ac}}(t)-qB{\zeta }_{x_{ac}}(t).$$

(73)

Here, the thermal equilibrium noise and the active noise (exponentially decaying correlation) are denoted by ${\zeta }_{i_{th}}(t)$ and ${\zeta }_{i_{ac}}(t)$, respectively, for $i=x,y$. These two types of noise are defined as

$$<{\zeta }_{i_{th}}(t){\zeta }_{i_{th}}(t^{\prime })>={\displaystyle \frac{{\zeta }_{i0,th}^2}{2}}|{\displaystyle \frac{t-t^{\prime }}{{\tau }_{th}}}{|}^{2h-2}, <{\zeta }_{i_{ac}}(t){\zeta }_{i_{ac}}(t^{\prime })>={\displaystyle \frac{{\zeta }_{i0,ac}^2}{2{\tau }_{ac}}}\mathit{exp}(-{\displaystyle \frac{|t-t^{\prime }|}{{\tau }_{ac}}}).$$

(74)

Here, ${\tau }_{\mathrm{th}}$ and ${\tau }_{\mathrm{ac}}$ denote the thermal and active correlation times, respectively. The thermal equilibrium noise is a fractional Gaussian noise coupled to the memory kernel via the fluctuation dissipation theorem, whereas the active noise is responsible for the nonequilibrium motion of a charged colloid.

Next, we derive the Fokker–Planck equation from the active fractional generalized Langevin equation. We first express the time derivatives of the joint probability densities for $p(x_{th},v_{x_{th}},t)\equiv p_{x,th}$, $p(y_{th},v_{y_{th}},t)\equiv p_{y,th}$, $p(x_{ac},v_{x_{ac}},t)\equiv p_{x_{ac}}$, and $p(y_{ac},v_{y_{ac}},t)\equiv p_{y_{ac}}$ as follows:

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,th}=-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{x}_{th}}}<{\displaystyle \frac{\mathit{\partial }{x}_{th}}{\mathit{\partial }t}}{\delta }_{x_{th}}{\delta }_{v_{x_{th}}}>-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_{x_{th}}}}<v_{x_{th}}(t){\delta }_{x_{th}}{\delta }_{v_{x_{th}}}>, {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,th}=-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{y}_{th}}}<{\displaystyle \frac{\mathit{\partial }{y}_{th}}{\mathit{\partial }t}}{\delta }_{y_{th}}{\delta }_{v_{y_{th}}}>-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_{y,th}}}<v_{y_{th}}(t){\delta }_{y_{th}}{\delta }_{v_{y_{th}}}>,$$

(75)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,ac}=-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{x}_{ac}}}<{\displaystyle \frac{\mathit{\partial }{x}_{ac}}{\mathit{\partial }t}}{\delta }_{x_{ac}}{\delta }_{v_{x_{ac}}}>-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_{x_{ac}}}}<v_{x_{ac}}(t){\delta }_{x_{ac}}{\delta }_{v_{x_{ac}}}>, {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,ac}=-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{y}_{ac}}}<{\displaystyle \frac{\mathit{\partial }{y}_{ac}}{\mathit{\partial }t}}{\delta }_{y_{ac}}{v}_{y_{ac}}-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_{y_{ac}}}}<v_{y_{ac}}(t){\delta }_{y_{ac}}{\delta }_{v_{y_{ac}}}>,$$

(76)

Here, ${\delta }_{x_i}(x_i-x_i(t))\equiv {\delta }_{x_i}$, ${\delta }_{v_i}(v_i-v_i(t))\equiv {\delta }_{v_i}$. As the persistent Hurst exponent h ranges to $1/2<h<1$, we introduce ${\int }_0^{t}dt\prime |t-t\prime {|}^{2h-2}{v}_x(t\prime )=[\mathit{\Gamma }(2h-1){]}^{-1}{\displaystyle \frac{d^{2-2h}}{d{t}^{2-2h}}}x(t)$. We assume that, in the joint probability density, the particle is initially at rest at $t=0$.

We define the double Fourier transforms of the joint probability density $p({\xi }_1,{\nu }_1,t)$ and $p({\xi }_2,{\nu }_2,t)$ as

$$p({\xi }_1,{\nu }_1,t)={\int }_{-\infty }^{+\infty }d{x}_i{\int }_{-\infty }^{+\infty }d{v}_{x_i}\mathit{exp}(-i{\xi }_1{x}_i-i{\nu }_1{v}_{x_i})p(x_i,v_{x_i},t), p({\xi }_2,{\nu }_2,t)={\int }_{-\infty }^{+\infty }d{y}_i{\int }_{-\infty }^{+\infty }d{v}_{y_i}\mathit{exp}(-i{\xi }_2{y}_i-i{\nu }_2{v}_{y_i})p(y_i,v_{y_i},t),$$

(77)

for $i=th,ac$. Inserting Equations (70)–(73) into Equations (75) and (76), the time derivatives of Fourier transform of probability densities, Equations (75) and (76) can be expressed as

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,th}({\xi }_1,{\nu }_1,t)=-{\gamma }_{1,th}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_1{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}{p}_{x,th}({\xi }_1,{\nu }_1,t)+[{\xi }_1-q^2{B}^2{\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}{p}_{x,th}({\xi }_1,{\nu }_1,t) \\ -[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_1{\nu }_1+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }_1^2]p_{x,th}({\xi }_1,{\nu }_1,t), \end{gathered}$$

(78)

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,th}({\xi }_2,{\nu }_2,t)=-{\gamma }_{2,th}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,th}({\xi }_2,{\nu }_2,t)+[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,th}({\xi }_2,{\nu }_2,t) \\ -[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_2{\nu }_2+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }_2^2]p_{y,th}({\xi }_2,{\nu }_2,t), \end{gathered}$$

(79)

and

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,ac}({\xi }_1,{\nu }_1,t)=-{\gamma }_{1,ac}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_1{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}{p}_{x,ac}({\xi }_1,{\nu }_1,t)+[{\xi }_1-q^2{B}^2{\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}{p}_{x,ac}({\xi }_1,{\nu }_1,t) \\ +[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][b(t){\xi }_1{\nu }_1-a(t){\nu }_1^2]p_{x,th}({\xi }_1,{\nu }_1,t), \end{gathered}$$

(80)

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,ac}({\xi }_2,{\nu }_2,t)=-{\gamma }_{2,ac}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,ac}({\xi }_2,{\nu }_2,t)+[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,ac}({\xi }_2,{\nu }_2,t) \\ +[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][b(t){\xi }_2{\nu }_2-a(t){\nu }_2^2]p_{y,ac}({\xi }_2,{\nu }_2,t). \end{gathered}$$

(81)

Here, $D_{2-2h}=d^{2-2h}/d{t}^{2-2h}$, and the parameters are

$${\overline{\alpha }}_1={\displaystyle \frac{{\zeta }_{x0,th}^2}{2}}, {\overline{\alpha }}_2={\displaystyle \frac{{\zeta }_{y0,ac}^2}{2}}, a(t)=1-\mathit{exp}(-{\displaystyle \frac{t}{{\tau }_{ac}}}), b(t)=(t+{\tau }_{ac})\mathit{exp}(-{\displaystyle \frac{t}{{\tau }_{ac}}})-{\tau }_{ac}.$$

(82)

It is therefore reasonable to call Equations (78)–(81) the Fourier-transformed Fokker–Planck equations, as mentioned in the Introduction. Further details regarding correlated Gaussian forces are provided in Ref. [34].

3.1. $p_{x,th}(x_{th},v_{x_{th}},t)$ with Thermal Noise

3.1.1. $p_{x,th}(x_{th},v_{x_{th}},t)$ with the Thermal Noise in Short-Time Domain

In this subsection, we obtain the solutions of the probability densities $p_{x,th}(x_{th},v_{x_{th}},t)$, and $p_{y,th}(y_{th},v_{y_{th}},t)$ in the short-time domain $t<<{\tau }_{th}$. To find the special solutions for ${\xi }_1$ and ${\nu }_1$ by separating variables from Equations (78)–(81), we write the four equations for displacement and velocity as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,th}\left({\xi }_1,t\right)=-{\gamma }_{1,th}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_1{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}{p}_{x,th}({\xi }_1,t)-[{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_1{\nu }_1+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }_1^2]-A]p_{x,th}({\xi }_1,t),$$

(83)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,th}({\nu }_1,t)=[{\xi }_1-q^2{B}^2{\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}{p}_{x,th}({\nu }_1,t)-[{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_1{\nu }_1+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }_1^2]+A]p_{x,th}({\nu }_1,t),$$

(84)

and

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,th}({\xi }_2,t)=-{\gamma }_{2,th}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,th}({\xi }_2,t)-[{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_2{\nu }_2+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }_2^2]-B]p_{y,th}({\xi }_2,t),$$

(85)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,th}({\nu }_2,t)=[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,th}({\nu }_2,t)-[{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_2{\nu }_2+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }_2^2]-B]p_{y,th}({\nu }_2,t),$$

(86)

where $A$ and B denote the separation constants.

Assuming ${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,th}({\xi }_1,t)=0$ in the steady state from Equation (83), we obtain the steady probability density

$$p_{x,th}^{st}({\xi }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2{\gamma }_{1,th}{D}_{2-2h}\nu }}[{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\displaystyle \frac{{\xi }_1^2}{2}}{\nu }_1+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }^2\xi -A{\xi }_1]].$$

(87)

Setting $\mathit{\Gamma }(2h-1)\cong 1$, and expressing $p_{x,th}({\xi }_1,t)=r_{x,th}({\xi }_1,t)q_{x,th}^{st}({\xi }_1,t)$, we can perform the Fourier transform including terms up to order $1/{\tau }_{th}^2$ as

$$p_{x,th}({\xi }_1,t)=q_{x,th}({\xi }_1,t)\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2{\gamma }_{1,th}{D}_{2-2h}{\nu }_1}}[{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\displaystyle \frac{{\xi }_1^2}{2}}{\nu }_1+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{{\nu }_1}^2{\xi }_1-A{\xi }_1]],$$

(88)

$$q_{x,th}({\xi }_1,t)=r_{x,th}({\xi }_1,t)\mathit{exp}[-{\displaystyle \frac{\left[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2\right]}{2({\gamma }_{1,th}{D}_{2-2h}{\nu }_1{)}^2}}[{\displaystyle \frac{t^{2h-1}}{4h^2{\tau }_{th}^{2h}}}{\displaystyle \frac{{\xi }_1^3}{6}}{\nu }_1+{\displaystyle \frac{t^{2h-2}}{(2h-1{)}^2{\tau }_{th}^{2h-1}}}{{\nu }_1}^2{\displaystyle \frac{{{\xi }_1}^2}{2}}]].$$

(89)

Taking the solutions as arbitrary functions of variable $t-{\xi }_1/{\gamma }_{1,th}{D}_{2-2h}{\nu }_1$, the arbitrary function $r_{x,th}({\xi }_1,t)$ becomes $\mathit{\Theta }[t-{\xi }_1/{\gamma }_{1,th}{D}_{2-2h}{\nu }_1]$. As a result, we find that

$$p_{x,th}({\xi }_1,t)=r_{x,th}({\xi }_1,t)q_{x,th}^{st}({\xi }_1,t)p_{x,th}^{st}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/{\gamma }_{1,th}{D}_{2-2h}{\nu }_1]q_{x,th}^{st}({\xi }_1,t)p_{x,th}^{st}({\xi }_1,t).$$

(90)

Using a similar method as in Equations (87)–(90), we also obtain the Fourier of the probability density for ${\nu }_1$ as

$$p_{x,th}({\nu }_1,t)=r_{x,th}({\nu }_1,t)q_{x,th}^{st}({\nu }_1,t)p_{x,th}^{st}({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/{\xi }_1-q^2{B}^2{\nu }_1]q_{x,th}^{st}({\nu }_1,t)p_{x,th}^{st}({\nu }_1,t).$$

(91)

Therefore, by calculating Equations (90) and (91), we get the Fourier transform of the joint probability density as

$$p_{x,th}({\xi }_1,{\nu }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}{6(2h-1{)}^2{\tau }_{th}^{2h}}}{{\xi }_1}^2-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h}}{4(2h-1{)}^2{\tau }_{th}^{2h-1}}}{{\nu }_1}^2].$$

(92)

Using the inverse Fourier transform, we get

$$p_{x,th}(x_{th},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}{3(2h-1{)}^2{\tau }_{th}^{2h}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3(2h-1{)}^2{\tau }_{th}^{2h}}{2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}}{x}_{th}^2],$$

(93)

$$p_{x,th}(v_{x_{th}},t)=[\pi {{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h}}{(2h-1){\tau }_{th}^{2h-1}}}}^2{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{(2h-1{)}^2{\tau }_{th}^{2h-1}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h}}}{v}_{x_{th}}^2].$$

(94)

Finally, the mean squared displacement and the mean squared displacement are given by

$$<x_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{3(2h-1{)}^2{\tau }_{th}^{2h-1}}}{t}^{2h+1}, <v_{x_{th}}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2(2h-1{)}^2{\tau }_{th}^{2h-1}}}{t}^{2h}.$$

(95)

3.1.2. $p_{x,th}(x_{th},v_{x_{th}},t)$ with Thermal Noise in the Long-Time Domain

Now we find the probability densities $p_{x,th}(x_{th},t)$ and $p_{x,th}(v_{x_{th}},t)$ in long-time domain $t>>{\tau }_{th}$. An approximate equation from Equation (83) can be written as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,th{\xi }_1}({\xi }_1,t)\cong -{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_1{\nu }_1+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }_1^2]p_{x,th{\xi }_1}({\xi }_1,t).$$

(96)

The Fourier transform of the probability density $p_{x,th{\xi }_1}({\xi }_1,t)$ from the above equation is calculated as

$$p_{x,th{\xi }_1}({\xi }_1,t)=\mathit{exp}[-{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][{\displaystyle \frac{t^{2h+1}}{2h(2h+1){\tau }_{th}^{2h}}}{\xi }_1{\nu }_1+{\displaystyle \frac{t^{2h}}{(2h-1)2h{\tau }_{th}^{2h-1}}}{\nu }_1^2]].$$

(97)

We then find the steady probability density $q_{x,th{\xi }_1}^{st}({\xi }_1,t)$ from ${p_{x,th}({\xi }_1,t)\equiv q}_{x,th{\xi }_1}({\xi }_1,t)p_{x,th{\xi }_1}({\xi }_1,t)$ as

$$q_{x,th{\xi }_1}^{st}({\xi }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2{\gamma }_{1,th}{\nu }_1{D}_{2-2h}}}[{\displaystyle \frac{1}{2h(2h+1){\tau }_{th}^{2h}}}{t}^{2h+1}{\displaystyle \frac{{{\xi }_1}^1}{2}}{\nu }_1]].$$

(98)

Since the Fourier transform of probability density $q_{x,th}({\xi }_1,t)$ in the short-time domain is given by $q_{x,th}({\xi }_1,t)=r_{x,th}({\xi }_1,t)q_{x,th}^{st}({\xi }_1,t)$, $p(\xi ,t)$ can be derived as

$$p_{x,th}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/{\gamma }_{1,th}{\nu }_1{D}_{2-2h}]q_{x,th{\xi }_1}^{st}({\xi }_1,t)p_{x,th}^{st}({\xi }_1,t),$$

(99)

where $r_{x,th}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/{\gamma }_{1,th}{\nu }_1{D}_{2-2h}]$. Applying Equation (84) for $p_{x,th}({\nu }_1,t)$ and using the similar method as in Equations (96)–(99), we also obtain the Fourier transforms of the probability density for velocity as

$$p_{x,th}({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[\xi -q^2{B}^2{\nu }_1]]q_{x,th{\nu }_1}^{st}({\nu }_1,t)p_{x,th}^{st}({\nu }_1,t).$$

(100)

From Equations (99) and (100), we have

$$p_{x,th}({\xi }_1,{\nu }_1,t)=p_{x,th}({\xi }_1,t)p_{x,th}({\nu }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+1}}{4(2h{)}^2{\tau }_{th}^{2h}}}{{\xi }_1}^2-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h}}{2(2h-1){\tau }_{th}^{2h-1}}}{{\nu }_1}^2]$$

(101)

Using the inverse Fourier transform, the probability density $p_{x,th}(x_{th},t)$ and $p_{x,th}(v_{x_{th}},t)$ are, respectively, expressed as

$$p_{x,th}(x_{th},t)=[\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+1}}{4h^2{\tau }_{th}^{2h}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{4h^2{\tau }_{th}^{2h}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+1}}}{x}_{th}^2],$$

(102)

$$p_{x,th}(v_{x_{th}},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h}}{(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{(2h-1){\tau }_{th}^{2h-1}}{2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h}}}{v}_{x_{th}}^2].$$

(103)

The mean squared values $<x_{th}^2(t)>$ and $<v_{x_{th}}^2(t)>$ for the probability densities $p_{x,th}(x_{th},t)$ and $p_{x,th}(v_{x_{th}},t)$ are, respectively, given by

$$<x_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{8h^2{\tau }_{th}^{2h}}}{t}^{2h+2},<v_{x_{th}}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}.$$

(104)

3.1.3. $p_{x,th}\left(x_{th},v_{x_{th}},t\right)$ with the Thermal Noise for ${\tau }_{th}=0$

In this subsection, we find the probability densities $p_{x,th}(x_{th},t)$ and $p_{x,th}(v_{x_{th}},t)$ in time domain ${\tau }_{th}=0$. The approximate equation from Equation (83) for this regime is written as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,th{\xi }_1}({\xi }_1,t)\cong -{\gamma }_{1,th}{\nu }_1{D}_{2-2h}{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}{p}_{x,th{\xi }_1}({\xi }_1,t)-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{\nu }^2{p}_{x,th{\xi }_1}({\xi }_1,t).$$

(105)

In the steady state, $p_{x,th{\xi }_1}^{st}(\xi ,t)$ can be calculated from Equation (105) as

$$p_{x,th{\xi }_1}^{st}({\xi }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2{\gamma }_{1,th}{\nu }_1{D}_{2-2h}}}{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{{\nu }_1}^2{\xi }_1].$$

(106)

The Fourier transform of probability density $p_{x,th}({\xi }_1,t)$ is given by

$$p_{x,th}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/{\gamma }_{1,th}{\nu }_1{D}_{2-2h}]p_{x,th{\xi }_1}^{st}({\xi }_1,t)$$

(107)

Using a similar procedure, we obtain the Fourier transform of the probability density $p_{x,th}({\nu }_1,t)$ as

$$p_{x,th}({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[{\xi }_1-q^2{B}^2{\nu }_1]p_{x,th{\nu }_1}^{st}({\nu }_1,t)$$

(108)

From Equations (107) and (108), we can calculate

$$p_{x,th}\left({\xi }_1,{\nu }_1,t\right)=p_{x,th}\left({\xi }_1,t\right)p_{x,th}\left({\nu }_1,t\right)=\mathit{exp}[-{\displaystyle \frac{\left[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2\right]t^{2h+1}}{4(2h{)}^2{\tau }_{th}^{2h}}}{{\xi }_1}^2-{\displaystyle \frac{\left[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2\right]t^{2h}}{2\left(2h-1\right){\tau }_{th}^{2h-1}}}{{\nu }_1}^2].$$

(109)

Using the inverse Fourier transform, the probability densities $p_{x,th}(x_{th},t)$ and $p_{x,th}(v_{x_{th}},t)$ are, respectively, expressed as

$$p_{x,th}(x_{th},t)=[\pi {\displaystyle \frac{\left[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2\right]t^{2h+1}}{4h^2{\tau }_{th}^{2h}}}{]}^{-{\displaystyle \frac{1}{2}}}\mathit{exp}[-{\displaystyle \frac{4h^2{\tau }_{th}^{2h}}{\left[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2\right]t^{2h+1}}}{x}_{th}^2],$$

(110)

$$p_{x,th}\left(v_{x_{th}},t\right)=\left[2\pi {\displaystyle \frac{\left[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2\right]t^{2h}}{\left(2h-1\right){\tau }_{th}^{2h-1}}}{]}^{-{\displaystyle \frac{1}{2}}}\mathit{exp}[-{\displaystyle \frac{\left(2h-1\right){\tau }_{th}^{2h-1}}{2\left[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2\right]t^{2h}}}{v}_{x_{th}}^2\right].$$

(111)

The mean squared displacement $<x_{th}^2(t)>$ and the mean squared velocity $<v_{x_{th}}^2(t)>$ are

$$<x_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{8h^2{\tau }_{th}^{2h}}}{t}^{2h+2}, <v_{x_{th}}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}.$$

(112)

3.2. $p_{y,th}(y_{th},v_{y,th},t)$ with the Thermal Noise

From the Fourier transforms of the Fokker–Planck equation for the y-component of a charged colloid,

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,th}({\xi }_2,{\nu }_2,t)=-{\gamma }_{2,th}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,th}({\xi }_2,{\nu }_2,t)+[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,th}({\xi }_2,{\nu }_2,t) \\ -[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_2{\nu }_2+{\displaystyle \frac{t^{2h-1}}{\left(2h-1\right){\tau }_{th}^{2h-1}}}{\nu }_2^2]p_{y,th}({\xi }_2,{\nu }_2,t). \end{gathered}$$

We can calculate the probability densities $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y_{th}},t)$ in three-time domains.

