Fractional Motion of an Active Particle in Fractional Generalized Langevin Equations

Abstract

We first investigate the dynamical behavior of an active Brownian particle influenced by a viscoelastic memory effect characterized by a power-law kernel, under the effects of thermal and active noises. We then analyze the dynamics of an active Brownian particle confined in a harmonic trap in the presence of the same noise sources. To derive the Fokker–Planck equation for the joint probability density of the active particle, we obtain analytical solutions for the joint probability density and its moments using double Fourier transforms in the limits $t\ll \tau$, $t\gg \tau$, and $\tau =0$. As a result, the mean squared displacement of an active Brownian particle driven by thermal noise exhibits a super-diffusive scaling of $t^{2h+1}$ in the short-time regime ($t\ll \tau$). In contrast, for a particle in a harmonic trap driven by active noise, the mean squared velocity scales linearly with $t$ when $\tau =0$. Moreover, the higher-order moments of an active Brownian particle in a harmonic trap with thermal noise scale with $t^{4h+2}$ in the long-time limit ($t\gg \tau$) and for $\tau =0$, consistent with our analytical results.

1. Introduction

Recently, the nonequilibrium motion of active particles has been extensively studied in various models, including the run-and-tumble model [1,2], active Brownian particles [3,4], active Langevin particles [5], active Ornstein–Uhlenbeck particles [6,7,8], self-propelled Janus colloids [9,10], and biological microswimmers [11]. Active diffusion, investigated through theoretical analyses, computer simulations, and experiments on active viscoelastic systems, has revealed both similarities and differences among these systems. Representative examples include the transport of passive tracers in active baths and living cells [12], chromosomal dynamics [13], lateral diffusion of membrane proteins [14], and tracer diffusion in dense colloidal suspensions [15]. Active particles have also been explored in polymeric environments such as actomyosin and endoplasmic reticulum networks [16], microtubule assemblies, and macromolecules bound to polymer strands, including DNA and chromosomes.

Over the past two decades, anomalous diffusion dynamics [17,18] have been widely discussed and extended in natural and complex scientific systems. In the typical form of anomalous diffusion, characterized by a scaling exponent $\alpha$, the system exhibits sub-diffusive behavior for $\alpha <1$, indicated by a sub-linear growth of the mean squared displacement over time, and super-diffusive behavior for $\alpha >1$, characterized by super-linear growth of the mean squared displacement. Sub-diffusion has been observed for endogenous submicron tracers in biological systems, such as living cells [19], artificially crowded biological environments [20], protein motion in supercomputing simulations [21], and dilute or protein-crowded lipid bilayer membranes. Super-diffusion has been reported in several cellular systems [22]. For instance, a restoring force applied by an optical tweezer in a biological cell [23,24] enabled the fractional Ornstein–Uhlenbeck process [25]. Active processes in the underdamped limit exhibit ballistic motion, while complex underdamped dynamics can lead to hyper-diffusion in the $t\ll \tau$ regime [26,27].

Furthermore, theories based on the active fractional Langevin equation have been developed to quantitatively describe transport phenomena in various active viscoelastic systems. These studies also introduced weak ergodicity breaking, a novel phenomenon not previously reported in other systems [28,29]. For Gaussian processes, all statistical properties can be inferred from the mean and covariance functions [30,31,32]. Indirect approaches to assessing the ergodic properties of Gaussian processes have been compared with the behaviors of the mean squared displacement and the time-averaged MSD [33,34].

Fractional Brownian motion is a type of Gaussian process characterized by a two-time autocovariance function [35], $⟨B_{\alpha }\left(t_1\right)B_{\alpha }\left(t_2\right)⟩=K\left[t_1^{\alpha }+t_2^{\alpha }-∣t_1-t_2{\mid }^{\alpha }\right], 0<\alpha <2,$ with the mean squared displacement given by $⟨B_{\alpha }^2(t)⟩=2K{t}^{\alpha }$ for $t=t_1=t_2$. The corresponding probability density function is $p(x,t)=[4\pi K{t}^{\alpha }{]}^{-1/2}\exp [-(x-x_0{)}^2/4K{t}^{\alpha }]$. For $\alpha >1$, increments are positively correlated, leading to super-diffusive motion, whereas for $\alpha <1$, increments are negatively correlated, resulting in sub-diffusive behavior. Standard Brownian motion with independent increments corresponds to normal diffusion for $\alpha =1$. Fractional Gaussian noise is defined as ${\eta }_{\alpha }\left(t\right)=\left[B_{\alpha }\left(t+\delta t\right)-B_{\alpha }\left(t\right)\right]/\delta t,$ where $\delta t$ is a small, finite time interval. The autocovariance function for an active particle driven by fractional Gaussian noise is then $⟨{\eta }_{\alpha }(\tau +\delta t){\eta }_{\alpha }(\tau )⟩=K\delta t^{-2}[\mid \tau +\delta t{\mid }^{\alpha }+\mid \tau -\delta t{\mid }^{\alpha }-2\mid \tau {\mid }^{\alpha }],$ which asymptotically satisfies $⟨{\eta }_{\alpha }(t+\tau ){\eta }_{\alpha }(t)⟩\sim \alpha (\alpha -1)K{\tau }^{\alpha -2}$ for $\tau \gg \delta t$. The variance is given by $⟨{\eta }_{\alpha }^2(t)⟩=2K(\delta t{)}^{\alpha -2}$, and in the ballistic limit ($\alpha =2$), the autocovariance reduces to $⟨{\eta }_{\alpha }(t+\tau ){\eta }_{\alpha }(t)⟩\sim 2K$.

Until now, most of these problems have been addressed numerically using perturbation methods, special functions, or approximations. This study builds on the classical formulations of Heinrichs [30], Athanassoulis et al. [31], and Mamis and Farazmand [32], but extends the framework in several significant ways. By considering thermal noise, active noise, viscoelastic memory effects, and optical trapping forces, we provide novel analytical and numerical comparisons that reveal new scaling relationships, correlation coefficients, entropy connections, and stability properties not covered in previous studies.

Recently, it has become more straightforward to describe the motion of active particles using equations of motion when they experience varying forces within a viscous medium. However, obtaining analytical solutions for the probability densities of displacement, velocity, orientation, and other dynamical quantities remains challenging. Previous studies have investigated the fractional generalized Langevin equation for a passive particle subjected to thermal equilibrium noise ${\zeta }_{\mathrm{t}\mathrm{h}}(t)$ and active noise ${\zeta }_{\mathrm{a}\mathrm{c}}(t)$ with exponentially decaying correlations. In these viscoelastic systems, the particle experiences two distinct types of noise: thermal noise, which satisfies the fluctuation–dissipation theorem, and active noise, modeled as an active Ornstein–Uhlenbeck process. In this work, we derive the Fokker–Planck equation for the joint probability density and obtain its solution using double Fourier transforms in three distinct time regimes. The organization of this paper is as follows. In Section 2, we derive the Fokker–Planck equation from the fractional generalized Langevin equation. In Section 3, using double Fourier transforms, we obtain approximate solutions for the joint probability density of an active Brownian particle subject to harmonic and viscous forces in three regimes of correlation times ${\tau }_{\mathrm{t}\mathrm{h}}$ and ${\tau }_{\mathrm{a}\mathrm{c}}$. Section 4 presents numerical calculations of the non-Gaussian parameter, correlation coefficients, and entropy. Finally, in Section 5, we summarize the key findings and provide concluding remarks.

2. Thermal and Active Fractional Generalized Langevin Equations

As a nonequilibrium dynamic model, our first model derives the Fokker–Planck equation for the two variables of displacement and velocity by introducing the viscoelastic memory effect, as well as thermal and active noises, in the fractional Langevin equations, thereby obtaining the joint probability density. The second model introduces the fractional Langevin equations that account for the viscoelastic memory effect, an optical trapped force, and thermal and active noises to obtain a new joint probability density for displacement and velocity.