By separating the variables ${\zeta }_2$ and ${\nu }_2$ as ${p_{y,th}\left({\zeta }_2,v_2,t\right)=p}_{y,th}({\zeta }_2,t)p_{y,th}({\nu }_2,t)$, Equation (80) becomes

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,th}({\xi }_2,t)=-{\gamma }_{2,th}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,th}({\xi }_2,t)-[{\displaystyle \frac{1}{2}}[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_2{\nu }_2+{\displaystyle \frac{t^{2h-1}}{2h-1){\tau }_{th}^{2h-1}}}{\nu }_2^2]-B]p_{y,th}({\xi }_2,t),$$

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,th}({\nu }_2,t)=[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,th}({\nu }_2,t)-[{\displaystyle \frac{1}{2}}[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_2{\nu }_2+{\displaystyle \frac{t^{2h-1}}{2h-1){\tau }_{th}^{2h-1}}}{\nu }_2^2]-B]p_{y,th}({\nu }_2,t).$$

Here, $B$ is the separation constant. Using the same method applied in Section 3.1 to calculate $p_{x,th}(x_{th},t)$ and $p_{x,th}(v_{x_{th}},t)$, we can determine the probability densities $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y,th},t)$ in three-time domains.

3.2.1. $p_{y,th}\left(y_{th},t\right)$ and $p_{y,th}\left(v_{y_{th}},t\right)$ in $t<<{\tau }_{th}$

The probability densities $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y,th},t)$ are expressed as

$$p_{y,th}(y_{th},t)=[2\pi {\displaystyle \frac{\left[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}\right]t^{2h+1}}{3(2h-1{)}^2{\tau }_{th}^{2h-1}}}{]}^{-{\displaystyle \frac{1}{2}}}\mathit{exp}[-{\displaystyle \frac{3(2h-1{)}^2{\tau }_{th}^{2h-1}}{2\left[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}\right]t^{2h+1}}}{y}_{th}^2],$$

(113)

$$p_{y,th}(v_{y_{th}},t)=[\pi {\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]t^{2h}}{(2h-1{)}^2{\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{(2h-1{)}^2{\tau }_{th}^{2h-1}}{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]t^{2h}}}{v}_{y_{th}}^2].$$

(114)

The corresponding mean squared values from $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{th},t)$ yield

$$<y_{th}^2(t)>={\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]}{3(2h-1{)}^2{\tau }_{th}^{2h-1}}}{t}^{2h+1}, <v_{y_{th}}^2(t)>={\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]}{2(2h-1{)}^2{\tau }_{th}^{2h-1}}}{t}^{2h}.$$

(115)

3.2.2. $p_{y,th}\left(y_{th},t\right)$ and $p_{y,th}\left(v_{y_{th}},t\right)$ in $t>>{\tau }_{th}$

By using the inverse Fourier transform, the probability densities $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y_{th}},t)$ are, respectively, given by

$$p_{y,th}(y_{th},t)=[\pi {\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]t^{2h+1}}{4h^2{\tau }_{th}^{2h}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{4h^2{\tau }_{th}^{2h}}{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]t^{2h+1}}}{y}_{th}^2],$$

(116)

$$p_{y,th}(v_{y_{th}},t)=[2\pi {\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]t^{2h}}{(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{(2h-1){\tau }_{th}^{2h-1}}{2[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]{\gamma }_{2,th}{t}^{2h}}}{v}_{y_{th}}^2].$$

(117)

The mean squared values $<y_{th}^2(t)>$ and $<v_{y,th}^2(t)>$ for $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y_{th}},t)$ are, respectively, given by

$$<y_{th}^2(t)>={\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]}{8h^2{\tau }_{th}^{2h}}}{t}^{2h+1}, <v_{y_{th}}^2(t)>={\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}.$$

(118)

3.2.3. $p_{y,th}\left(y_{th},t\right)$ and $p_{y,th}\left(v_{y_{th}},t\right)$ in the Time Domain $\tau =0$

The probability densities $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y_{th}},t)$ are, respectively, expressed as

$$p_{y,th}(y_{th},t)=[\pi {\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]t^{2h+1}}{4h^2{\tau }_{th}^{2h}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{4h^2{\tau }_{th}^{2h}}{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]t^{2h+1}}}{y}_{th}^2],$$

(119)

$$p_{y,th}(v_{y_{th}},t)=[2\pi {\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]t^{2h}}{(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{(2h-1){\tau }_{th}^{2h-1}}{2[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]t^{2h}}}{v}_{y_{th}}^2].$$

(120)

The mean squared displacement $<y_{th}^2(t)>$ and the mean squared velocity $<v_{y_{th}}^2(t)>$ are

$$<y_{th}^2(t)>={\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]}{8h^2{\tau }_{th}^{2h}}}{t}^{2h+1},<v_{y_{th}}^2(t)>={\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}.$$

(121)

3.3. $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ with the Active Noise

3.3.1. $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ with the Active Noise in the Short-Time Domain

In this subsection, we determine the solutions of the probability density $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ in the short-time domain $t<<{\tau }_{ac}$. To find the special solutions by separating variables from Equations (80) and (81), the Fourier transformed equations of displacement and velocity are given as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,ac}({\xi }_1,t)=-{\gamma }_{1,ac}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_1{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}{p}_{x,ac}({\xi }_1,t)+{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][b(t){\xi }_1{\nu }_1-a(t){\nu }_1^2]p_{x,th}({\xi }_1,t)+C{p}_{x,th}({\xi }_1,t),$$

(122)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,ac}({\nu }_1,t)=[{\xi }_1-q^2{B}^2{\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}{p}_{x,ac}({\nu }_1,t)+{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][b(t){\xi }_1{\nu }_1-a(t){\nu }_1^2]p_{x,th}({\xi }_1,{\nu }_1,t)-C{p}_{x,th}({\xi }_1,{\nu }_1,t),$$

(123)

and

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,ac}({\nu }_2,t)=-{\gamma }_{2,ac}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,ac}({\xi }_2,t)+{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][b(t){\xi }_2{\nu }_2-a(t){\nu }_2^2]p_{y,ac}({\xi }_2,t)+E{p}_{y,ac}({\xi }_2,t),$$

(124)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,ac}({\nu }_2,t)=[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,ac}({\nu }_2,t)+{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][b(t){\xi }_2{\nu }_2-a(t){\nu }_2^2]p_{y,ac}({\nu }_2,t)-E{p}_{y,ac}({\nu }_2,t).$$

(125)

Here, C and E denote separation constants.

From Equation (122) the steady state solution $p_{x,ac}^{st}({\xi }_1,t)$ is obtained as

$$p_{x,ac}^{st}({\xi }_1,t)=\mathit{exp}[{\displaystyle \frac{1}{2{\gamma }_{1,ac}\nu D_{2-2h}}}[[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][{\displaystyle \frac{b(t)}{2}}{\nu }_1{{\xi }_1}^2-a(t){{\nu }_1}^2{\xi }_1]+2C{\xi }_1]].$$

(126)

Letting ${q_{ac}({\xi }_1,t)\equiv r}_{ac}({\xi }_1,t)q_{ac}^{st}({\xi }_1,t)$, and expanding the Fourier transform of the probability density up to order $1/{\tau }_{ac}^2$, we have

$$p_{x,ac}({\xi }_1,t)=q_{x,ac}({\xi }_1,t)\mathit{exp}[{\displaystyle \frac{1}{2{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}}}[[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][{\displaystyle \frac{b(t)}{2}}{\nu }_1{{\xi }_1}^2-a(t){{\nu }_1}^2{\xi }_1]+2C\xi ]],$$

(127)

$$q_{x,ac}({\xi }_1,t)=r_{x,ac}({\xi }_1,t)\mathit{exp}[{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2[{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}{]}^3}}[{\displaystyle \frac{b\prime (t)}{6}}{\nu }_1{{\xi }_1}^3-{\displaystyle \frac{a\prime (t)}{2}}{{\nu }_1}^2{{\xi }_1}^2]],$$

(128)

$$r_{x,ac}({\xi }_1,t)=s_{x,ac}({\xi }_1,t)\mathit{exp}[{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2[{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}{]}^3}}[{\displaystyle \frac{b_2″(t)}{24}}{\nu }_1{{\xi }_1}^4-{\displaystyle \frac{a_2″(t)}{6}}{{\nu }_1}^2{{\xi }_1}^3]],$$

(129)

$$s_{x,ac}({\xi }_1,t)=t_{x,ac}({\xi }_1,t)\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2[{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}{]}^4}}{\displaystyle \frac{b‴(t)}{120}}{\nu }_1{{\xi }_1}^5].$$

(130)

Assuming the solutions are arbitrary functions of variable $t-{\xi }_1/{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}$, the arbitrary function $t_{x,ac}({\xi }_1,t)$ becomes $\mathit{\Theta }[t-{\xi }_1/{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}]$. Consequently, we obtain

$$\begin{gathered} p_{x,ac}({\xi }_1,t)=t_{x,ac}({\xi }_1,t)s_{x,ac}^{st}({\xi }_1,t)r_{x,ac}^{st}({\xi }_1,t)q_{x,ac}^{st}({\xi }_1,t)p_{x,ac}^{st}({\xi }_1,t) \\ =\mathit{\Theta }[t-{\xi }_1/{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}]s_{x,ac}^{st}({\xi }_1,t)r_{x,ac}^{st}({\xi }_1,t)q_{x,ac}^{st}({\xi }_1,t)p_{x,ac}^{st}({\xi }_1,t). \end{gathered}$$

(131)

Using a similar method for $p_{x,ac}({\nu }_1,t)$, we also obtain the Fourier transform of the velocity probability density as

$$\begin{gathered} p_{x,ac}({\nu }_1,t)=t_{x,ac}({\nu }_1,t)s_{x,ac}^{st}({\nu }_1,t)r_{x,ac}^{st}({\nu }_1,t)q_{x,ac}^{st}({\nu }_1,t)p_{x,ac}^{st}({\nu }_1,t) \\ =\mathit{\Theta }[t+{\nu }_1/[{\xi }_1-q^2{B}^2{\nu }_1]]s_{x,ac}^{st}({\nu }_1,t)r_{x,ac}^{st}({\nu }_1,t)q_{x,ac}^{st}({\nu }_1,t)p_{x,ac}^{st}({\nu }_1,t). \end{gathered}$$

(132)

From Equations (131) and (132), we find the Fourier transform of the joint probability density as

$$p_{x,ac}({\xi }_1,{\nu }_1,t)=\mathit{exp}[-{\displaystyle \frac{1}{16{\tau }_{ac}}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^4{{\xi }_1}^2-{\displaystyle \frac{{\gamma }_{2,ac}}{4}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+1}{\nu }_1^2].$$

(133)

Using the inverse Fourier transform, we obtain

$$p_{x,ac}(x_{ac},t)=[\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^4}{4{\tau }_{ac}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{4{\tau }_{ac}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^4}}{x}_{ac}^2],$$

(134)

$$p_{x,ac}(v_{x_{ac}},t)=[\pi [{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{2,ac}{t}^{2h+1}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{v_{x_{ac}}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{2,ac}{t}^{2h+1}}}].$$

(135)

The mean squared displacement and the mean squared displacement for $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ are, respectively, given by

$$<x_{ac}^2(t)>={\displaystyle \frac{1}{8{\tau }_{ac}}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^4, <v_{x_{ac}}^2(t)>={\displaystyle \frac{{\gamma }_{2,ac}}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+1}.$$

(136)

3.3.2. $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ with the Active Noise in the Long-Time Domain

Now we determine the probability densities $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ in long-time domain $t>>{\tau }_{ac}$. An approximate equation from Equation (122) as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,ac{\xi }_1}({\xi }_1,t)\cong {\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2[b(t){\xi }_1\nu -a(t){\nu }^2]p_{x,ac{\xi }_1}({\xi }_1,t).$$

(137)

The Fourier transform of the probability density $p_{x,ac{\xi }_1}({\xi }_1,t)$ from the above equation is calculated as

$$p_{x,ac{\xi }_1}({\xi }_1,t)\cong \mathit{exp}[{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\int }_0^{t}dt[b(t){\xi }_1{\nu }_1-a(t){{\nu }_1}^2]].$$

(138)

The steady probability density $q_{x,ac}^{st}(\xi ,t)$ for $t\to \infty$ is obtained from ${p_{x,ac}({\xi }_1,t)\equiv q}_{x,ac{\xi }_1}({\xi }_1,t)p_{x,ac{\xi }_1}({\xi }_1,t)$ as

$$q_{x,ac{\xi }_1}^{st}({\xi }_1,t)=\mathit{exp}[-{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\int }_0^{t}dt[b(t){\xi }_1{\nu }_1-a(t){{\nu }_1}^2]].$$

(139)

Since the Fourier transform of probability density $q_{x,ac}^{st}({\xi }_1,t)$ in the short-time domain is given by $q_{x,ac}({\xi }_1,t)=r_{x,ac}({\xi }_1,t)q_{x,ac{\xi }_1}^{st}({\xi }_1,t)$, we derive

$$p_{x,ac}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}]q_{x,ac{\xi }_1}^{st}({\xi }_1,t)p_{x,ac}^{st}({\xi }_1,t),$$

(140)

where $r_{x,ac}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}]$ and Equation (126) hold. Applying to the same method as in Equations (137)–(140) of $p_{x,ac}({\xi }_1,t)$ derived, we obtain Fourier transforms of the velocity probability density as

$$p_{x,ac}({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[\xi -q^2{B}^2{\nu }_1]]q_{x,ac{\nu }_1}^{st}({\nu }_1,t)p_{x,ac}^{st}({\nu }_1,t).$$

(141)

From Equations (140) and (141), we find

$$p_{x,ac}({\xi }_1,{\nu }_1,t)=p_{x,ac}({\xi }_1,t)p_{x,ac}({\nu }_1,t)=\mathit{exp}[-{\displaystyle \frac{t^3}{6}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{{\xi }_1}^2-{\displaystyle \frac{{\gamma }_{2,ac}{t}^{2h+1}}{4}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{{\nu }_1}^2].$$

(142)

By using the inverse Fourier transform, the probability density $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ are, respectively, expressed as

$$p_{x,ac}(x_{ac},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3}{3}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3x_{ac}^2}{2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3}}],$$

(143)

$$p_{x,ac}(v_{x_{ac}},t)=[\pi [{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{2,ac}{t}^{2h+1}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{v_{x_{ac}}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{2,ac}{t}^{2h+1}}}].$$

(144)

The mean squared values $<x_{ac}^2(t)>$ and $<v_{x_{ac}}^2(t)>$ are given by

$$<x_{ac}^2(t)>={\displaystyle \frac{1}{3}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3, <v_{x_{ac}}^2(t)>={\displaystyle \frac{{\gamma }_{2,ac}}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+1}.$$

(145)

3.3.3. $p_{x,ac}\left(x_{ac},t\right)$ and $p_{x,ac}\left(v_{x_{ac}},t\right)$ with the Active Noise for ${\tau }_{ac}=0$

In this subsection, we find $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ for time domain ${\tau }_{ac}=0$. The approximate equation from Equation (122) for ${\xi }_1$ is written as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,ac{\xi }_1}\left({\xi }_1,t\right)\cong -{\gamma }_{ac}{\nu }_1{D}_{2-2h}{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}{p}_{x,ac{\xi }_1}\left({\xi }_1,t\right)-{\displaystyle \frac{1}{2}}\left[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2\right]a\left(t\right){{\nu }_1}^2{p}_{x,ac{\xi }_1}\left({\xi }_1,t\right).$$

(146)

In the steady state, $p_{x,ac{\xi }_1}^{st}({\xi }_1,t)$ can be calculated from Equation (146) as

$$p_{x,ac{\xi }_1}^{st}({\xi }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}}}{{\nu }_1}^2{\xi }_1].$$

(147)

Here, $a(t)=1$, $b(t)=0$ for ${\tau }_{ac}=0$. The Fourier transform of the probability density $p_{x,ac}({\xi }_1,t)$ is given by

$$p_{x,ac}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}]p_{x,ac{\xi }_1}^{st}({\xi }_1,t).$$