In this section, we introduce a class of nonequilibrium dynamic models referred to as the fractional Langevin equation ${\displaystyle \frac{d}{dt}}v\left(t\right)=-\gamma {\int }_0^{t}d{t}^{\prime }K\left(t-t^{\prime }\right)v\left(t^{\prime }\right)+\zeta \left(t\right),$ which incorporates a viscoelastic memory effect [26] with a power-law kernel $K(t-t^{\prime })=|(t-t^{\prime })/\tau {|}^{2h-2}$.

The active fractional Langevin equation, a class of nonequilibrium dynamic models, is presented; 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 fractional generalized Langevin equations in our model are expressed as

$${\displaystyle \frac{d}{dt}}{x}_{th}(t)=v_{th}(t), {\displaystyle \frac{d}{dt}}{v}_{th}(t)=-{\gamma }_{th}{\int }_0^{t}d{t}^{\prime }|{\displaystyle \frac{t-t^{\prime }}{{\tau }_{th}}}{|}^{2h-2}{v}_{th}(t^{\prime })+{\zeta }_{th}(t),$$

(1)

$${\displaystyle \frac{d}{dt}}{x}_{ac}(t)=v_{ac}(t), {\displaystyle \frac{d}{dt}}{v}_{ac}(t)=-{\gamma }_{ac}{\int }_0^{t}d{t}^{\prime }|{\displaystyle \frac{t-t^{\prime }}{{\tau }_{ac}}}{|}^{2h-2}{v}_{ac}(t^{\prime })+{\zeta }_{ac}(t),$$

(2)

where ${\zeta }_{th}(t)$ and ${\zeta }_{ac}(t)$ denote the thermal equilibrium noise and the active noise (with exponentially decaying correlation), respectively. We introduce these two types of noise as

$$<{\zeta }_{th}(t){\zeta }_{th}(t^{\prime })>={\zeta }_{th}^2{\zeta }_{th}(t-t^{\prime })={\displaystyle \frac{{\zeta }_{th}^2}{2}}|{\displaystyle \frac{t-t^{\prime }}{{\tau }_{th}}}{|}^{2h-2},$$

(3)

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

(4)

where the persistent Hurst exponent h ranges as $1/2<h<1$ and the correlation times are ${\tau }_{th}$ and ${\tau }_{ac}$. The thermal equilibrium noise is a fractional Gaussian noise coupled to the memory kernel via the fluctuation–dissipation theorem. The joint probability density $p(x_i,v_i,t)$ for the displacement $x_i$ and the velocity $v_i$ is defined by

$$p(x_i,v_i,t)=<\delta (x_i-x_i(t))\delta (v_i-v_i(t))>$$

(5)

for $i=th,ac$. By taking time derivatives in the joint probability density and inserting Equations (1) and (2) into Equation (5), we can write the time derivative of the joint probability equation for $p(x_{th}(t),v_{th}(t),t)\equiv p_{th}$ and $p(x_{ac}(t),v_{ac}(t),t)\equiv p_{ac}$ as

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th}=-{\displaystyle \frac{\partial }{\partial x_{th}}}<{\displaystyle \frac{\partial x_{th}}{\partial t}}{\delta }_{x_{th}}{\delta }_{v_{th}}>-{\displaystyle \frac{\partial }{\partial v_{x_{th}}}}<[-{\gamma }_{th}{\int }_0^{t}d{t}^{\prime }|{\displaystyle \frac{t-t^{\prime }}{{\tau }_{th}}}{|}^{2h-2}{v}_{th}(t^{\prime })+{\zeta }_{th}(t)]{\delta }_{x_{th}}{\delta }_{v_{th}}>,$$

(6)