(148)

Using a similar procedure for $p_{x,ac}({\xi }_1,t)$, we obtain the Fourier transform of the probability density $p_{x,ac}({\nu }_1,t)$ as

$$p_{x,ac}({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[\xi -q^2{B}^2{\nu }_1]]p_{x,ac{\nu }_1}^{st}({\nu }_1,t).$$

(149)

We can calculate $p_{x,ac}({\xi }_1,{\nu }_1,t)$ from Equations (149) and (150) as

$$p_{x,ac}({\xi }_1,{\nu }_1,t)=p_{x,ac}({\xi }_1,t)p_{x,ac}({\nu }_1,t)=\mathit{exp}[-{\displaystyle \frac{\left[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2\right]t^3}{6}}{{\xi }_1}^2-[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t{{\nu }_1}^2].$$

(150)

Using the inverse Fourier transform, the probability densities $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ are, respectively, given by

$$p_{x,ac}(x_{ac},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3}{3}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3}{2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3}}{x}_{ac}^2].$$

(151)

$$p_{x,ac}(v_{x_{ac}},t)=[4\pi [{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{1}{4[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t}}{v}_{x_{ac}}^2].$$

(152)

The mean squared displacement and the mean squared velocity are, respectively,

$$<x_{ac}^2(t)>={\displaystyle \frac{1}{3}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3, <v_{x_{ac}}^2(t)>=2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t.$$

(153)

3.4. $p_{y,ac}\left(y_{ac},t\right)$ and $p_{y,ac}\left(v_{y_{ac}},t\right)$ with the Active Noise ${\zeta }_{ac}\left(t\right)$

From the Fourier transforms of the Fokker–Planck equation for the y-component of a charged colloid, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,ac}({\xi }_2,{\nu }_2,t)=& -{\gamma }_{2,ac}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,ac}({\xi }_2,{\nu }_2,t)+[{\xi }_2-q^2{B}^2{\nu }_2]\hfill \\ & {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,ac}({\xi }_2,{\nu }_2,t)+[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}][b(t){\xi }_2{\nu }_2-a(t){\nu }_2^2]p_{y,ac}({\xi }_2,{\nu }_2,t).\end{array}$$

We can calculate the probability densities $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y,ac},t)$ in three-time domains. By separating the variables ${\zeta }_2$ and ${\nu }_2$ as ${p_{y,ac}\left({\zeta }_2,v_2,t\right)=p}_{y,ac}({\zeta }_2,t)p_{y,ac}({\nu }_2,t)$, Equation (81) becomes

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,ac}({\nu }_2,t)=-{\gamma }_{2,ac}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,ac}({\xi }_2,t)+{\displaystyle \frac{1}{2}}[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}][b(t){\xi }_2{\nu }_2-a(t){\nu }_2^2]p_{y,ac}({\xi }_2,t)+E{p}_{y,ac}({\xi }_2,t),$$

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,ac}({\nu }_2,t)=[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,ac}({\nu }_2,t)+{\displaystyle \frac{1}{2}}[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}][b(t){\xi }_2{\nu }_2-a(t){\nu }_2^2]p_{y,th}({\nu }_2,t)-E{p}_{y,th}({\nu }_2,t),$$

Here, $E$ is the separation constant. By applying the same method used to calculate the probability densities $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ in Section 3.2, we can determine the probability densities $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y_{ac}},t)$ in three-time domains.

3.4.1. $p_{y,ac}\left(y_{ac},t\right)$ and $p_{y,ac}\left(v_{y_{ac}},t\right)$ with the Active Noise in the Short-Time Domain $t<<{\tau }_{ac}$

The probability densities $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y_{ac}},t)$ are given by

$$p_{y,ac}(x_{ac},t)=[\pi {\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^4}{4{\tau }_{ac}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{4{\tau }_{ac}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^4}}{y}_{ac}^2],$$

(154)

$$p_{y,ac}\left(v_{y_{ac}},t\right)=\left[\pi \left[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1\right]{\gamma }_{2,ac}{t}^{2h+1}{]}^{-{\displaystyle \frac{1}{2}}}\mathit{exp}[-{\displaystyle \frac{v_{y_{ac}}^2}{\left[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1\right]{\gamma }_{2,ac}{t}^{2h+1}}}\right].$$

(155)

with the mean squared values

$$<y_{ac}^2(t)>={\displaystyle \frac{1}{8{\tau }_{ac}}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^4, <v_{y_{ac}}^2(t)>={\displaystyle \frac{{\gamma }_{2,ac}}{2}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+1}.$$

(156)

3.4.2. $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y_{ac}},t)$ with the Active Noise in the Long-Time Domain

Using the inverse Fourier transform, the probability densities $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y,ac},t)$ are obtained by

$$p_{y,ac}(y_{ac},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3}{3}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3y_{ac}^2}{2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3}}],$$

(157)

$$p_{y,ac}(v_{y_{ac}},t)=[\pi [{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,ac}{t}^{2h+1}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{v_{y_{ac}}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,ac}{t}^{2h+1}}}].$$

(158)

The mean squared values $<y_{ac}^2(t)>$ and $<v_{y_{ac}}^2(t)>$ for $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y_{ac}},t)$ are, respectively, given by

$$<x_{ac}^2(t)>={\displaystyle \frac{1}{3}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3,<v_{y_{ac}}^2(t)>={\displaystyle \frac{{\gamma }_{2,ac}}{2}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+1}.$$

(159)

3.4.3. $p_{y,ac}\left(y_{ac},t\right)$ and $p_{y,ac}\left(v_{y_{ac}},t\right)$ with the Active Noise for ${\tau }_{ac}=0$

The probability densities $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y_{ac}},t)$ are expressed as

$$p_{y,ac}(y_{ac},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3}{3}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3}{2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3}}{y}_{ac}^2],$$

(160)

$$p_{y,ac}(v_{y_{ac}},t)=[4\pi [{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{1}{4[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t}}{v}_{y_{ac}}^2].$$

(161)

The corresponding mean squared displacement $<y_{ac}^2(t)>$ and the mean squared velocity $<v_{y_{ac}}^2(t)>$ are

$$<y_{ac}^2(t)>={\displaystyle \frac{1}{3}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3, <v_{y_{ac}}^2(t)>=2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t.$$

(162)

4. Thermal and Active Fractional Generalized Langevin Equation

We consider a nonequilibrium dynamic model referred to as the active fractional Langevin equation having a viscoelastic memory effect with a power-law kernel $k(t)=|{\displaystyle \frac{t-t\prime }{\tau }}{|}^{2h-2}$ with an optical trapped force and the random noise.

From Equations (70) and (71), the thermal fractional generalized Langevin equations with a harmonic force and a viscous force in our model are expressed as

$${\displaystyle \frac{d}{dt}}{v}_{x_{th}}(t)=-q^2{B}^2{v}_{x_{th}}(t)-k_1{x}_{th}(t)-{\gamma }_{1,th}{\int }_0^{t}dt\prime |{\displaystyle \frac{t-t\prime }{{\tau }_{th}}}{|}^{2h-2}{v}_{x_{th}}(t\prime )+{\zeta }_{x_{th}}(t)+qB{\zeta }_{y_{th}}(t),$$

(163)

$${\displaystyle \frac{d}{dt}}{v}_{y_{th}}(t)=-q^2{B}^2{v}_{y_{th}}(t)-k_2{y}_{th}(t)-{\gamma }_{2,th}{\int }_0^{t}dt\prime |{\displaystyle \frac{t-t\prime }{{\tau }_{th}}}{|}^{2h-2}{v}_{y_{th}}(t\prime )+{\zeta }_{y_{th}}(t)-qB{\zeta }_{x_{th}}(t).$$

(164)

The active fractional generalized Langevin equations from Equations (72) and (74) are is given by

$${\displaystyle \frac{d}{dt}}{v}_{x_{ac}}(t)=-q^2{B}^2{v}_{x_{ac}}(t)-k_1{x}_{ac}(t)-{\gamma }_{1,ac}{\int }_0^{t}dt\prime |{\displaystyle \frac{t-t\prime }{{\tau }_{ac}}}{|}^{2h-2}{v}_{x_{ac}}(t\prime )+{\zeta }_{x_{ac}}(t)+qB{\zeta }_{y_{ac}}(t),$$

(165)

$${\displaystyle \frac{d}{dt}}{v}_{y_{ac}}(t)=-q^2{B}^2{v}_{y_{ac}}(t)-k_2{y}_{ac}(t)-{\gamma }_{2,ac}{\int }_0^{t}dt\prime |{\displaystyle \frac{t-t\prime }{{\tau }_{ac}}}{|}^{2h-2}{v}_{y_{ac}}(t\prime )+{\zeta }_{y_{ac}}(t)-qB{\zeta }_{x_{ac}}(t),$$

(166)

Here, the thermal equilibrium noise and the active noise denote ${\zeta }_{\mathrm{i},\mathrm{th}}(t)$ and ${\zeta }_{\mathrm{i},,\mathrm{ac}}(t)$ for $i=x,y$, respectively. Two noises, the thermal noise and the active noise, used to Equation (74) are given by

$${\zeta }_{i_{th}}(t){\zeta }_{i_{th}}(t^{\prime })>={\displaystyle \frac{{\zeta }_{i0,th}^2}{2}}|{\displaystyle \frac{t-t^{\prime }}{{\tau }_{th}}}{|}^{2h-2}, <{\zeta }_{i_{ac}}(t){\zeta }_{i_{ac}}(t^{\prime })>={\displaystyle \frac{{\zeta }_{i0,ac}^2}{2{\tau }_{ac}}}\mathit{exp}(-{\displaystyle \frac{|t-t^{\prime }|}{{\tau }_{ac}}}).$$

(167)

Next, we next derive the Fokker–Planck equation from the active fractional generalized Langevin equation. The joint probability density $p(x_i,v_{x_i},t)$ for displacement $x_i$ and velocity $v_{x_i}$ is defined by $p(x_i,v_{x_i},t)=<\delta (x_i-x_i(t))\delta (v_{x_i}-v_{x_i}(t))>$ for $i=th,ac$. The time derivatives of the joint probability equation for $p_{x,th}(x_{th},v_{x_{th}},t)\equiv p_{x,th}$, $p_{y,th}(y_{th},v_{y_{th}},t)\equiv p_{y,th}$, $p_{x,ac}(x_{ac},v_{x_{ac}},t)\equiv p_{x,ac}$, and $p_{y,ac}$ $(y_{ac},v_{y_{ac}},t)\equiv p_{y,ac}$ are expressed as follows:

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,th}=-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{x}_{th}}}<{\displaystyle \frac{\mathit{\partial }{x}_{th}}{\mathit{\partial }t}}{\delta }_{x_{th}}{\delta }_{v_{x_{th}}}>-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_{x_{th}}}}<v_{x_{th}}(t){\delta }_{x_{th}}{\delta }_{v_{x_{th}}}> \\ {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,th}=-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{y}_{th}}}<{\displaystyle \frac{\mathit{\partial }{y}_{th}}{\mathit{\partial }t}}{\delta }_{y_{th}}{\delta }_{v_{y_{th}}}>-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_{y_{th}}}}<v_{y_{th}}(t){\delta }_{y_{th}}{\delta }_{v_{y_{th}}}>, \end{gathered}$$

(168)

and

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,ac}=-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{x}_{ac}}}<{\displaystyle \frac{\mathit{\partial }{x}_{ac}}{\mathit{\partial }t}}{\delta }_{x_{ac}}{\delta }_{v_{x_{ac}}}>-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_{x_{ac}}}}<v_{x_{ac}}(t){\delta }_{x_{ac}}{\delta }_{v_{x_{ac}}}> \\ {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,ac}=-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{y}_{ac}}}<{\displaystyle \frac{\mathit{\partial }{y}_{ac}}{\mathit{\partial }t}}{\delta }_{y_{ac}}{\delta }_{v_{y_{ac}}}>-{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{v}_{y_{ac}}}}<v_{y_{ac}}(t){\delta }_{y_{ac}}{\delta }_{v_{y_{ac}}}>. \end{gathered}$$

(169)

Here, ${\delta }_{x_i}(x_i-x_i(t))\equiv {\delta }_{x_i}$ and ${\delta }_{v_{x_i}}(v_{x_i}-v_{x_i}(t))\equiv {\delta }_{v_{x_i}}$. As the persistent Hurst exponent h ranges to $1/2<h<1$ is the same value as Section 3, we introduce ${\int }_0^{t}dt\prime |t-t\prime {|}^{2h-2}v(t\prime )=[\mathit{\Gamma }(2h-1){]}^{-1}{\displaystyle \frac{d^{2-2h}}{d{t}^{2-2h}}}x(t)$ in Equations (163)–(166). We assume that the particle is initially at rest at $t=0$.

We define the double Fourier transforms of the joint probability densities as

$$\begin{gathered} p_{x,th}({\xi }_1,{\nu }_1,t)={\int }_{-\infty }^{+\infty }d{x}_{th}{\int }_{-\infty }^{+\infty }d{v}_{x_{th}}\mathit{exp}(-i{\xi }_1{x}_{th}-i{\nu }_1{v}_{x_{th}})p(x_x,v_{x_{th}},t) \\ p_{y,th}({\xi }_2,{\nu }_2,t)={\int }_{-\infty }^{+\infty }d{y}_{th}{\int }_{-\infty }^{+\infty }d{v}_{y_{th}}\mathit{exp}(-i{\xi }_2{y}_{th}-i{\nu }_2{v}_{y_{th}})p(y_{th},v_{y_{th}},t), \end{gathered}$$

(170)

and

$$\begin{gathered} p_{x,ac}({\xi }_1,{\nu }_1,t)={\int }_{-\infty }^{+\infty }d{x}_{ac}{\int }_{-\infty }^{+\infty }d{v}_{x_{ac}}\mathit{exp}(-i{\xi }_1{x}_{ac}-i{\nu }_1{v}_{x_{ac}})p(x_x,v_{x_{ac}},t) \\ p_{y,ac}({\xi }_2,{\nu }_2,t)={\int }_{-\infty }^{+\infty }d{y}_{ac}{\int }_{-\infty }^{+\infty }d{v}_{y_{ac}}\mathit{exp}(-i{\xi }_2{y}_{ac}-i{\nu }_2{v}_{y_{ac}})p(y_{ac},v_{y_{ac}},t). \end{gathered}$$

(171)

Then, the time derivatives of the Fourier transform of joint probability densities for the x- and y-component of a colloid particle are derived as

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,th}({\xi }_1,{\nu }_1,t)=-[k_1{\nu }_1+{\gamma }_{1,th}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}{p}_{x,th}({\xi }_1,{\nu }_1,t)+[{\xi }_1-q^2{B}^2{\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}{p}_{x,th}({\xi }_1,{\nu }_1,t) \\ -[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_1{\nu }_1+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }_1^2]p_{x,th}({\xi }_1,{\nu }_1,t), \end{gathered}$$

(172)

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,th}({\xi }_2,{\nu }_2,t)=-[k_2{\nu }_2+{\gamma }_{2,th}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,th}({\xi }_2,{\nu }_2,t)+[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,th}({\xi }_2,{\nu }_2,t) \\ -[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_2{\nu }_2+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }_2^2]p_{y,th}({\xi }_2,{\nu }_2,t), \end{gathered}$$

(173)

and

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,ac}({\xi }_1,{\nu }_1,t)=-[k_1{\nu }_1+{\gamma }_{1,ac}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}{p}_{x,ac}({\xi }_1,{\nu }_1,t)+[{\xi }_1-q^2{B}^2{\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}{p}_{x,ac}({\xi }_1,{\nu }_1,t) \\ +[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][b(t){\xi }_1{\nu }_1-a(t){\nu }_1^2]p_{x,th}({\xi }_1,{\nu }_1,t), \end{gathered}$$

(174)

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,ac}({\xi }_2,{\nu }_2,t)=-[k_2{\nu }_2+{\gamma }_{2,ac}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,ac}({\xi }_2,{\nu }_2,t)+[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,ac}({\xi }_2,{\nu }_2,t) \\ +\left[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1\right]\left[b\left(t\right){\xi }_2{\nu }_2-a\left(t\right){\nu }_2^2\right]p_{y,ac}\left({\xi }_2,{\nu }_2,t\right). \end{gathered}$$

(175)

Here, $D_{2-2h}=d^{2-2h}/d{t}^{2-2h}$. The parameters ${\overline{\alpha }}_1$, ${\overline{\alpha }}_2$, $a(t)$, and $b(t)$ are ${\overline{\alpha }}_1={\displaystyle \frac{{\zeta }_{x0,th}^2}{2}}$, ${\overline{\alpha }}_2={\displaystyle \frac{{\zeta }_{y0,ac}^2}{2}}$, $a(t)=1-\mathit{exp}(-{\displaystyle \frac{t}{{\tau }_{ac}}})$, and $b(t)=(t+{\tau }_{ac})\mathit{exp}(-{\displaystyle \frac{t}{{\tau }_{ac}}})-{\tau }_{ac}$. It is apparent to call Equations (172)–(175) the Fourier-transformed Fokker–Planck equations, as mentioned as Introduction.

4.1. $p_{x,th}(x_{th},t)$ and $p_{x,th}(v_{x_{th}},t)$ with the Thermal Noise

4.1.1. $p_{x,th}(x_{th},t)$ and ${ p}_{x,th}(v_{x_{th}},t)$ with the Thermal Noise in the Short-Time Domain

In this subsection, we obtain the solutions of the probability densities $p_{x,th}(x_{th},t)$ and $p_{x,th}(v_{x_{th}},t)$ in the short-time domain $t<<{\tau }_{th}$. To find the special solutions for ${\xi }_1$ and ${\nu }_1$ by variable-separating from Equation (172), the two equations for the displacement and the velocity are given by

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,th}({\xi }_1,t)=-[k_1{\nu }_1+{\gamma }_{1,th}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}{p}_{x,th}({\xi }_1,t)-[{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_1{\nu }_1+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }_1^2]-A]p_{x,th}({\xi }_1,t),$$

(176)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,th}({\nu }_1,t)=[{\xi }_1-q^2{B}^2{\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}{p}_{x,th}({\nu }_1,t)-[{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_1{\nu }_1+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }_1^2]+A]p_{x,th}({\nu }_1,t).$$

(177)

Here, $A$ denotes the separation constant.