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac}=-{\displaystyle \frac{\partial }{\partial x_{ac}}}<{\displaystyle \frac{\partial x_{ac}}{\partial t}}{\delta }_{x_{ac}}{\delta }_{v_{ac}}>-{\displaystyle \frac{\partial }{\partial v_{ac}}}<[-{\gamma }_{ac}{\int }_0^{t}d{t}^{\prime }|{\displaystyle \frac{t-t^{\prime }}{{\tau }_{ac}}}{|}^{2h-2}{v}_{ac}(t^{\prime })+{\zeta }_{ac}(t)]{\delta }_{x_{ac}}{\delta }_{v_{ac}}>.$$

(7)

where ${\delta }_{x_{th}}(x_{th}-x_{th}(t))\equiv {\delta }_{x_{th}}$, ${\delta }_{v_{th}}(v_{th}-v_{th}(t))\equiv {\delta }_{v_{th}}$, ${\delta }_{x_{ac}}(x_{ac}-x_{ac}(t))\equiv {\delta }_{x_{ac}}$, and ${\delta }_{v_{ac}}(v_{ac}-v_{ac}(t))\equiv {\delta }_{v_{ac}}$. The memory effect is expressed as ${\int }_0^{t}d{t}^{\prime }{\left|{\displaystyle \frac{t-t^{\prime }}{{\tau }_i}}\right|}^{2h-2}{v}_i(t^{\prime })={\tau }_i^{2h-2}[\Gamma (2h-1){]}^{-1}{\displaystyle \frac{d^{2-2h}}{d{t}^{2-2h}}}{x}_i(t)$ for $i=th,ac$. Assuming that the particle is initially at rest at $t=0$, the forms of the joint probability density [30,31,32] can be written as

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th}=[-v_{th}{\displaystyle \frac{\partial }{\partial x_{th}}}+{\gamma }_{th}{D}_{2-2h}{\displaystyle \frac{\partial }{\partial v_{th}}}{x}_{th}]p_{th}+{\alpha }_1[{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\displaystyle \frac{{\partial }^2}{\partial v_{th}\partial x_{th}}}+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\displaystyle \frac{{\partial }^2}{\partial v_{th}^2}}]p_{th},$$

(8)

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac}=[-v_{ac}{\displaystyle \frac{\partial }{\partial x_{ac}}}+{\gamma }_{ac}{D}_{2-2h}{\displaystyle \frac{\partial }{\partial v_{ac}}}{x}_{ac}]p_{ac}+{\alpha }_2[-b_2(t){\displaystyle \frac{{\partial }^2}{\partial x_{ac}\partial v_{ac}}}+a_2(t){\displaystyle \frac{{\partial }^2}{\partial v_{ac}^2}}]p_{ac},$$

(9)

where $D_{2-2h}=d^{2-2h}/d{t}^{2-2h}$. The parameters are

$${\alpha }_1={\displaystyle \frac{{\zeta }_{th}^2}{2}}, {\alpha }_2={\displaystyle \frac{{\zeta }_{ac}^2}{2}}, a_2(t)=1-\mathit{exp}(-{\displaystyle \frac{t}{{\tau }_{ac}}}), b_2(t)=(t+{\tau }_{ac})\mathit{exp}(-{\displaystyle \frac{t}{{\tau }_{ac}}})-{\tau }_{ac}.$$

(10)

It is apparent that Equations (8) and (9) are the Fokker–Planck equations, as mentioned in the Introduction. Later, the time evolutions of the joint probability densities $p(x,v_x,t)$ and $p(y,v_y,t)$, Equations (8) and (9), will be derived in Appendix A.