For the short-time domain, assuming ${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,th}({\xi }_1,t)=0$, we obtain $p_{x,th}^{st}({\xi }_1,t)$ in the steady state as

$$p_{th}^{st}{\xi }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{[k_1{\nu }_1+2{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}]}}[{\displaystyle \frac{1}{4h{\tau }_{th}^{2h}}}{t}^{2h}{\displaystyle \frac{{{\xi }_1}^2}{2}}{\nu }_1+{\displaystyle \frac{1}{2(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{\xi }_1{{\nu }_1}^2+A{\xi }_1]].$$

(178)

Here, we assume that $[k_1{\nu }_1+2{\gamma }_{1,th}{D}_{2-2h}{]}^{-1}\cong (k_1{\nu }_1{)}^{-1}[1-2{\gamma }_{1,th}{D}_{2-2h}/k_1]$. In order to get the solution of the probability density for ${\xi }_1$ from ${q_{x,th}({\xi }_1,t)\equiv r}_{x,th}({\xi }_1,t)q_{x,th}^{st}({\xi }_1,t)$, we calculate the Fourier transform of the probability density after including terms up to order $1/{\tau }_{th}^2$ as

$$p_{x,th}({\xi }_1,t)=q_{x,th}({\xi }_1,t)\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2k_1{\nu }_1}}[1-2\gamma D_{2-2h}/k_1][{\displaystyle \frac{1}{2h{\tau }_{th}^{2h}}}{t}^{2h}{\displaystyle \frac{{{\xi }_1}^2}{2}}{\nu }_1+{\displaystyle \frac{1}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{\xi }_1{{\nu }_1}^2+A{\xi }_1]],$$

(179)

$$q_{x,th}({\xi }_1,t)=r_{x,th}({\xi }_1,t)\mathit{exp}[{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2(k_1{\nu }_1{)}^2}}[1-2\gamma D_{2-2h}/k_1][{\displaystyle \frac{1}{(2h{)}^2{\tau }_{th}^{2h}}}{t}^{2h-1}{\displaystyle \frac{{{\xi }_1}^2}{6}}{\nu }_1+{\displaystyle \frac{1}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{\displaystyle \frac{{{\xi }_1}^2}{2}}{{\nu }_1}^2]].$$

(180)

Taking the solutions as arbitrary functions of variable $t-{\xi }_1/[k_1{\nu }_1+2\gamma {\nu }_1{D}_{2-2h}]$, the arbitrary function $r_{x,th}({\xi }_1,t)$ becomes $\mathit{\Theta }[t-{\xi }_1/[k_1{\nu }_1+2\gamma {\nu }_1{D}_{2-2h}]]$. As a result, we find that

$$p_{x,th}^{st}({\xi }_1,t)=r_{x,th}^{st}({\xi }_1,t)q_{x,th}^{st}({\xi }_1,t)p_{x,th}^{st}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/[k_1{\nu }_1+2\gamma {\nu }_1{D}_{2-2h}]]q_{x,th}^{st}({\xi }_1,t)p_{x,th}^{st}({\xi }_1,t).$$

(181)

Using a similar method as in Equations (178)–(181), we also obtain the Fourier transform of the probability density for ${\nu }_1$ as

$$p_{x,th}^{st}({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[{\xi }_1-q^2{B}^2{\nu }_1]]q_{x,th}^{st}({\nu }_1,t)p_{x,th}^{st}({\nu }_1,t).$$

(182)

Therefore, by calculating Equations (181) and (182), we find the Fourier transform of the joint probability density as

$$p_{x,th}({\xi }_1,{\nu }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}{6(2h-1){\tau }_{th}^{2h}}}{{\xi }_1}^2-{\displaystyle \frac{{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{t}^{2h}}{2(2h-1){\tau }_{th}^{2h-1}}}{{\nu }_1}^2].$$

(183)

Using the inverse Fourier transform, we get

$$p_{x,th}(x_{th},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}{3(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3(2h-1){\tau }_{th}^{2h-1}}{2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}}{x}_{th}^2],$$

(184)

$$p_{x,th}(v_{x_{th}},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}{(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{(2h-1){\tau }_{th}^{2h-1}}{2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}}{v}_{x_{th}}^2].$$

(185)

The mean squared displacement $p_{x,th}(x_{th},t)$ and the mean squared displacement $p_{x,th}(v_{x_{th}},t)$ from Equations (184) and (185) are, respectively, given by

$$<x_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{3(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h+2}, <v_{x_{th}}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h+2}.$$

(186)

4.1.2. $p_{x,th}(x_{th},t)$ and $p_{x,th}(v_{x_{th}},t)$ with the Thermal Noise in the Long-Time Domain

Now we find the probability densities $p_{x,th}(x_{th},t)$ and $p_{x,th}(v_{x_{th}},t)$ in long-time domain $t>>{\tau }_{th}$. An approximate equation from Equation (176) can be written as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{th\xi }({\xi }_1,t)\cong -{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2h{\tau }_{th}^{2h}}}{t}^{2h}{\xi }_1{\nu }_1{p}_{th\xi }({\xi }_1,t)-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{{\nu }_1}^2{p}_{th\xi }({\xi }_1,t).$$

(187)

The Fourier transform of the probability density $p_{th\xi }({\xi }_1,t)$ from Equation (187) is calculated as

$$p_{th\xi }({\xi }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{8h^2{\tau }_{th}^{2h}}}{t}^{2h+1}{\xi }_1{\nu }_1-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{4h(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}{{\nu }_1}^2].$$

(188)

We find the Fourier transform of the steady probability density $q_{th}^{st}({\xi }_1,t)$ for ${\xi }_1$, from ${p_{th}({\xi }_{1,}t)\equiv q}_{th\xi }({\xi }_1,t)p_{th\xi }({\xi }_1,t)$,

$$q_{th\xi }^{st}({\xi }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2k_1{\nu }_1}}[{\displaystyle \frac{1}{2h(2h+1){\tau }_{th}^{2H}}}{t}^{2h+1}{\displaystyle \frac{{{\xi }_1}^3}{3}}+{\displaystyle \frac{1}{h(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}{\nu }_1{\displaystyle \frac{{{\xi }_1}^2}{2}}].$$

(189)

As the Fourier transform of probability density $q({\xi }_1,t)$ in the short-time domain is given by $q_{th}({\xi }_1,t)=r_{th}({\xi }_1,t)q_{th}^{st}({\xi }_1,t)$, then $p_{x,th}({\xi }_1,t)$ is derived as

$$p_{x,th}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/[k_1{\nu }_1+{\gamma }_{1,th}{\nu }_1{D}_{2-2h}]]q_{th\xi }^{st}({\xi }_1,t)p_{th}^{st}({\xi }_1,t),$$

(190)

where $r_{th}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/[k_1{\nu }_1+{\gamma }_{1,th}{\nu }_1{D}_{2-2h}]]$. Applying Equation (177) for $p_{x,th}({\xi }_1,t)$ to the similar method as in Equations (187)–(190), we also obtain the Fourier transforms of the probability density for velocity as

$$p_{x,th}({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[{\xi }_1-q^2{B}^2{\nu }_1]]q_{th\nu }^{st}({\nu }_1,t)p_{th}^{st}({\nu }_1,t).$$

(191)

From Equations (190) and (191), we have

$$p_{x,th}({\xi }_1,{\nu }_1,t)=p_{x,th}({\xi }_1,t)p_{x,th}({\nu }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}{4h(2h+1){\tau }_{th}^{2h}}}{{\xi }_1}^2-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}_1{\gamma }_{1,th}{t}^{2h}}{2h(2h-1){\tau }_{th}^{2h-1}}}{{\nu }_1}^2].$$

(192)

Using the inverse Fourier transform, the probability density $p_{x,th}(x_{th},t)$ and $p_{x,th}(v_{x_{th}},t)$ are, respectively, expressed as

$$p_{x,th}(x_{th},t)=[\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}_1{t}^{2h+2}}{h(2h+1){\tau }_{th}^{2h}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{h(2h+1){\tau }_{th}^{2h}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}}{x}_{th}^2],$$

(193)

$$p_{x,th}(v_{x_{th}},t)=[\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{1,th}{t}^{2h}}{h(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{h(2h-1){\tau }_{th}^{2h-1}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{1,th}{t}^{2h}}}{v}_{x_{th}}^2].$$

(194)

The mean squared values $<x_{th}^2(t)>$ and $<v_{x_{th}}^2(t)>$ for probability densities $p_{th}(x_{th},t)$ and $p_{th}(v_{x_{th}},t)$ are, respectively, given by

$$<x_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2h(2h+1){\tau }_{th}^{2h}}}{t}^{2h+2}, <v_{x_{th}}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{1,th}}{2h(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}.$$

(195)

4.1.3. $p_{th}\left(x_{th},t\right)$ and $p_{th}\left(v_{x_{th}},t\right)$ with the Thermal Noise for ${\tau }_{th}=0$

In this subsection, we find $p_{th}(x_{th},t)$ and $p_{th}(v_{x_{th}},t)$ in time domain ${\tau }_{th}=0$. An approximate equation from Equation (176) for $\xi$ can be written as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{th\xi }({\xi }_1,t)\cong -[k_1{\nu }_1{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}+{\gamma }_{1,th}{\nu }_1{D}_{2-2h}]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}{p}_{th\xi }({\xi }_1,t)-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{{\nu }_1}^2{p}_{th\xi }({\xi }_1,t).$$

(196)

In the steady state, we calculate $p_{th}^{st}({\xi }_1,t)$ from Equation (196) as

$$p_{th\xi }^{st}({\xi }_1,t)=\mathit{exp}[-{\displaystyle \frac{\left[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2\right]}{2k_1{\nu }_1}}[1-{\gamma }_{1,th}{D}_{2-2h}/k_1][{\displaystyle \frac{t^{2h-1}}{\left(2h-1\right){\tau }_{th}^{2h-1}}}{\xi }_1{{\nu }_1}^2]].$$

(197)

We find the Fourier transform of probability density $p_{th}({\xi }_1,t)$ as

$$p_{th}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/[k_1{\nu }_1+{\gamma }_{1,th}{\nu }_1{D}_{2-2h}]]p_{th\xi }^{st}({\xi }_1,t).$$

(198)

Using a similar method for $p_{th}({\xi }_1,t)$, we obtain the Fourier transform of the probability density $p_{th}({\nu }_1,t)$ as

$$p_{th}({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/{\xi }_1]p_{th\nu }^{st}({\nu }_1,t).$$

(199)

We calculate $p_{th}({\xi }_1,{\nu }_1,t)$ from Equations (198) and (199) as

$$p_{th}({\xi }_1,{\nu }_1,t)=p_{th}({\xi }_1,t)p_{th}({\nu }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}{6(2h-1){\tau }_{th}^{2H-1}}}{{\xi }_1}^2-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h}}{2(2h-1){\tau }_{th}^{2h-1}}}{{\nu }_1}^2].$$

(200)

Using the inverse Fourier transform, the probability densities $p_{th}(x_{th},t)$ and $p_{th}(v_{x_{th}},t)$ are, respectively, calculated as

$$p_{th}(x_{th},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}{3(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3(2h-1){\tau }_{th}^{2h-1}}{2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+2}}}{x}_{th}^2],$$

(201)

$$p_{th}(v_{x_{th}},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h}}{(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{(2h-1){\tau }_{th}^{2h-1}}{2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h}}}{v}_{x_{th}}^2].$$

(202)

Consequently, we have the mean squared displacement $<x_{th}^2(t)>$ and the mean squared velocity $<v_{x_{th}}^2(t)>$ as

$$<x_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{3(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h+2}, <v_{x_{th}}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}.$$

(203)

4.2. $p_{y,th}(y_{th},v_{y_{th}},t)$ with Thermal Noise

From the Fourier transforms of the Fokker–Planck equation for the y-component of a charged colloid,

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,th}({\xi }_2,{\nu }_2,t)=-[k_2{\nu }_2+{\gamma }_{2,th}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,th}({\xi }_2,{\nu }_2,t)+[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,th}({\xi }_2,{\nu }_2,t) \\ -[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_2{\nu }_2+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }_2^2]p_{y,th}({\xi }_2,{\nu }_2,t), \end{gathered}$$

We calculate the probability densities $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y,th},t)$ in three-time domains.

By separating the variables ${\zeta }_2$ and ${\nu }_2$ as ${p_{y,th}\left({\zeta }_2,v_2,t\right)=p}_{y,th}({\zeta }_2,t)p_{y,th}({\nu }_2,t)$, Equation (172) has

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,th}({\xi }_2,t)=-[k_2{\nu }_2+{\displaystyle \frac{{\gamma }_{2,th}\mathit{\Gamma }(2h-1)}{{\tau }^{2h-3}}}{D}_{2-2h}{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,th}({\xi }_2,t)-[{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{2}}[{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_2{\nu }_2+{\displaystyle \frac{t^{2h-1}}{2h-1){\tau }_{th}^{2h-1}}}{\nu }_2^2]-F]p_{y,th}({\xi }_2,t),$$

(204)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,th}({\nu }_2,t)=[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,th}({\nu }_2,t)-[{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\xi }_2{\nu }_2+{\displaystyle \frac{t^{2h-1}}{2h-1){\tau }_{th}^{2h-1}}}{\nu }_2^2]-F]p_{y,th}({\nu }_2,t),$$

(205)

Here, $F$ is the separation constant. By applying the same method used to calculate the probability densities $p_{x,th}(x_{th},t)$ and $p_{x,th}(v_{x,th},t)$ in Section 4.1, we determine the probability densities $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y,th},t)$ in three-time domains.

4.2.1. $p_{y,th}\left(y_{th},t\right)$ and $p_{y,th}\left(v_{y_{th}},t\right)$ in the Time Domain $t<<{\tau }_{th}$

The probability densities $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y_{th}},t)$ are expressed as

$$p_{y,th}(y_{th},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+2}}{3(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3(2h-1){\tau }_{th}^{2h-1}}{2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+2}}}{y}_{th}^2],$$

(206)

$$p_{y,th}(v_{y_{th}},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+2}}{(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{(2h-1){\tau }_{th}^{2h-1}}{2[{\overline{\alpha }}_2-qB{{\overline{\alpha }}_1}_2]t^{2h+2}}}{v}_{y_{th}}^2].$$

(207)

Then, the mean squared values $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y_{th}},t)$ yield

$$<y_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{3(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h+2}, <v_{y_{th}}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h+2}.$$

(208)

4.2.2. $p_{y,th}\left(y_{th},t\right)$ and $p_{y,th}\left(v_{y_{th}},t\right)$ in the Time Domain $t>>{\tau }_{th}$

By using the inverse Fourier transform, the probability densities $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y_{th}},t)$ are, respectively, expressed as

$$p_{y,th}(y_{th},t)=[\pi {\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+2}}{h(2h+1){\tau }_{th}^{2h}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{h(2h+1){\tau }_{th}^{2h}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+2}}}{y}_{th}^2],$$

(209)

$$p_{y,th}(v_{y_{th}},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,th}{t}^{2h}}{h(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{h(2h-1){\tau }_{th}^{2h-1}}{2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,th}{t}^{2h}}}{v}_{y_{th}}^2].$$

(210)

The mean squared values $<y_{th}^2(t)>$ and $<v_{y_{th}}^2(t)>$ for $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y_{th}},t)$ are, respectively, given by

$$<y_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{2h(2h+1){\tau }_{th}^{2h}}}{t}^{2h+2}, <v_{y_{th}}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,th}}{h(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}.$$

(211)

4.2.3. $p_{y,th}\left(y_{th},t\right)$ and $p_{y,th}\left(v_{y_{th}},t\right)$ in the Time Domain ${\tau }_{th}=0$

The probability densities $p_{y,th}(y_{th},t)$ and $p_{y,th}(v_{y_{th}},t)$ are, respectively, expressed as

$$p_{y,th}(y_{th},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^{2h+2}}{3(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3(2h-1){\tau }_{th}^{2h-1}}{2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+2}}}{y}_{th}^2],$$

(212)

$$p_{y,th}(v_{y_{th}},t)=[2\pi {\displaystyle \frac{[qB{\overline{\alpha }}_1-{\overline{\alpha }}_2]t^{2h}}{(2h-1){\tau }_{th}^{2h-1}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{(2h-1){\tau }_{th}^{2h-1}}{2[qB{\overline{\alpha }}_1-{\overline{\alpha }}_2]t^{2h}}}{v}_{y_{th}}^2].$$

(213)

Lastly, the mean squared displacement $<y_{th}^2(t)>$ and the mean squared velocity $<v_{y_{th}}^2(t)>$ are

$$<y_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{3(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h+2}, <v_{y_{th}}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}.$$

(214)

4.3. $p_{x,ac}(x_{ac},t)$ and $p_{ac}(v_{x_{ac}},t)$ with Active Noise

4.3.1. $p_{x,ac}(x_{ac},t)$ and $p_{ac}(v_{x_{ac}},t)$ with Active Noise in the Short-Time Domain

In this subsection, we find the solutions of the probability density $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{x_{ac}},t)$ in the short-time domain $t<<{\tau }_{ac}$. To find the special solutions for ${\xi }_1$ and ${\nu }_1$ by the variable separation from Equation (174), we write the two equations for displacement and velocity as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,ac}({\xi }_1,t)=-[k_1{\nu }_1+{\gamma }_{1,ac}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}{p}_{x,ac}({\xi }_1,t)+[{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][b(t){\xi }_1{\nu }_1-a(t){\nu }_1^2]+F]p_{x,ac}({\xi }_1,t),$$

(215)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,ac}({\nu }_1,t)=[{\xi }_1-q^2{B}^2{\nu }_1]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_1}}{p}_{x,ac}({\nu }_1,t)+[{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2][b(t){\xi }_1{\nu }_1-a(t){\nu }_1^2]+F]p_{x,ac}({\xi }_1,t).$$

(216)

Here, F denotes the separation constant. Assuming ${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{x,ac}({\xi }_1,t)=0$, the Fourier transform of the steady probability density $p_{x,ac}^{st}({\xi }_1,t)$ as

$$p_{ac}^{st}({\xi }_1,t)=\mathit{exp}[{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2k_1{\nu }_1}}[1-2{\gamma }_{1,ac}{D}_{2-2h}/k_1][{\displaystyle \frac{b(t)}{2}}{\nu }_1{{\xi }_1}^2-a(t){{\nu }_1}^2{\xi }_1+F{\xi }_1]].$$

(217)

To obtain the solution of the probability density for ${\xi }_1$ from ${q_{x,ac}({\xi }_1,t)\equiv r}_{x,ac}({\xi }_1,t)q_{x,ac}^{st}({\xi }_1,t)$, we calculate the Fourier transform of the probability density including terms up to order $1/{\tau }_{ac}^2$ as

$$p_{x,ac}({\xi }_1,t)=q_{x,ac}({\xi }_1,t)\mathit{exp}[{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2k_1{\nu }_1}}[1-2{\gamma }_{1,ac}{D}_{2-2h}/k_1][{\displaystyle \frac{b(t)}{2}}{\nu }_1{{\xi }_1}^2-a(t){{\nu }_1}^2{\xi }_1+B{\xi }_1]],$$