Some derivations related to the correlated Gaussian force are given in Ref. [30]. We define the double Fourier transform of the joint probability density $p_i(\xi ,\nu ,t)$ as

$$p_i(\xi ,\nu ,t)={\int }_{-\infty }^{+\infty }d{x}_i{\int }_{-\infty }^{+\infty }d{v}_i\mathit{exp}(-i\xi x_i-i\nu v_i)p_i(x_i,v_i,t), i=th, ac.$$

(11)

The Fourier transforms of the Fokker–Planck equation, Equations (8) and (9), are expressed as

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th}(\xi ,\nu ,t)=[\xi {\displaystyle \frac{\partial }{\partial \nu }}-{\gamma }_{th}\nu D_{2-2h}{\displaystyle \frac{\partial }{\partial \xi }}-{\displaystyle \frac{{\alpha }_1}{2h{\tau }_{th}^{2h}}}{t}^{2h}\xi \nu -{\displaystyle \frac{{\alpha }_1}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{\nu }^2]p_{th}(\xi ,\nu ,t),$$

(12)

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac}(\xi ,\nu ,t)=[\xi {\displaystyle \frac{\partial }{\partial \nu }}-{\gamma }_{ac}\nu D_{2-2h}{\displaystyle \frac{\partial }{\partial \xi }}]p_{ac}(\xi ,\nu ,t)+{\alpha }_2[b_2(t)\xi \nu -a_2(t){\nu }^2]p_{ac}(\xi ,\nu ,t).$$

(13)

2.1. $p_{th}(x_{th},t)$ and $p_{th}(v_{th},t)$ with Thermal Noise ${\zeta }_{th}(t)$

2.1.1. $p_{th}(x_{th},t)$ and $p_{th}(v_{th},t)$ with Thermal Noise ${\zeta }_{th}(t)$ in $t<<{\tau }_{th}$

In this subsection, we derive the solutions of the probability densities $p_{th}(x_{th},t)$ and $p_{th}(v_{th},t)$ in the short-time domain $t<<{\tau }_{th}$. To find the special solutions for $\xi$ and $\nu$ by the variable separation from Equation (12), the two equations for the displacement and the velocity are written as

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th}(\xi ,t)=-{\gamma }_{th}\nu D_{2-2h}{\displaystyle \frac{\partial }{\partial \xi }}{p}_{th}(\xi ,t)-[{\displaystyle \frac{{\alpha }_1}{2}}[{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}\xi \nu +{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\nu }^2]-A]p_{th}(\xi ,t),$$

(14)

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

(15)

where $A$ denotes the separation constant. Taking ${\displaystyle \frac{\partial }{\partial t}}{p}_{th}(\xi ,t)=0$ in the steady state, we get $p_{th}^{st}(\xi ,t)$ as

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

(16)

To find the solution of the probability density for $\xi$ from $q_{th}^{st}\left(\xi ,t\right)\equiv r_{th}(\xi ,t)q_{th}^{st}(\xi ,t)$, we include terms up to order $1/{\tau }_{th}^2$ and write

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

(17)

$$q_{th}(\xi ,t)=r_{th}(\xi ,t)\mathit{exp}[{\displaystyle \frac{{\alpha }_1}{2({\gamma }_{th}\nu D_{2-2h}{)}^2}}[{\displaystyle \frac{t^{2h-1}}{4h^2{\tau }_{th}^{2h}}}\nu {\displaystyle \frac{{\xi }^2}{6}}+{\displaystyle \frac{t^{2h-2}}{(2h-1{)}^2{\tau }_{th}^{2h-1}}}{\nu }^2{\displaystyle \frac{{\xi }^2}{2}}]].$$

(18)

Assuming arbitrary functions of variable $t-\xi /{\gamma }_{th}\nu D_{2-2h}$, the probability density $r_{th}(\xi ,t)$ becomes $\Theta [t-\xi /{\gamma }_{th}\nu D_{2-2h}]$. Therefore, we have

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

(19)

By a similar method, from Equation (16) to Equation (19), for $\xi$, we also get the Fourier transform of the probability density for $\nu$ as

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

(20)

By calculating Equations (19) and (20), the Fourier transform of the joint probability density is

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

(21)

By taking the inverse Fourier transform, we obtain

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

(22)

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

(23)

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

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

(24)

2.1.2. $p_{th}(x_{th},t)$ and $p_{th}(v_{th},t)$ with Thermal Noise ${\zeta }_{th}(t)$ in $t>>{\tau }_{th}$