(218)

$$q_{x,ac}({\xi }_1,t)=r_{x,ac}({\xi }_1,t)\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2(k_1{\nu }_1{)}^2}}[1-2{\gamma }_{1,ac}{D}_{2-2h}/k_1][{\displaystyle \frac{b^{\prime }(t)}{6}}{\nu }_1{{\xi }_1}^3-{\displaystyle \frac{a^{\prime }(t)}{2}}{{\nu }_1}^2{{\xi }_1}^2]],$$

(219)

$$r_{x,ac}({\xi }_1,t)=s_{x,ac}({\xi }_1,t)\mathit{exp}[{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2(k_1{\nu }_1{)}^3}}[1-2{\gamma }_{1,ac}{D}_{2-2h}/k_1][{\displaystyle \frac{b^{″}(t)}{24}}{\nu }_1{{\xi }_1}^4-{\displaystyle \frac{a^{″}(t)}{6}}{{\nu }_1}^2{{\xi }_1}^3]],$$

(220)

$$s_{x,ac}({\xi }_1,t)=t_{x,ac}({\xi }_1,t)\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2(k_1{\nu }_1{)}^4}}[1-2{\gamma }_{1,ac}{D}_{2-2h}/k_1]{\displaystyle \frac{b^{‴}(t)}{120}}{\nu }_1{{\xi }_1}^5].$$

(221)

Taking the solutions as arbitrary functions of $t-{\xi }_1/[k_1{\nu }_1+{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}]$, the arbitrary function $t_{x,ac}({\xi }_1,t)$ becomes $\mathit{\Theta }[t-{\xi }_1/[k_1{\nu }_1+{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}]]$. As a result, we find that

$$p_{x,ac}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/[k_1{\nu }_1+{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}]]s_{x,ac}^{st}({\xi }_1,t)r_{x,ac}^{st}({\xi }_1,t)q_{x,ac}^{st}({\xi }_1,t)p_{x,ac}^{st}({\xi }_1,t).$$

(222)

Using a similar method as in Equations (217)–(222), we also obtain the Fourier transform of the probability density for the velocity as

$$p_{x,ac}({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[{\xi }_1-q^2{B}^2{\nu }_1]]s_{x,ac}^{st}({\nu }_1,t)r_{x,ac}^{st}({\nu }_1,t)q_{x,ac}^{st}({\nu }_1,t)p_{x,ac}^{st}({\nu }_1,t).$$

(223)

Therefore, by calculating Equations (222) and (223), we get the Fourier transform of the joint probability density as

$$p_{x,ac}({\xi }_1,{\nu }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^4}{16{\tau }_{ac}}}{{\xi }_1}^2-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]\gamma t^{2h+1}}{4}}{\nu }_1^2]$$

(224)

Using the inverse Fourier transform, we have

$$\begin{gathered} p_{x,ac}(x_{ac},t)=[\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^4}{4{\tau }_{ac}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{4{\tau }_{ac}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^4}}{x}_{ac}^2] \\ p_{x,ac}(v_{x_{ac}},t)=[\pi [{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{1,ac}{t}^{2h+1}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{v_{x_{ac}}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{1,ac}{t}^{2h+1}}}], \end{gathered}$$

(225)

The mean squared displacement and the mean squared displacement for $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ are, respectively,

$$<x_{ac}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{8{\tau }_{ac}}}{t}^4,<v_{x_{ac}}^2(t)>={\displaystyle \frac{1}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{1,ac}{t}^{2h+1}.$$

(226)

4.3.2. $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{x_{ac}},t)$ with the Active Noise in the Long-Time Domain

Now we find the probability densities $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ in long-time domain $t>>{\tau }_{ac}$. We write an approximate equation from Equation (215) as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{ac\xi }({\xi }_1,t)\cong {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2}}[b(t)\xi \nu -a(t){{\nu }_1}^2]p_{ac\xi }({\xi }_1,t).$$

(227)

The Fourier transform of the probability density $p_{ac\xi }({\xi }_1,t)$ from Equation (125) is expressed as

$$p_{ac\xi }({\xi }_1,t)\cong \mathit{exp}[{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2}}{\int }_0^{t}dt[b(t){\xi }_1{\nu }_1-a(t){{\nu }_1}^2]]$$

(228)

We find the steady probability density $q_{ac}^{st}({\xi }_1,t)$ for ${\xi }_1$, from ${p_{ac}({\xi }_1,t)\equiv q}_{ac\xi }({\xi }_1,t)p_{ac\xi }({\xi }_1,t)$,

$$q_{ac\xi }^{st}({\xi }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2}}{\int }_0^{t}dt[b(t){\xi }_1{\nu }_1-a(t){{\nu }_1}^2]]$$

(229)

As the Fourier transform of probability density $q_{x,ac}^{st}({\xi }_1,t)$ in the short-time domain is given by $q_{x,ac}({\xi }_1,t)=r_{x,ac}({\xi }_1,t)q_{ac\xi }^{st}({\xi }_1,t)$, then $p_{ac}({\xi }_1,t)$ is derived as

$$p_{x,ac}({\xi }_1,t)=r_{x,ac}({\xi }_1,t)q_{ac\xi }^{st}({\xi }_1,t)p_{x,ac}^{st}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/[k_1{\nu }_1+{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}]]q_{ac\xi }^{st}({\xi }_1,t)p_{x,ac}^{st}({\xi }_1,t),$$

(230)

where $r_{x,ac}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/[k_1{\nu }_1+{\gamma }_{1,ac}{\nu }_1{D}_{2-2H}]]$ and $p_{x,ac}^{st}({\xi }_1,t)$ is equal to Equation (217). Applying Equation (217) for ${\xi }_1$ to the similar method from Equations (227)–(230) for $p_{x,ac}({\xi }_1,t)$, we also obtain the Fourier transform of the probability density for velocity as

$$p_{x,ac}({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[{\xi }_1-q^2{B}^2{\nu }_1]]q_{ac\nu }^{st}({\nu }_1,t)p_{x,ac}^{st}({\nu }_1,t).$$

(231)

From Equations (230) and (231), we have

$$p_{x,ac}({\xi }_1,{\nu }_1,t)=p_{x,ac}({\xi }_1,t)p_{x,ac}({\nu }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3}{6}}{{\xi }_1}^2-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]\gamma t^{2H+1}}{4}}{{\nu }_1}^2].$$

(232)

Using the inverse Fourier transform, the probability densities $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ are, respectively, expressed as

$$p_{x,ac}(x_{ac},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3}{3}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3}{2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3}}{x}_{ac}^2],$$

(233)

$$p_{x,ac}(v_{x_{ac}},t)=[\pi [{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{1,ac}{t}^{2H+1}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{1}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{1,ac}{t}^{2H+1}}}{v}_{x_{ac}}^2].$$

(234)

Consequently, the mean squared values $<x_{ac}^2(t)>$ and $<v_{x_{ac}}^2(t)>$ for probability densities $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ are

$$<x_{ac}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{3}}{t}^3, <v_{x_{ac}}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{1,ac}}{2}}{t}^{2H+1}.$$

(235)

4.3.3. $p_{ac}\left(x_{ac},t\right)$ and $p_{ac}\left(v_{x_{ac}},t\right)$ with the Active Noise for ${\tau }_{ac}=0$

In this subsection, we find $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{x_{ac}},t)$ in time domain ${\tau }_{ac}=0$. An approximate equation from Equation (215) for ${\xi }_1$ can be written as

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{ac\xi }({\xi }_1,t)\cong -[k_1{\nu }_1+{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_1}}{p}_{ac\xi }({\xi }_1,t)-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2}}a(t){{\nu }_1}^2{p}_{ac\xi }({\xi }_1,t).$$

(236)

In the steady state, we calculate $p_{ac}^{st}({\xi }_1,t)$ from Equation (236) as

$$p_{ac\xi }^{st}({\xi }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2k_1{\nu }_1}}[1-{\gamma }_{1,ac}{D}_{2-2h}/k]{{\nu }_1}^2{\xi }_1].$$

(237)

Here, $a(t)=1,b(t)=0$ for ${\tau }_{ac}=0$. We find the Fourier transform of probability density $p_{x,ac}({\xi }_1,t)$ as

$$p_{x,ac}({\xi }_1,t)=\mathit{\Theta }[t-{\xi }_1/[k_1{\nu }_1+{\gamma }_{1,ac}{\nu }_1{D}_{2-2h}]]p_{ac\varsigma }^{st}({\xi }_1,t).$$

(238)

Using a similar method for $p_{x,ac}({\xi }_1,t)$, we obtain the Fourier transform of the probability density $p_{ac}({\nu }_1,t)$ as

$$p_{x,ac}({\nu }_1,t)=\mathit{\Theta }[t+{\nu }_1/[{\xi }_1-q^2{B}^2{\nu }_1]]p_{ac\nu }^{st}({\nu }_1,t).$$

(239)

We calculate $p(\xi ,\nu ,t)$ from Equations (238) and (239) as

$$p_{x,ac}({\xi }_1,{\nu }_1,t)=p_{x,ac}({\xi }_1,t)p_{x,ac}({\nu }_1,t)=\mathit{exp}[-{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3}{6}}{{\xi }_1}^2-[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t{{\nu }_1}^2].$$

(240)

Using the inverse Fourier transform, the probability densities $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ are, respectively, expressed as

$$p_{x,ac}(x_{ac},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3}{3}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3}{2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3}}{x}_{ac}^2],$$

(241)

$$p_{x,ac}(v_{x_{ac}},t)=[4\pi [{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{1}{4[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t}}{v}_{x_{ac}}^2].$$

(242)

The mean squared displacement and the mean squared velocity are, respectively, given by

$$<x_{ac}^2(t)>={\displaystyle \frac{1}{3}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3,<v_{x_{ac}}^2(t)>=2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t.$$

(243)

4.4. $p_{y,ac}(y_{ac},v_{y_{ac}},t)$ with Thermal Noise

From the Fourier transforms of the Fokker–Planck equation for the y-component of a charged colloid,

$$\begin{gathered} {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,ac}({\xi }_2,{\nu }_2,t)=-[k_2{\nu }_2+{\gamma }_{2,ac}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,ac}({\xi }_2,{\nu }_2,t)+[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,ac}({\xi }_2,{\nu }_2,t) \\ +[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][b(t){\xi }_2{\nu }_2-a(t){\nu }_2^2]p_{y,ac}({\xi }_2,{\nu }_2,t). \end{gathered}$$

We calculate the probability densities $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y,ac},t)$ in three-time domains.

By separating the variables ${\zeta }_2$ and ${\nu }_2$ as ${p_{y,ac}\left({\zeta }_2,v_2,t\right)=p}_{y,ac}({\zeta }_2,t)p_{y,ac}({\nu }_2,t)$, Equation (175) becomes

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,ac}({\xi }_2,t)=-[k_2{\nu }_2+{\gamma }_{2,th}\mathit{\Gamma }(2h-1){\tau }^{3-2h}{D}_{2-2h}{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\xi }_2}}{p}_{y,ac}({\xi }_2,t)+[{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][b(t){\xi }_2{\nu }_2-a(t){\nu }_2^2]+G]p_{y,ac}({\xi }_2,t),$$

(244)

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}{p}_{y,ac}({\nu }_2,t)=[{\xi }_2-q^2{B}^2{\nu }_2]{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }{\nu }_2}}{p}_{y,ac}({\nu }_2,t)+[{\displaystyle \frac{1}{2}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][b(t){\xi }_2{\nu }_2-a(t){\nu }_2^2]-G]p_{y,ac}({\xi }_2,t).$$

(245)

Here, $G$ is the separation constant. Applying the same method used to calculate the probability densities $p_{x,ac}(x_{ac},t)$ and $p_{x,ac}(v_{x_{ac}},t)$ in Section 4.3, we can determine the probability densities $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y_{ac}},t)$ in three-time domains.

4.4.1. $p_{y,ac}\left(y_{ac},t\right)$ and $p_{y,ac}\left(v_{y_{ac}},t\right)$ in the Time Domain $t<<{\tau }_{ac}$

The probability densities $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y_{ac}},t)$ are expressed as

$$p_{y,ac}(y_{ac},t)=[\pi {\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^4}{4{\tau }_{ac}}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{4{\tau }_{ac}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^4}}{y}_{ac}^2],$$

(246)

$$p_{y,ac}(v_{y_{ac}},t)=[\pi [{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,ac}{t}^{2h+1}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{v_{y_{ac}}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,ac}{t}^{2h+1}}}]$$

(247)

with the mean squared values

$$<y_{ac}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{8{\tau }_{ac}}}{t}^4,<v_{y_{ac}}^2(t)>={\displaystyle \frac{1}{2}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,ac}{t}^{2h+1}.$$

(248)

4.4.2. $p_{y,ac}\left(y_{ac},t\right)$ and $p_{y,ac}\left(v_{y_{ac}},t\right)$ in the Time Domain $t>>{\tau }_{ac}$

By using the inverse Fourier transform, the probability densities $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y_{ac}},t)$ are, respectively, given by

$$p_{y,ac}(y_{ac},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3}{3}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3}{2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3}}{y}_{ac}^2],$$

(249)

$$p_{y,ac}(v_{y_{ac}},t)=[\pi [{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,ac}{t}^{2H+1}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{1}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,ac}{t}^{2H+1}}}{v}_{y_{ac}}^2].$$

(250)

Thus, the mean squared values $<y_{ac}^2(t)>$ and $<v_{y_{ac}}^2(t)>$ for $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y_{ac}},t)$ are, respectively,

$$<y_{ac}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{3}}{t}^3,<v_{y_{ac}}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,ac}}{2}}{t}^{2h+1}.$$

(251)

4.4.3. $p_{y,ac}\left(y_{ac},t\right)$ and $p_{y,ac}\left(v_{y_{ac}},t\right)$ in the Time Domain ${\tau }_{ac}=0$

Lastly, the probability densities $p_{y,ac}(y_{ac},t)$ and $p_{y,ac}(v_{y_{ac}},t)$ are, respectively, expressed as

$$p_{y,ac}(y_{ac},t)=[2\pi {\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3}{3}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3}{2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3}}{y}_{ac}^2],$$

(252)

$$p_{y,ac}(v_{y_{ac}},t)=[4\pi [{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{1}{4[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t}}{v}_{y_{ac}}^2].$$

(253)

The mean squared displacement $<y_{th}^2(t)>$ and the mean squared velocity $<v_{y_{ac}}^2(t)>$ from Equations (252) and (253) are, respectively, calculated as

$$<y_{ac}^2\left(t\right)>={\displaystyle \frac{1}{3}}\left[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1\right]t^3,<v_{y_{ac}}^2(t)>=2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t$$

(254)

4.5. Mean Squared Displacement and Mean Squared Velocity

In this subsection, we summarize the analytical results for the mean squared displacement (MSD) and mean squared velocity (MSV) obtained in Section 2, Section 3 and Section 4, as listed in

Table 1. Summary of analytical results obtained in Section 2, Section 3 and Section 4.

Section	Model	Time Domain	MSD ⟨x2(t)⟩, ⟨y2(t)⟩	$\mathbf{MSV} \mathbf{⟨}{\mathit{v}}_{\mathit{x}}^2\mathbf{(}\mathit{t}\mathbf{)}\mathbf{⟩}\mathbf{,} \mathbf{⟨}{\mathit{v}}_{\mathit{y}}^2\mathbf{(}\mathit{t}\mathbf{)}\mathbf{⟩}$
		$t\ll \tau$	$<x^2\left(t\right)>=\left({\alpha }_1+qB{\alpha }_2\right)t^3/6k_1\tau$ $<y^2(t)>=({\alpha }_2-qB{\alpha }_1)t^3/6k_2\tau$	$<v_x^2\left(t\right)>=3/2(q^2{B}^2+{\gamma }_1{)}^2 \left({\alpha }_1+qB{\alpha }_2\right)t^3$ $<v_y^2(t)>=3/2(q^2{B}^2+{\gamma }_2{)}^2({\alpha }_2-qB{\alpha }_1)t^3$
Section 2.2 and Section 2.3	A charged colloid with a harmonic trap	$t\gg \tau$	$<x^2(t)>=1/3({\alpha }_1+qB{\alpha }_2)t^4$ $<y^2(t)>=1/3({\alpha }_2-qB{\alpha }_1)t^4$	$<v_x^2(t)>=2(q^2{B}^2+{\gamma }_1{)}^2({\alpha }_1+qB{\alpha }_2)t^3$ $<v_y^2(t)>=2(q^2{B}^2+{\gamma }_2{)}^2({\alpha }_2-qB{\alpha }_1{)t}^3$
		$\tau =0$	$<x^2(t)>=1/4(q^2{B}^2+{\gamma }_1)({\alpha }_1+qB{\alpha }_2)t^4$ $<y^2(t)>=1/4(q^2{B}^2+{\gamma }_2)({\alpha }_2-qB{\alpha }_1)t^4$	$<v_x^2(t)>=({\alpha }_1+qB{\alpha }_2)t$ $<v_y^2(t)>=({\alpha }_2-qB{\alpha }_1)t$
		$t\ll {\tau }_{th}$	$<x_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{3(2h-1{)}^2{\tau }_{th}^{2h-1}}}{t}^{2h+1}$ $<y_{th}^2(t)>={\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]}{3(2h-1{)}^2{\tau }_{th}^{2h-1}}}{t}^{2h+1}$	$<v_{x,th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2(2h-1{)}^2{\tau }_{th}^{2h-1}}}{t}^{2h}$ $<v_{y,th}^2(t)>={\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]}{2(2h-1{)}^2{\tau }_{th}^{2h-1}}}{t}^{2h}$
Section 3.1 and Section 3.2	Thermal fractional Langevin equation with viscoelastic kernel	$t\gg {\tau }_{th}$	$<x_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{8h^2{\tau }_{th}^{2h}}}{t}^{2h+2}$ $<y_{th}^2(t)>={\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]}{8h^2{\tau }_{th}^{2h}}}{t}^{2h+1}$	$<v_{x,th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}$ $<v_{y,th}^2(t)>={\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}$
		${\tau }_{th}=0$	$<x_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{8h^2{\tau }_{th}^{2h}}}{t}^{2h+2}$ $<y_{th}^2(t)>={\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]}{8h^2{\tau }_{th}^{2h}}}{t}^{2h+1}$	$v_{x,th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}$ $<v_{th}^2(t)>={\displaystyle \frac{[\overline{{\alpha }_2}-qB\overline{{\alpha }_1}]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}$
		$t\ll {\tau }_{ac}$	$<x_{ac}^2(t)>=1/8{\tau }_{ac}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^4$ $<y_{ac}^2(t)>=1/8{\tau }_{ac}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^4$	$<v_{x,ac}^2(t)>={\displaystyle \frac{{\gamma }_{2,ac}}{2}}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+1}$ $<v_{y,ac}^2(t)>={\displaystyle \frac{{\gamma }_{2,ac}}{2}}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+1}$
Section 3.3 and Section 3.4	Active fractional Langevin equation	$t\gg {\tau }_{ac}$	$<x_{ac}^2(t)>=1/3[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3$ $<x_{ac}^2(t)>=1/3[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3$	$<v_{x,ac}^2(t)>={\gamma }_{2,ac}/2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+1}$ $<v_{y,ac}^2(t)>={\gamma }_{2,ac}/2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+1}$
	with viscoelastic kernel	${\tau }_{ac}=0$	$<x_{ac}^2(t)>=1/3[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3$ $<y_{ac}^2(t)>=1/3[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3$	$<v_{x,ac}^2\left(t\right)\ge 2\left[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2\right]t$ $<v_{y,ac}^2(t)>=2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t$
		$t\ll {\tau }_{th}$	$<x_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{3(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h+2}$ $<y_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{3(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h+2}$	$<v_{x,th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h+2}$ $<v_{y,th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h+2}$
Section 4.1 and Section 4.2	Thermal fractional Langevin equation	$t\gg {\tau }_{th}$	$<x_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{2h(2h+1){\tau }_{th}^{2h}}}{t}^{2h+2}$ $<y_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{2h(2h+1){\tau }_{th}^{2h}}}{t}^{2h+2}$	$<v_{x,th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{1,th}}{2h(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}$ $<v_{y,th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,th}}{h(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}$
		${\tau }_{th}=0$	$<x_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{3(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h+2}$ $<y_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{3(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h+2}$	$<v_{th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}$ $<v_{y,th}^2(t)>={\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h}$
		$t\ll {\tau }_{ac}$	$<x_{ac}^2\left(t\right)>=\left[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2\right]t^4/8{\tau }_{ac}$ $<y_{ac}^2(t)>=[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^4/8{\tau }_{ac}$	$<v_{x,ac}^2\left(t\right)>=1/2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{1,ac}{t}^{2h+1}$ $<v_{y,ac}^2(t)>=1/2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,ac}{t}^{2h+1}$
Section 4.3 and Section 4.4	Active fractional Langevin equation	$t\gg {\tau }_{ac}$	$<x_{ac}^2(t)>=1/3[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3$ $<y_{ac}^2(t)>=1/3[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3$	$v_{x,ac}^2(t)>=1/2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]{\gamma }_{1,ac}{t}^{2H+1}$ $<v_{y,ac}^2(t)>=1/2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]{\gamma }_{2,ac}{t}^{2h+1}$
		${\tau }_{ac}=0$	$<x_{ac}^2(t)>=1/3[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3$ $<y_{ac}^2(t)>=1/3[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3$	$<v_{x,ac}^2(t)>=2[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t$ $<v_{y,ac}^2(t)>=2[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t$