We now consider the long-time domain $t>>{\tau }_{th}$. From Equation (14), an approximate equation for the Fourier-transformed probability density is

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th\xi }(\xi ,t)\cong -{\displaystyle \frac{{\alpha }_1}{4h{\tau }_{th}^{2h}}}{t}^{2h}\xi \nu p_{th\xi }(\xi ,t)-{\displaystyle \frac{{\alpha }_1}{2(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{\nu }^2{p}_{th\xi }(\xi ,t),$$

(25)

The Fourier transform of the probability density $p_{th\xi }(\xi ,t)$ from Equation (25) is calculated as

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

(26)

The steady probability density $q_{th}^{st}(\xi ,t)$ for $\xi$, from ${p_{th}(\xi ,t)\equiv q}_{th\xi }(\xi ,t)p_{th\xi }(\xi ,t)$, is

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

(27)

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

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

(28)

where $r_{th}(\xi ,t)=\Theta [t-\xi /{\gamma }_{th}\nu D_{2-2h}]$. Applying Equation (15) for $\nu$ to a similar method from Equation (25) to Equation (28) of $p_{th}(\xi ,t)$ derived, we also get the Fourier transforms of the probability density for velocity as

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

(29)

By calculating Equations (28) and (29), we have

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

(30)

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

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

(31)

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

(32)

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

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

(33)

2.1.3. $p_{th}(x_{th},t)$ and $p_{th}(v_{th},t)$ with Thermal Noise ${\zeta }_{th}(t)$ for ${\tau }_{th} = 0$

In this subsection, we find the probability densities $p_{th}(x_{th},t)$ and $p_{th}(v_{th},t)$ in the time domain ${\tau }_{th}=0$. The approximate equation from Equation (14) for $\xi$ is written as

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th\xi }(\xi ,t)\cong -{\gamma }_{th}\nu D_{2-2h}{\displaystyle \frac{\partial }{\partial \xi }}{p}_{th\xi }(\xi ,t)-{\displaystyle \frac{{\alpha }_1}{2(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{\nu }^2{p}_{th\xi }(\xi ,t).$$

(34)

In the steady state, we can calculate $p_{th}^{st}(\xi ,t)$ from Equation (34) as

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

(35)

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

$$p_{th}(\xi ,t)=\Theta [t-\xi /{\gamma }_{th}\nu D_{2-2h}]p_{th\xi }^{st}(\xi ,t).$$

(36)

Using a similar procedure of $p_{th}\left(\xi ,t\right)$, we get the Fourier transform of the probability density $p_{th}(\nu ,t)$ as

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

(37)

We can calculate $p_{th}(\xi ,\nu ,t)$ by calculating Equations (36) and (37) as

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

(38)

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

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

(39)

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

(40)

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

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

(41)

2.2. $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ with Active Noise ${\zeta }_{ac}(t)$

2.2.1. $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ with Active Noise ${\zeta }_{ac}(t)$ in the Short-Time Domain $t<<{\tau }_{ac}$

In this subsection, we settle the solutions of the probability densities $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ in the short-time domain $t<<{\tau }_{ac}$. In order to find the special solutions for $\xi ,\nu$, with the variable separation from Equation (13), the two equations for the displacement and the velocity are given by

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac}(\xi ,t)=-{\gamma }_{ac}\nu D_{2-2h}{\displaystyle \frac{\partial }{\partial \xi }}{p}_{ac}(\xi ,t)+{\displaystyle \frac{{\alpha }_2}{2}}[b_2(t)\xi \nu -a_2(t){\nu }^2+B]p_{ac}(\xi ,t),$$

(42)

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac}(\nu ,t)=[\xi {\displaystyle \frac{\partial }{\partial \nu }}+{\displaystyle \frac{{\alpha }_2}{2}}[b_2(t)\xi \nu -a_2(t)]{\nu }^2-B]p_{ac}(\nu ,t).$$