. Here, MSD denotes the mean squared displacement, and MSV represents the mean squared velocity.

In Section 3.3 and Section 3.4, the MSD and MSV in the regime $t\ll {\tau }_{\mathrm{t}\mathrm{h}}$ exhibit a ballistic motion dominated by the viscoelastic memory kernel, whereas those in the long-time limit $t\gg {\tau }_{\mathrm{t}\mathrm{h}}$ show fractional diffusive behavior. Both MSD and MSV display superdiffusion depending on the Hurst exponent $h$, as demonstrated in Section 4.1 and Section 4.2. Furthermore, in Section 4.3 and Section 4.4, the MSD and MSV in the regime $t\ll {\tau }_{\mathrm{a}\mathrm{c}}$ indicate a transient ballistic scaling, while those in the limit $t\gg {\tau }_{\mathrm{a}\mathrm{c}}$ exhibit an enhanced long-time diffusion corresponding to the instantaneous active correlation limit. In addition, symbols (nomenclature), being a method of naming and calling a specific object, are shown in Appendix C.

4.6. Comparison with Existing Active and Viscoelastic Model

To clarify the novelty and physical significance of our model, we here compare its predictions with those of the existing active Ornstein–Uhlenbeck process (AOUP) and fractional Ornstein–Uhlenbeck models.

In the AOUP framework, the mean squared displacement for a particle confined in a harmonic potential with stiffness $\beta$ is given by

$$⟨x^2(t){⟩}_{\mathrm{AOUP}}={\displaystyle \frac{D_a}{\beta }}(1-e^{-2\beta t/{\tau }_a}),$$

(255)

where $D_a$ and ${\tau }_a$ denote the active diffusivity and the noise correlation time, respectively. This leads to the short- and long-time scaling behaviors

$$⟨x^2(t){⟩}_{\mathrm{AOUP}}\sim \{\begin{array}{cc}{D}_a{t}^2,& t\ll {\tau }_a,\\ D_a/\beta ,& t\gg {\tau }_a.\end{array}$$

(256)

These indicate a crossover from ballistic ($t^2$) to confined (saturated) motion. In contrast, our fractional generalized Langevin model with a magnetic coupling term and power-law correlated active noise yields the following asymptotic form:

$$⟨x^2(t)⟩\sim \{\begin{array}{cc}{t}^{2h+1},& t\ll \tau ,\\ t^{4h-2},& t\gg \tau ,\end{array}$$

(257)

where $h$ is the Hurst exponent ($1/2<h<1$). For specific ranges of $h$, the model predicts novel scaling regimes such as

$$⟨x^2(t)⟩\propto t^3 \mathrm{and}{ t}^4,$$

(258)

which do not appear in standard AOUP or OU dynamics. These new regimes emerge from the combined effects of long-memory active noise and magnetic coupling, leading to enhanced nonequilibrium correlations between the position and velocity variables. Furthermore, the joint probability density $p(x,v,t)$ exhibits a non-Gaussian structure with non-trivial kurtosis and coupled entropy growth, distinguishing our model from Gaussian AOUP dynamics. The entropy production rate and time-dependent variance satisfy scaling relations of the form

$$S_{x,v}(t)\sim \mathrm{l}\mathrm{n}(t^{4h+4})$$

(259)

consistent with the analytical predictions in Section 4.4. Finally, we confirm that in the limiting cases $B\to 0,$ and/or $h\to 1/2,$ our model smoothly reduces to the standard AOUP and fractional Langevin results. This guarantees theoretical consistency and shows that the present formulation extends, rather than contradicts, the established active and viscoelastic stochastic frameworks. Thus, our model bridges the gap between magnetic active particle dynamics and viscoelastic active matter, providing a unified description capable of capturing multiple diffusion regimes observed in nonequilibrium colloidal systems.

5. Statistical Quantities

Now, multiplying and integrating both sides of the Fokker–Planck equations by $x^m{v}^n$, the moment equations for a charged colloid from Equations (8) and (9) are expressed as

$${\displaystyle \frac{d{\mu }_{m,n}}{dt}}=-m{\mu }_{m-1,n+1}+(q^2{B}^2+{\gamma }_1)n{\mu }_{m,n}+k_{1}n{\mu }_{m+1,n-1}+({\alpha }_1+qB{\alpha }_2)[-mn{b}_1(t){\mu }_{m-1,n-1}+n(n-1)a_1(t){\mu }_{m,n-2}],$$

(260)

$${\displaystyle \frac{d{\overline{\mu }}_{m,n}}{dt}}=-m{\overline{\mu }}_{m-1,n+1}+(q^2{B}^2+{\gamma }_2)n{\overline{\mu }}_{m,n}+k_{2}n{\overline{\mu }}_{m+1,n-1}+({\alpha }_2-qB{\alpha }_1)[-mn{b}_2(t){\overline{\mu }}_{m-1,n-1}+n(n-1)a_2(t){\overline{\mu }}_{m,n-2}].$$

(261)

Here,${\mu }_{m,n}={\int }_{-\infty }^{+\infty }dx{\int }_{-\infty }^{+\infty }d{v}_x{x}^m{v}_x^{n}p\left(x,v_x,t\right),$ ${\overline{\mu }}_{m,n}={\int }_{-\infty }^{+\infty }dy{\int }_{-\infty }^{+\infty }d{v}_y{y}^m{v}_y^{n}p\left(y,v_y,t\right).$ The moment equations from Equations (75) and (76) have

$${\displaystyle \frac{d{\mu }_{m,n}^{th}}{dt}}=-m{\mu }_{m-1,n+1}^{th}+n{\gamma }_{1,th}{D}_{2-2h}{\mu }_{m+1,n-1}^{th}-[{\alpha }_1+qB{\alpha }_2][mn{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\mu }_{m-1,n-1}^{th}+n(n-1){\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\mu }_{m,n-2}^{th}],$$

(262)

$${\displaystyle \frac{d{\overline{\mu }}_{m,n}^{th}}{dt}}=-m{\overline{\mu }}_{m-1,n+1}^{th}+n{\gamma }_{2,th}{D}_{2-2h}{\overline{\mu }}_{m+1,n-1}^{th}-[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][mn{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\overline{\mu }}_{m-1,n-1}^{th}+n(n-1){\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\overline{\mu }}_{m,n-2}^{th}],$$

(263)

and

$${\displaystyle \frac{d{\mu }_{m,n}^{ac}}{dt}}=-m{\mu }_{m-1,n+1}^{ac}+n{\gamma }_{1,th}{D}_{2-2h}{\mu }_{m+1,n-1}^{ac}+({\overline{\alpha }}_1+qB{\overline{\alpha }}_2)[-mnb(t){\mu }_{m-1,n-1}^{ac}+n(n-1)a(t){\mu }_{m,n-2}^{ac}],$$

(264)

$${\displaystyle \frac{d{\overline{\mu }}_{m,n}^{ac}}{dt}}=-m{\overline{\mu }}_{m-1,n+1}^{ac}+n{\gamma }_{2,th}{D}_{2-2h}]{\overline{\mu }}_{m+1,n-1}^{ac}+({\overline{\alpha }}_2-qB{\overline{\alpha }}_1)[-mnb(t){\overline{\mu }}_{m-1,n-1}^{ac}+n(n-1)a(t){\overline{\mu }}_{m,n-2}^{ac}].$$

(265)

Here, the moments are simply given by ${\mu }_{m,n}^i={\int }_{-\infty }^{+\infty }d{x}_i{\int }_{-\infty }^{+\infty }d{v}_{x,i}{x}_i^m{v}_{x,i}^{n}p\left(x_i,v_{x,i},t\right),$ ${\overline{\mu }}_{m,n}^i={\int }_{-\infty }^{+\infty }d{y}_i{\int }_{-\infty }^{+\infty }d{v}_{y,i}{y}_i^m{v}_{y,i}^{n}p(y_i,v_{y,i},t)$ for $i=th,ac$.The moment equations for an active particle with optical trap from Equations (168) and (169) are, respectively, derived as

$${\displaystyle \frac{d{\mu }_{m,n}^{th}}{dt}}=-m{\mu }_{m-1,n+1}^{th}++n[k_1+{\gamma }_{1,th}{D}_{2-2h}]{\mu }_{m+1,n-1}^{th}-[{\alpha }_1+qB{\alpha }_2][mn{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\mu }_{m-1,n-1}^{th}+n(n-1){\alpha }_1{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\mu }_{m,n-2}^{th}],$$

(266)

$${\displaystyle \frac{d{\overline{\mu }}_{m,n}^{th}}{dt}}=-m{\overline{\mu }}_{m-1,n+1}^{th}+n[k_2+{\gamma }_{2,th}{D}_{2-2h}]{\overline{\mu }}_{m+1,n-1}^{th}-[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1][mn{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}} {\overline{\mu }}_{m-1,n-1}^{th}+n(n-1){\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\overline{\mu }}_{m,n-2}^{th}],$$

(267)

and

$${\displaystyle \frac{d{\mu }_{m,n}^{ac}}{dt}}=-m{\mu }_{m-1,n+1}^{ac}++n[k_1+{\gamma }_{1,th}{D}_{2-2h}]{\mu }_{m+1,n-1}^{ac}+({\overline{\alpha }}_1+qB{\overline{\alpha }}_2)[-mnb(t){\mu }_{m-1,n-1}^{th}+n(n-1)a(t){\mu }_{m,n-2}^{ac}],$$

(268)

$${\displaystyle \frac{d{\overline{\mu }}_{m,n}^{ac}}{dt}}=-m{\overline{\mu }}_{m-1,n+1}^{ac}++n[k_2+{\gamma }_{2,th}{D}_{2-2h}]{\overline{\mu }}_{m+1,n-1}^{ac}+({\overline{\alpha }}_2-qB{\overline{\alpha }}_1)[-mnb(t){\overline{\mu }}_{m-1,n-1}^{ac}+n(n-1)a(t){\overline{\mu }}_{m,n-2}^{ac}],$$

(269)

Next, the entropy, the combined entropy, the non-Gaussian parameter, and the correlation coefficient are numerically calculated. The entropies $S(x_i,t)$ and $S(v_i,t)$ are, respectively, calculated as

$$S(x_i,t)=-p(x_i,t)\mathit{ln}p(x_i,t), S(v_i,t)=-p(v_i,t)\mathit{ln}p(v_i,t).$$

(270)

The combined entropy is defined by

$$S(x_i,v_i,t)\equiv -\int dx\int dvp(x_i,t)p(v_i,t)\mathit{ln}p(x_i,t)p(v_i,t).$$

(271)

Entropy refers to a numerical representation of the reliability or quantity of information that has a probability distribution; in a probability distribution, as the probability of a particular value increases and the probability of the remaining value decreases, the entropy decreases.

The non-Gaussian parameters for displacement and velocity are, respectively, given by

$$K_{x_i}=<x_i^4>/3<x_i^2{>}^2-1, K_{v_i}=<v_i^4>/3<v_i^2{>}^2-1.$$

(272)

Non-Gaussian parameter refers to the degree to which there is a tail in the probability distribution of a real-valued variable, and provides insight into certain properties of probability distributions in probability theory and statistics. The correlation coefficient is defined as

$${{\rho }_{x_i,v}}_i=<(x_i-<x_i>)<(v_i-<v_i>)/{\sigma }_{x_i}{{\sigma }_v}_i.$$

(273)

As is well known, the correlation coefficient refers to a quantity describing the strength and direction of the relationship between two variables $x_i(t)$ and $v_i(t)$. Here, we assume that a passive particle is initially at $x_i=x_{i0}$ and at $v_i=v_{i0}$ for $i=th,ac$. The parameters ${\sigma }_{x_i}$ and ${\sigma }_{v_i}$ denote the root-mean-squared displacement and the root-mean-squared velocity of the joint probability density, respectively.

Table 2. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density in x-component with the harmonic force and the correlated Gaussian force ${\eta }_x(t)+qB{\eta }_y(t)$ in the three-time domains. Here, we assume that a passive particle is initially at $x=x_0$ and at $v_x=v_{x0}$.

Time	$\mathit{x},{\mathit{v}}_{\mathit{x}}$	${\mathit{K}}_{\mathit{x}},{\mathit{K}}_{{\mathit{v}}_{\mathit{x}}}$	${\mathit{\rho }}_{\mathit{x},{\mathit{v}}_{\mathit{x}}}$	${\mathit{\mu }}_{\mathrm{2,2}}$	$\mathit{S}(\mathit{x},\mathit{t})$, $\mathit{S}({\mathit{v}}_{\mathit{x}},\mathit{t})$	$\mathit{S}(\mathit{x},{\mathit{v}}_{\mathit{x}},\mathit{t})$
$t<<\tau$	$x$	${\displaystyle \frac{k_1^2{\tau }^2{x}_0^4}{[{\alpha }_1+qB{\alpha }_2{]}^2}}{t}^{-6}+{\displaystyle \frac{k_1\tau x_0^2}{[{\alpha }_1+qB{\alpha }_2]}}{t}^{-3}$	${\displaystyle \frac{[q^2{B}^2+{\gamma }_1{]}^{-1}{x}_0{v}_{x0}}{[{\alpha }_1+qB{\alpha }_2]k_1^{-1/2}{\tau }^{-1/2}}}{t}^{-3}$	${\displaystyle \frac{[{\alpha }_1+qB{\alpha }_2{]}^2}{k_1{\tau }^2}}{t}^5$	$\mathit{ln}{\displaystyle \frac{[{\alpha }_1+qB{\alpha }_2]}{k_1\tau }}{t}^3$	$\mathit{ln}{\displaystyle \frac{[{\alpha }_1+qB{\alpha }_2{]}^2}{[q^2{B}^2+{\gamma }_1{]}^{-2}{k}_1\tau }}{t}^6$
	$v_x$	${\displaystyle \frac{[q^2{B}^2+{\gamma }_1{]}^{-4}{v}_{x0}^4}{[{\alpha }_1+qB{\alpha }_2{]}^2}}{t}^{-6}+{\displaystyle \frac{[q^2{B}^2+{\gamma }_1{]}^{-2}{v}_{x0}^2}{[{\alpha }_1+qB{\alpha }_2]}}{t}^{-3}$			$\mathit{ln}{\displaystyle \frac{[{\alpha }_1+qB{\alpha }_2]}{[q^2{B}^2+{\gamma }_1{]}^{-2}}}{t}^3$
$t>>\tau$	$x$	${\displaystyle \frac{x_0^4}{[{\alpha }_1+qB{\alpha }_2{]}^2}}{t}^{-8}+{\displaystyle \frac{x_0^2}{[{\alpha }_1+qB{\alpha }_2]}}{t}^{-4}$	${\displaystyle \frac{[q^2{B}^2+{\gamma }_1{]}^{-1}{x}_0{v}_{x0}}{[{\alpha }_1+qB{\alpha }_2]}}{t}^{-7/2}$	${\displaystyle \frac{[{\alpha }_1+qB{\alpha }_2{]}^2}{\tau }}{t}^6$	$\mathit{ln}[{\alpha }_1+qB{\alpha }_2]t^4$	$\mathit{ln}{\displaystyle \frac{[{\alpha }_1+qB{\alpha }_2{]}^2}{[q^2{B}^2+{\gamma }_1{]}^{-2}}}{t}^7$
	$v_x$	${\displaystyle \frac{[q^2{B}^2+{\gamma }_1{]}^{-4}{v}_{x0}^4}{[{\alpha }_1+qB{\alpha }_2{]}^2}}{t}^{-6}+{\displaystyle \frac{[q^2{B}^2+{\gamma }_1{]}^{-2}{v}_{x0}^2}{[{\alpha }_1+qB{\alpha }_2]}}{t}^{-3}$			$\mathit{ln}{\displaystyle \frac{[{\alpha }_1+qB{\alpha }_2]}{[q^2{B}^2+{\gamma }_1{]}^{-2}}}{t}^3$
$\tau =0$	$x$	${\displaystyle \frac{[q^2{B}^2+{\gamma }_1{]}^{-4}{x}_0^4}{[{\alpha }_1+qB{\alpha }_2{]}^2}}{t}^{-8}+{\displaystyle \frac{[q^2{B}^2+{\gamma }_1{]}^{-2}{x}_0^2}{[{\alpha }_1+qB{\alpha }_2]}}{t}^{-4}$	${\displaystyle \frac{[q^2{B}^2+{\gamma }_1{]}^{-1/2}{x}_0{v}_{x0}}{[{\alpha }_1+qB{\alpha }_2]}}{t}^{-5/2}$	${\displaystyle \frac{[{\alpha }_1+qB{\alpha }_2{]}^2}{[q^2{B}^2+{\gamma }_1{]}^{-1}\tau }}{t}^6$	$\mathit{ln}{\displaystyle \frac{[{\alpha }_1+qB{\alpha }_2]}{[q^2{B}^2+{\gamma }_1{]}^{-1}}}{t}^4$	$\mathit{ln}{\displaystyle \frac{[{\alpha }_1+qB{\alpha }_2{]}^2}{[q^2{B}^2+{\gamma }_1{]}^{-1}}}{t}^5$
	$v_x$	${\displaystyle \frac{v_{x0}^4}{[{\alpha }_1+qB{\alpha }_2{]}^2}}{t}^{-2}+{\displaystyle \frac{v_{x0}^2}{[{\alpha }_1+qB{\alpha }_2]}}{t}^{-1}$			$\mathit{ln}[{\alpha }_1+qB{\alpha }_2]t$

and

Table 3. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density in y-component with the harmonic force and the correlated Gaussian force ${\eta }_y(t)-qB{\eta }_x(t)$ in the three-time domains.

Time	$\mathit{y},{\mathit{v}}_{\mathit{y}}$	${\mathit{K}}_{\mathit{y}},{\mathit{K}}_{{\mathit{v}}_{\mathit{y}}}$	${\mathit{\rho }}_{\mathit{y},{\mathit{v}}_{\mathit{y}}}$	${\mathit{\mu }}_{\mathrm{2,2}}$	$\mathit{S}(\mathit{y},\mathit{t})$, $\mathit{S}({\mathit{v}}_{\mathit{y}},\mathit{t})$	$\mathit{S}(\mathit{y},{\mathit{v}}_{\mathit{y}},\mathit{t})$
$t<<\tau$	$y$	${\displaystyle \frac{k_2^2{\tau }^2{y}_0^4}{[{\alpha }_2-qB{\alpha }_1{]}^2}}{t}^{-6}+{\displaystyle \frac{k_2\tau y_0^2}{[{\alpha }_2-qB{\alpha }_1]}}{t}^{-3}$	${\displaystyle \frac{[q^2{B}^2+{\gamma }_2{]}^{-1}{y}_0{v}_{y0}}{[{\alpha }_2-qB{\alpha }_1]k_2^{-1/2}{\tau }^{-1/2}}}{t}^{-3}$	${\displaystyle \frac{[{\alpha }_2-qB{\alpha }_1{]}^2}{k_2{\tau }^2}}{t}^5$	$\mathit{ln}{\displaystyle \frac{[{\alpha }_2-qB{\alpha }_1]}{k_2\tau }}{t}^3$	$\mathit{ln}{\displaystyle \frac{[{\alpha }_2-qB{\alpha }_1{]}^2}{[q^2{B}^2+{\gamma }_2{]}^{-2}{k}_2\tau }}{t}^6$
	$v_y$	${\displaystyle \frac{[q^2{B}^2+{\gamma }_2{]}^{-4}{v}_{y0}^4}{[{\alpha }_2-qB{\alpha }_1{]}^2}}{t}^{-6}+{\displaystyle \frac{[q^2{B}^2+{\gamma }_2{]}^{-2}{v}_{y0}^2}{[{\alpha }_2-qB{\alpha }_1]}}{t}^{-3}$			$\mathit{ln}{\displaystyle \frac{[{\alpha }_2-qB{\alpha }_1]}{[q^2{B}^2+{\gamma }_2{]}^{-2}}}{t}^3$
$t>>\tau$	$y$	${\displaystyle \frac{y_0^4}{[{\alpha }_2-qB{\alpha }_1{]}^2}}{t}^{-8}+{\displaystyle \frac{y_0^2}{[{\alpha }_2-qB{\alpha }_1]}}{t}^{-4}$	${\displaystyle \frac{[q^2{B}^2+{\gamma }_2{]}^{-1}{y}_0{v}_{y0}}{[{\alpha }_2-qB{\alpha }_1]}}{t}^{-7/2}$	${\displaystyle \frac{[{\alpha }_2-qB{\alpha }_1{]}^2}{\tau }}{t}^6$	$\mathit{ln}[{\alpha }_2-qB{\alpha }_1]t^4$	$\mathit{ln}{\displaystyle \frac{[{\alpha }_2-qB{\alpha }_1{]}^2}{[q^2{B}^2+{\gamma }_2{]}^{-2}}}{t}^7$
	$v_y$	${\displaystyle \frac{[q^2{B}^2+{\gamma }_2{]}^{-4}{v}_{y0}^4}{[{\alpha }_2-qB{\alpha }_1{]}^2}}{t}^{-6}+{\displaystyle \frac{[q^2{B}^2+{\gamma }_2{]}^{-2}{v}_{y0}^2}{[{\alpha }_2-qB{\alpha }_1]}}{t}^{-3}$			$\mathit{ln}{\displaystyle \frac{[{\alpha }_2-qB{\alpha }_1]}{[q^2{B}^2+{\gamma }_2{]}^{-2}}}{t}^3$
$\tau =0$	$y$	${\displaystyle \frac{[q^2{B}^2+{\gamma }_2{]}^{-4}{y}_0^4}{[{\alpha }_2-qB{\alpha }_1{]}^2}}{t}^{-8}+{\displaystyle \frac{[q^2{B}^2+{\gamma }_2{]}^{-2}{y}_0^2}{[{\alpha }_2-qB{\alpha }_1]}}{t}^{-4}$	${\displaystyle \frac{[q^2{B}^2+{\gamma }_2{]}^{-1/2}{y}_0{v}_{y0}}{[{\alpha }_2-qB{\alpha }_1]}}{t}^{-5/2}$	${\displaystyle \frac{[{\alpha }_2-qB{\alpha }_1{]}^2}{[q^2{B}^2+{\gamma }_2{]}^{-1}\tau }}{t}^6$	$\mathit{ln}{\displaystyle \frac{[{\alpha }_2-qB{\alpha }_1]}{[q^2{B}^2+{\gamma }_2{]}^{-1}}}{t}^4$	$\mathit{ln}{\displaystyle \frac{[{\alpha }_2-qB{\alpha }_1{]}^2}{[q^2{B}^2+{\gamma }_2{]}^{-1}}}{t}^5$
	$v_y$	${\displaystyle \frac{v_{y0}^4}{[{\alpha }_2-qB{\alpha }_1{]}^2}}{t}^{-2}+{\displaystyle \frac{v_{y0}^2}{[{\alpha }_2-qB{\alpha }_1]}}{t}^{-1}$			$\mathit{ln}[{\alpha }_2-qB{\alpha }_1]t$

, respectively, summarized the values of the entropy, the non-Gaussian parameter, and the correlation coefficient for the joint probability density with a harmonic trap and a viscous force in x- and y-components in the limits of $t<<\tau$, $t<<\tau$, and for $\tau =0$. We calculate the statistical quantities such as the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for a charged colloid with thermal equilibrium noise ${\zeta }_{th}(t)$ and an active noise ${\zeta }_{ac}(t)$ in the three-time domains in

Table 4. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density in x-component with ${\zeta }_{x,th}(t)$ in the three-time domains. Here, we assume that a passive particle is initially at $x_{th}=x_{th0}$ and at $v_{x,th}=v_{x,th0}$.

Time	${\mathit{x}}_{\mathit{t}\mathit{h}},$ ${\mathit{v}}_{\mathit{x},\mathit{t}\mathit{h}}$	${\mathit{K}}_{{\mathit{x}}_{\mathit{t}\mathit{h}}},{\mathit{K}}_{{\mathit{v}}_{\mathit{x},\mathit{t}\mathit{h}}}$	${\mathit{\rho }}_{{\mathit{x}}_{\mathit{t}\mathit{h}},{\mathit{v}}_{\mathit{x},\mathit{t}\mathit{h}}}$	$\left|{\mathit{\mu }}_{\mathrm{2,2}}\right|$	$\mathit{S}({\mathit{x}}_{\mathit{t}\mathit{h}},\mathit{t})$, $\mathit{S}({\mathit{v}}_{\mathit{x},\mathit{t}\mathit{h}},\mathit{t})$	$\mathit{S}({\mathit{x}}_{\mathit{t}\mathit{h}},{\mathit{v}}_{\mathit{x},\mathit{t}\mathit{h}},\mathit{t})$
$t<<{\tau }_{th}$	$x_{th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{x}_{th0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{x}_{th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-1}$	${\displaystyle \frac{{\tau }_{th}^{2h-1}{x}_{th0}{v}_{x,th0}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-1/2}$	${\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}^{4h-2}}}{t}^{4h+1}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{th}^{2h-1}}}{t}^{2h+1}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}^{4h-2}}}{t}^{4h+1}$
	$v_{x,th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{v}_{x,th0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{v}_{x,th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h}$			$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{th}^{2h-1}}}{t}^{2h}$
$t>>{\tau }_{th}$	$x_{th}$	${\displaystyle \frac{{\tau }_{th}^{4h}{x}_{th0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\tau }_{th}^{2h}{x}_{th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-1}$	${\displaystyle \frac{{\tau }_{th}^{2h-1/2}{x}_{th0}{v}_{x,th0}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-1/2}$	${\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}^{4h-1}}}{t}^{4h+1}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{th}^{2h}}}{t}^{2h+1}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}^{4h-1}}}{t}^{4h+1}$
	$v_{x,th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{v}_{x,th0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{v}_{x,th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h}$			$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{th}^{2h-1}}}{t}^{2h}$
${\tau }_{th}=0$	$x_{th}$	${\displaystyle \frac{{\tau }_{th}^{4h}{x}_{th0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\tau }_{th}^{2h}{x}_{th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-1}$	${\displaystyle \frac{{\tau }_{th}^{2h-1/2}{x}_{th0}{v}_{x,th0}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-1/2}$	${\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}^{4h-1}}}{t}^{4h+1}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{th}^{2h}}}{t}^{2h+1}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}^{4h-1}}}{t}^{4h+1}$
	$v_{x,th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{v}_{x,th0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{v}_{x,th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h}$			$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{th}^{2h-1}}}{t}^{2h}$

,

Table 5. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density in y-component with ${\zeta }_{y,th}(t)$ in the three-time domains. Here, we assume that a passive particle is initially at ${ y}_{th}=y_{th0}$ and at $v_{y,th}=v_{y,th0}$.

Time	${\mathit{y}}_{\mathit{t}\mathit{h}},$ ${\mathit{v}}_{\mathit{y},\mathit{t}\mathit{h}}$	${\mathit{K}}_{{\mathit{y}}_{\mathit{t}\mathit{h}}},{\mathit{K}}_{{\mathit{v}}_{\mathit{y},\mathit{t}\mathit{h}}}$	${\mathit{\rho }}_{{\mathit{y}}_{\mathit{t}\mathit{h}},{\mathit{v}}_{\mathit{y},\mathit{t}\mathit{h}}}$	$\left|{\mathit{\mu }}_{\mathrm{2,2}}\right|$	$\mathit{S}({\mathit{y}}_{\mathit{t}\mathit{h}},\mathit{t})$, $\mathit{S}({\mathit{v}}_{\mathit{y},\mathit{t}\mathit{h}},\mathit{t})$	$\mathit{S}({\mathit{y}}_{\mathit{t}\mathit{h}},{\mathit{v}}_{\mathit{y},\mathit{t}\mathit{h}},\mathit{t})$
$t<<{\tau }_{th}$	$y_{th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{y}_{th0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{y}_{th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-1}$	${\displaystyle \frac{{\tau }_{th}^{2h-1}{y}_{th0}{v}_{y,th0}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-1/2}$	${\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}^{4h-2}}}{t}^{4h}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{th}^{2h-1}}}{t}^{2h+1}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}^{4h-1}}}{t}^{4h+1}$
	$v_{y,th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{v}_{y,th0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{v}_{y,th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h}$			$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{th}^{2h-1}}}{t}^{2h}$
$t>>{\tau }_{th}$	$y_{th}$	${\displaystyle \frac{{\tau }_{th}^{4h}{y}_{th0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\tau }_{th}^{2h}{y}_{th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-1}$	${\displaystyle \frac{{\tau }_{th}^{2h-1/2}{y}_{th0}{v}_{y,th0}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-1/2}$	${\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}^{4h-1}}}{t}^{4h}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{th}^{2h}}}{t}^{2h+1}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}^{4h-1}}}{t}^{4h+1}$
	$v_{y,th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{v}_{y,th0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{v}_{y,th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h}$			$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{th}^{2h-1}}}{t}^{2h}$
${\tau }_{th}=0$	$y_{th}$	${\displaystyle \frac{{\tau }_{th}^{4h}{y}_{th0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\tau }_{th}^{2h}{y}_{th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-1}$	${\displaystyle \frac{{\tau }_{th}^{2h-1/2}{y}_{th0}{v}_{y,th0}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-1/2}$	${\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}^{4h-1}}}{t}^{4h}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{th}^{2h}}}{t}^{2h+1}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}^{4h-2}}}{t}^{4h+2}$
	$v_{y,th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{v}_{y,th0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{v}_{y,th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h}$			$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{th}^{2h-1}}}{t}^{2h}$

,

Table 6. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density in x-component with ${\zeta }_{x,ac}(t)$ in the three-time domains. Here, we assume that a passive particle is initially at $x_{ac}=x_{ac0}$ and at $v_{x,ac}=v_{x,ac0}$.

Time	${\mathit{x}}_{\mathit{a}\mathit{c}},$ ${\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}}$	${\mathit{K}}_{{\mathit{x}}_{\mathit{a}\mathit{c}}},{\mathit{K}}_{{\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}}}$	${\mathit{\rho }}_{{\mathit{x}}_{\mathit{a}\mathit{c}},{\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}}}$	${\mathit{\mu }}_{\mathrm{2,2}}$	$\mathit{S}({\mathit{x}}_{\mathit{a}\mathit{c}},\mathit{t})$, $\mathit{S}({\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}},\mathit{t})$	$\mathit{S}({\mathit{x}}_{\mathit{a}\mathit{c}},{\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}},\mathit{t})$
$t<<{\tau }_{ac}$	$x_{ac}$	${\displaystyle \frac{{\tau }_{ac}^2{x}_{ac0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-8}+{\displaystyle \frac{{\tau }_{ac}{x}_{ac0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-4}$	${\displaystyle \frac{{\gamma }_{1,ac}^{-1/2}{\tau }_{ac}^{1/2}{x}_{ac0}{v}_{x,ac0}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-h-5/2}$	${\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{ac}^2}}{t}^6$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{ac}}}{t}^4$	$\mathit{ln}{\displaystyle \frac{{\gamma }_{1,ac}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}}}{t}^{2h+5}$
	$v_{x,ac}$	${\displaystyle \frac{{\gamma }_{1,ac}^2{\tau }_{ac}^{4h-2}{v}_{x,ac0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\gamma }_{1,ac}{\tau }_{th}^{2h-1}{v}_{x,th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-1}$			$\mathit{ln}{\gamma }_{1,ac}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+1}$
$t>>{\tau }_{ac}$	$x_{ac}$	${\displaystyle \frac{{\tau }_{ac}^2{x}_{ac0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-6}+{\displaystyle \frac{{\tau }_{ac}{x}_{ac0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-3}$	${\displaystyle \frac{{\gamma }_{1,ac}^{-1/2}{\tau }_{ac}^{1/2}{x}_{ac0}{v}_{x,ac0}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-h-2}$	${\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{ac}}}{t}^5$	$\mathit{ln}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\gamma }_{1,th}^{-1}}}{t}^{2h+4}$
	$v_{x,ac}$	${\displaystyle \frac{{\gamma }_{1,ac}^2{\tau }_{ac}^{4h-2}{v}_{x,ac0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\gamma }_{1,ac}{\tau }_{ac}^{2h-1}{v}_{x,ac0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-1}$			$\mathit{ln}{\gamma }_{1,ac}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+1}$
${\tau }_{ac}=0$	$x_{ac}$	${\displaystyle \frac{{\tau }_{ac}^2{x}_{ac0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-8}+{\displaystyle \frac{{\tau }_{ac}{x}_{ac0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-4}$	${\displaystyle \frac{x_{ac0}{v}_{x,ac0}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2}$	${\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{ac}}}{t}^5$	$\mathit{ln}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3$	$\mathit{ln}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2{t}^4$
	$v_{x,ac}$	${\displaystyle \frac{v_{x,ac0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-2}+{\displaystyle \frac{v_{x,ac0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-1}$			$\mathit{ln}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t$

and

Table 7. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density in y-component with ${\zeta }_{y,ac}(t)$ in the three-time domains. Here, we assume that a passive particle is initially at $y_{ac}=y_{ac0}$ and at $v_{y,ac}=v_{y,ac0}$.

Time	${\mathit{x}}_{\mathit{a}\mathit{c}},$ ${\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}}$	${\mathit{K}}_{{\mathit{x}}_{\mathit{a}\mathit{c}}},{\mathit{K}}_{{\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}}}$	${\mathit{\rho }}_{{\mathit{x}}_{\mathit{a}\mathit{c}},{\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}}}$	${\mathit{\mu }}_{\mathrm{2,2}}$	$\mathit{S}({\mathit{x}}_{\mathit{a}\mathit{c}},\mathit{t})$, $\mathit{S}({\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}},\mathit{t})$	$\mathit{S}({\mathit{x}}_{\mathit{a}\mathit{c}},{\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}},\mathit{t})$
$t<<{\tau }_{ac}$	$x_{ac}$	${\displaystyle \frac{{\tau }_{ac}^2{x}_{ac0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-8}+{\displaystyle \frac{{\tau }_{ac}{x}_{ac0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-4}$	${\displaystyle \frac{{\gamma }_{1,ac}^{-1/2}{\tau }_{ac}^{1/2}{x}_{ac0}{v}_{x,ac0}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-h-5/2}$	${\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{ac}^2}}{t}^6$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{ac}}}{t}^4$	$\mathit{ln}{\displaystyle \frac{{\gamma }_{1,ac}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}}}{t}^{2h+5}$
	$v_{x,ac}$	${\displaystyle \frac{{\gamma }_{1,ac}^2{\tau }_{ac}^{4h-2}{v}_{x,ac0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\gamma }_{1,ac}{\tau }_{th}^{2h-1}{v}_{x,th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-1}$			$\mathit{ln}{\gamma }_{1,ac}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+1}$
$t>>{\tau }_{ac}$	$x_{ac}$	${\displaystyle \frac{{\tau }_{ac}^2{x}_{ac0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-6}+{\displaystyle \frac{{\tau }_{ac}{x}_{ac0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-3}$	${\displaystyle \frac{{\gamma }_{1,ac}^{-1/2}{\tau }_{ac}^{1/2}{x}_{ac0}{v}_{x,ac0}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-h-2}$	${\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{ac}}}{t}^5$	$\mathit{ln}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\gamma }_{1,th}^{-1}}}{t}^{2h+4}$
	$v_{x,ac}$	${\displaystyle \frac{{\gamma }_{1,ac}^2{\tau }_{ac}^{4h-2}{v}_{x,ac0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\gamma }_{1,ac}{\tau }_{ac}^{2h-1}{v}_{x,ac0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-1}$			$\mathit{ln}{\gamma }_{1,ac}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+1}$
${\tau }_{ac}=0$	$x_{ac}$	${\displaystyle \frac{{\tau }_{ac}^2{x}_{ac0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-8}+{\displaystyle \frac{{\tau }_{ac}{x}_{ac0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-4}$	${\displaystyle \frac{x_{ac0}{v}_{x,ac0}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2}$	${\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{ac}}}{t}^5$	$\mathit{ln}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3$	$\mathit{ln}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2{t}^4$
	$v_{x,ac}$	${\displaystyle \frac{v_{x,ac0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-2}+{\displaystyle \frac{v_{x,ac0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-1}$			$\mathit{ln}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t$

.

Table 8. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density in x-component with an optical trap force and ${\zeta }_{x,th}(t)$ in the three-time domains. Here, we assume that a passive particle is initially at $x_{th}=x_{th0}$ and at $v_{x,th}=v_{x,th0}$.

Time	${\mathit{x}}_{\mathit{t}\mathit{h}},$ ${\mathit{v}}_{\mathit{x},\mathit{t}\mathit{h}}$	${\mathit{K}}_{{\mathit{x}}_{\mathit{t}\mathit{h}}},{\mathit{K}}_{{\mathit{v}}_{\mathit{x},\mathit{t}\mathit{h}}}$	${\mathit{\rho }}_{{\mathit{x}}_{\mathit{t}\mathit{h}},{\mathit{v}}_{\mathit{x},\mathit{t}\mathit{h}}}$	$\left|{\mathit{\mu }}_{\mathrm{2,2}}\right|$	$\mathit{S}({\mathit{x}}_{\mathit{t}\mathit{h}},\mathit{t})$, $\mathit{S}({\mathit{v}}_{\mathit{x},\mathit{t}\mathit{h}},\mathit{t})$	$\mathit{S}({\mathit{x}}_{\mathit{t}\mathit{h}},{\mathit{v}}_{\mathit{x},\mathit{t}\mathit{h}},\mathit{t})$
$t<<{\tau }_{th}$	$x_{th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{x}_{th0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h-4}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{x}_{th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-2}$	${\displaystyle \frac{{\tau }_{th}^{2h-1}{x}_{th0}{v}_{x,th0}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-2}$	${\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}^{4h-2}}}{t}^{4h+2}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{th}^{2h-1}}}{t}^{2h+2}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}^{4h-2}}}{t}^{4h+4}$
	$v_{x,th}$	${\displaystyle \frac{v_{x,th0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h-4}+{\displaystyle \frac{v_{x,th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-2}$			$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{th}^{2h-1}}}{t}^{2h+2}$
$t>>{\tau }_{th}$	$x_{th}$	${\displaystyle \frac{{\tau }_{th}^{4h}{x}_{th0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h-4}+{\displaystyle \frac{{\tau }_{th}^{2h}{x}_{th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-2}$	${\displaystyle \frac{{\tau }_{th}^{2h-1/2}{x}_{th0}{v}_{x,th0}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-1}$	${\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}^{4h-1}}}{t}^{4h+2}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{th}^{2h}}}{t}^{2h+2}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}^{4h-1}}}{t}^{4h+2}$
	$v_{x,th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{v}_{x,th0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{v}_{x,th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h}$			$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{th}^{2h-1}}}{t}^{2h}$
${\tau }_{th}=0$	$x_{th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{x}_{th0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h-4}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{x}_{th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-2}$	${\displaystyle \frac{{\tau }_{th}^{2h-1}{x}_{th0}{v}_{x,th0}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-1}$	${\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}^{4h-2}}}{t}^{4h+2}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{th}^{2h-1}}}{t}^{2h+2}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}^{4h-2}}}{t}^{4h+2}$
	$v_{x,th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{v}_{x,th0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{v}_{x,th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h}$			$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{th}^{2h-1}}}{t}^{2h}$

,

Table 9. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density in y-component with an optical trap force and ${\zeta }_{y,th}(t)$ in the three-time domains. Here, we assume that a passive particle is initially at $y_{th}=y_{th0}$ and at $v_{y,th}=v_{y,th0}$.

Time	${\mathit{y}}_{\mathit{t}\mathit{h}},$ ${\mathit{v}}_{\mathit{y},\mathit{t}\mathit{h}}$	${\mathit{K}}_{{\mathit{y}}_{\mathit{t}\mathit{h}}},{\mathit{K}}_{{\mathit{v}}_{\mathit{y},\mathit{t}\mathit{h}}}$	${\mathit{\rho }}_{{\mathit{y}}_{\mathit{t}\mathit{h}},{\mathit{v}}_{\mathit{y},\mathit{t}\mathit{h}}}$	$\left|{\mathit{\mu }}_{\mathrm{2,2}}\right|$	$\mathit{S}({\mathit{y}}_{\mathit{t}\mathit{h}},\mathit{t})$, $\mathit{S}({\mathit{v}}_{\mathit{y},\mathit{t}\mathit{h}},\mathit{t})$	$\mathit{S}({\mathit{y}}_{\mathit{t}\mathit{h}},{\mathit{v}}_{\mathit{y},\mathit{t}\mathit{h}},\mathit{t})$
$t<<{\tau }_{th}$	$y_{th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{y}_{th0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h-4}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{y}_{th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-2}$	${\displaystyle \frac{{\tau }_{th}^{2h-1}{y}_{th0}{v}_{y,th0}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-2}$	${\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}^{4h-2}}}{t}^{4h+2}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{th}^{2h-1}}}{t}^{2h+2}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}^{4h-2}}}{t}^{4h+4}$
	$v_{y,th}$	${\displaystyle \frac{v_{y,th0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h-4}+{\displaystyle \frac{v_{y,th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-2}$			$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{th}^{2h-1}}}{t}^{2h+2}$
$t>>{\tau }_{th}$	$y_{th}$	${\displaystyle \frac{{\tau }_{th}^{4h}{y}_{th0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h-4}+{\displaystyle \frac{{\tau }_{th}^{2h}{y}_{th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-2}$	${\displaystyle \frac{{\tau }_{th}^{2h-1/2}{y}_{th0}{v}_{y,th0}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-1}$	${\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}^{4h-1}}}{t}^{4h+2}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{th}^{2h}}}{t}^{2h+2}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}^{4h-1}}}{t}^{4h+2}$
	$v_{y,th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{v}_{y,th0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{v}_{y,th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h}$			$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{th}^{2h-1}}}{t}^{2h}$
${\tau }_{th}=0$	$y_{th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{y}_{th0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h-4}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{y}_{th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-2}$	${\displaystyle \frac{{\tau }_{th}^{2h-1}{y}_{th0}{v}_{y,th0}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-1}$	${\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}^{4h-2}}}{t}^{4h+2}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{th}^{2h-1}}}{t}^{2h+2}$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}^{4h-2}}}{t}^{4h+2}$
	$v_{y,th}$	${\displaystyle \frac{{\tau }_{th}^{4h-2}{v}_{y,th0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h}+{\displaystyle \frac{{\tau }_{th}^{2h-1}{v}_{y,th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h}$			$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{th}^{2h-1}}}{t}^{2h}$

,

Table 10. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density in x-component with an optical trap force and ${\zeta }_{x,ac}(t)$ in the three-time domains. We assume that a passive particle is initially at $x_{ac}=x_{ac0}$ and at $v_{x,ac}=v_{x,ac0}$.

Time	${\mathit{x}}_{\mathit{a}\mathit{c}},$ ${\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}}$	${\mathit{K}}_{{\mathit{x}}_{\mathit{a}\mathit{c}}},{\mathit{K}}_{{\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}}}$	${\mathit{\rho }}_{{\mathit{x}}_{\mathit{a}\mathit{c}},{\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}}}$	${\mathit{\mu }}_{\mathrm{2,2}}$	$\mathit{S}({\mathit{x}}_{\mathit{a}\mathit{c}},\mathit{t})$, $\mathit{S}({\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}},\mathit{t})$	$\mathit{S}({\mathit{x}}_{\mathit{a}\mathit{c}},{\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}},\mathit{t})$
$t<<{\tau }_{ac}$	$x_{ac}$	${\displaystyle \frac{{\tau }_{ac}^2{x}_{ac0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-8}+{\displaystyle \frac{{\tau }_{ac}{x}_{ac0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-4}$	${\displaystyle \frac{{\gamma }_{1,ac}^{-1/2}{\tau }_{ac}^{1/2}{x}_{th0}{v}_{x,ac0}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-h-5/2}$	${\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{ac}^2}}{t}^6$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}{{\tau }_{ac}}}{t}^4$	$\mathit{ln}{\displaystyle \frac{{\gamma }_{1,ac}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{th}}}{t}^{2h+5}$
	$v_{x,ac}$	${\displaystyle \frac{{\gamma }_{1,ac}^{-2}{v}_{x,ac0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\gamma }_{1,ac}^{-1}{v}_{x,th0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-1}$			$\mathit{ln}{\gamma }_{1,ac}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+1}$
$t>>{\tau }_{ac}$	$x_{ac}$	${\displaystyle \frac{x_{ac0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-6}+{\displaystyle \frac{x_{ac0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-3}$	${\displaystyle \frac{{\gamma }_{1,ac}^{-1/2}{x}_{ac0}{v}_{x,ac0}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-h-2}$	${\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{ac}}}{t}^5$	$\mathit{ln}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\gamma }_{1,th}^{-1}}}{t}^{2h+4}$
	$v_{x,ac}$	${\displaystyle \frac{{\gamma }_{1,ac}^{-2}{v}_{x,ac0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\gamma }_{1,ac}^{-1}{v}_{x,ac0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2h-1}$			$\mathit{ln}{\gamma }_{1,ac}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^{2h+1}$
${\tau }_{ac}=0$	$x_{ac}$	${\displaystyle \frac{x_{ac0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-6}+{\displaystyle \frac{x_{ac0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-3}$	${\displaystyle \frac{x_{ac0}{v}_{x,ac0}}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-2}$	${\displaystyle \frac{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}{{\tau }_{ac}}}{t}^5$	$\mathit{ln}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t^3$	$\mathit{ln}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2{t}^4$
	$v_{x,ac}$	${\displaystyle \frac{v_{x,ac0}^4}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2{]}^2}}{t}^{-2}+{\displaystyle \frac{v_{x,ac0}^2}{[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]}}{t}^{-1}$			$\mathit{ln}[{\overline{\alpha }}_1+qB{\overline{\alpha }}_2]t$

and

Table 11. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density in y-component with an optical trap force and ${\zeta }_{y,ac}\left(t\right)$

Time	${\mathit{x}}_{\mathit{a}\mathit{c}},$ ${\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}}$	${\mathit{K}}_{{\mathit{x}}_{\mathit{a}\mathit{c}}},{\mathit{K}}_{{\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}}}$	${\mathit{\rho }}_{{\mathit{x}}_{\mathit{a}\mathit{c}},{\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}}}$	${\mathit{\mu }}_{\mathrm{2,2}}$	$\mathit{S}({\mathit{x}}_{\mathit{a}\mathit{c}},\mathit{t})$, $\mathit{S}({\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}},\mathit{t})$	$\mathit{S}({\mathit{x}}_{\mathit{a}\mathit{c}},{\mathit{v}}_{\mathit{x},\mathit{a}\mathit{c}},\mathit{t})$
$t<<{\tau }_{ac}$	$x_{ac}$	${\displaystyle \frac{{\tau }_{ac}^2{y}_{ac0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-8}+{\displaystyle \frac{{\tau }_{ac}{y}_{ac0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-4}$	${\displaystyle \frac{{\gamma }_{2,ac}^{-1/2}{\tau }_{ac}^{1/2}{y}_{ac0}{v}_{y,ac0}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-h-5/2}$	${\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{ac}^2}}{t}^6$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}{{\tau }_{ac}}}{t}^4$	$\mathit{ln}{\displaystyle \frac{{\gamma }_{2,ac}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{th}}}{t}^{2h+5}$
	$v_{x,ac}$	${\displaystyle \frac{{\gamma }_{2,ac}^{-2}{v}_{y,ac0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\gamma }_{2,ac}^{-1}{v}_{y,th0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-1}$			$\mathit{ln}{\gamma }_{2,ac}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+1}$
$t>>{\tau }_{ac}$	$x_{ac}$	${\displaystyle \frac{y_{ac0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-6}+{\displaystyle \frac{y_{ac0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-3}$	${\displaystyle \frac{y_{ac0}{v}_{y,ac0}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-h-2}$	${\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{ac}}}{t}^5$	$\mathit{ln}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3$	$\mathit{ln}{\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\gamma }_{2,th}^{-1}}}{t}^{2h+4}$
	$v_{x,ac}$	${\displaystyle \frac{{\gamma }_{2,ac}^{-2}{v}_{y,ac0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-4h-2}+{\displaystyle \frac{{\gamma }_{2,ac}^{-1}{v}_{y,ac0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2h-1}$			$\mathit{ln}{\gamma }_{2,ac}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^{2h+1}$
${\tau }_{ac}=0$	$x_{ac}$	${\displaystyle \frac{y_{ac0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-6}+{\displaystyle \frac{y_{ac0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-3}$	${\displaystyle \frac{y_{th0}{v}_{y,ac0}}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-2}$	${\displaystyle \frac{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}{{\tau }_{ac}}}{t}^5$	$\mathit{ln}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t^3$	$\mathit{ln}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2{t}^4$
	$v_{x,ac}$	${\displaystyle \frac{v_{y,ac0}^4}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1{]}^2}}{t}^{-2}+{\displaystyle \frac{v_{y,ac0}^2}{[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]}}{t}^{-1}$			$\mathit{ln}[{\overline{\alpha }}_2-qB{\overline{\alpha }}_1]t$

show, respectively, the statistical values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for a charged colloid with a harmonic trap, a viscous force, and thermal and active noises in the three-time domains.

6. Conclusions

In conclusion, we have derived the Fokker–Planck equation for a charged colloid confined in a harmonic trap under the influence of a magnetic field. We further extended this framework to include the effects of both thermal and active noises acting on a charged colloid. Analytical solutions for the joint probability density were obtained in the limits of $t\ll \tau$, $t\gg \tau$, and $\tau =0$, where $\tau$ denotes the correlation time.

The main findings are summarized as follows:

(1) Superdiffusive behavior: For $1/2<h<1$, a charged colloid subject to a harmonic trap, a viscous force, and correlated Gaussian noise, the mean squared displacements in both x- and y-components in the limits of $t<<\tau$, $t>>\tau$ and for $\tau =0$ scales as $\sim t^{2h+2}$ in all time limits, indicating superdiffusive motion. In particular, the mean squared velocity scales as $\sim t$ for $\tau =0$.

(2) Thermal noise effects: For $h\to 1$, the x- and y-component mean squared velocities for a charged colloid with ${\zeta }_{th}(t)$, scale as $\sim t^2$ in the short-time domain $t<<\tau$, whereas $\tau =0$, those exhibit normal diffusions ($\sim t$), consistent with previous studies [49,50].

(3) Active noise effects: For a charged colloid with a harmonic trap and ${\zeta }_{ac}(t)$, the x- and y-component mean squared velocities $<v_{ac}^2(t)>$ for $\tau =0$ are the same as a Gaussian form.

(4) Entropy and statistical moments: The entropy of the joint probability density associated with thermal noise ${\zeta }_{\mathrm{th}}(t)$ approaches the same value as that for active noise ${\zeta }_{\mathrm{ac}}(t)$ when $h\to 1/2$ in both $t\gg \tau$ and $\tau =0$ limits. Moreover, the combined entropy for the thermal case is larger than that for the active case in each time domain. For $h\to 1$, the coefficient $K_x$ of the joint probability density under active noise scales as $t^{-8}$ in the short-time limit ($t\ll \tau$), while the corresponding velocity coefficient $K_v$ scales as $t^{-3}$ for $h\to 1/2$ at $\tau =0$.

Although the present study mainly focuses on analytical derivations, the need for future numerical validation is clearly recognized. In this regard, the Virtual Element Method (VEM) proposed by Falletta and collaborators [51,52] provides a modern computational framework for time-domain and time-harmonic problems. While not directly applied here, such approaches could guide future numerical implementation of the proposed model, helping to extend its analytical findings toward practical validation.

The observed $t^3$ and $t^4$ scalings arise from the interplay between the Lorentz force induced by the magnetic field and the long-range correlations introduced by active noise. Specifically, the magnetic field couples the velocity components, while the active noise drives persistent motion with memory effects characterized by the Hurst exponent $h$. Their combined effect produces enhanced temporal correlations in the displacement resulting in super-diffusion behavior as a new finding.

Recent studies have demonstrated experimental evidence of similar systems, including magnetically driven colloidal microrollers under electric control [53], and active colloids in non-Newtonian fluids exhibiting persistent motion and long-range correlations [54]. These works suggest that the physical ingredients of our model—magnetic coupling, optical trapping, and active fluctuations—can be realized independently and potentially combined in future experiments to validate the predicted scaling laws and probability distributions. From the derived Fokker–Planck equation, we obtained approximate analytical expressions for the joint probability density, which were further validated by numerical simulations based on the generalized Langevin equation linking passive and active particles [55,56,57]. It is expected that these theoretical results can be experimentally verified in future studies of active colloidal systems. Moreover, extending the present model to generalized Langevin equations or other equations of motion involving different types of forces may provide further insights into the statistical and transport properties of complex active systems, allowing for direct comparison with other theoretical, numerical, and experimental results.