(43)

where $B$ denotes the separation constant. As we take ${\displaystyle \frac{\partial }{\partial t}}{p}_{ac}(\xi ,t)=0$ in the steady state, we get $p_{ac}^{st}(\xi ,t)$ as

$$p_{ac}^{st}(\xi ,t)=\mathit{exp}[{\displaystyle \frac{1}{2{\gamma }_{ac}\nu D_{2-2h}}}[{\alpha }_2[{\displaystyle \frac{b_2(t)}{2}}\nu {\xi }^2-a_2(t){\nu }^2\xi ]+B\xi ]].$$

(44)

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

$$p_{ac}(\xi ,t)=q_{ac}(\xi ,t)\mathit{exp}[{\displaystyle \frac{1}{2{\gamma }_{ac}\nu D_{2-2h}}}[{\alpha }_2[{\displaystyle \frac{b_2(t)}{2}}\nu {\xi }^2-a_2(t){\nu }^2\xi ]+B\xi ]],$$

(45)

$$q_{ac}(\xi ,t)=r_{ac}(\xi ,t)\mathit{exp}[-{\displaystyle \frac{{\alpha }_2}{2({\gamma }_{ac}\nu D_{2-2h}{)}^2}}[{\displaystyle \frac{{b_2}^{\prime }(t)}{6}}\nu {\xi }^3-{\displaystyle \frac{{a_2}^{\prime }(t)}{2}}{\nu }^2{\xi }^2]],$$

(46)

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

(47)

$$s_{\mathit{ac}}\left(\xi ,t\right)=t_{\mathit{ac}}(\xi ,t)\mathit{exp}[-{\displaystyle \frac{{\alpha }_2}{2({\gamma }_{ac}\nu D_{2-2h}{)}^4}}{\displaystyle \frac{{b_2}^{‴}(t)}{120}}\nu {\xi }^5].$$

(48)

Taking the solutions as arbitrary functions of variable $t-\xi /\gamma \nu D_{2-2h}$, the arbitrary function $t_{ac}(\xi ,t)$ becomes $\Theta [t-\xi /\gamma \nu D_{2-2h}]$. As a result, we find that

$$p_{ac}(\xi ,t)=t_{ac}(\xi ,t)s_{ac}^{st}(\xi ,t)r_{ac}^{st}(\xi ,t)q_{ac}^{st}(\xi ,t)p_{ac}^{st}(\xi ,t)=\Theta [t-\xi /{\gamma }_{ac}\nu D_{2-2h}]s_{ac}^{st}(\xi ,t)r_{ac}^{st}(\xi ,t)q_{ac}^{st}(\xi ,t)p_{ac}^{st}(\xi ,t).$$

(49)

Using a similar method from Equation (44) to Equation (49) for $\nu$, we also get the Fourier transform of the probability density for the velocity as

$$p_{ac}(\nu ,t)=t_{ac}(\nu ,t)s_{ac}^{st}(\nu ,t)r_{ac}^{st}(\nu ,t)q_{ac}^{st}(\nu ,t)p_{ac}^{st}(\nu ,t)=\Theta [t+\nu /\xi ]s_{ac}^{st}(\nu ,t)r_{ac}^{st}(\nu ,t)q_{ac}^{st}(\nu ,t)p_{ac}^{st}(\nu ,t).$$

(50)

Therefore, by calculating Equations (49) and (50), we get the Fourier transform of the joint probability density as

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

(51)

Using the inverse Fourier transform, we get

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

(52)

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

(53)

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

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

(54)

2.2.2. $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ with Active Noise ${\zeta }_{ac}(t)$ in the Long-Time Domain $t>>{\tau }_{ac}$

We find the probability densities $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ in the long-time domain $t>>{\tau }_{ac}$. We write an approximate equation from Equation (42) as

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac\xi }(\xi ,t)\cong {\displaystyle \frac{{\alpha }_2}{2}}[b_2(t)\xi \nu -a_2(t){\nu }^2]p_{ac\xi }(\xi ,t).$$

(55)

The Fourier transform of the probability density $p_{ac\xi }(\xi ,t)$ from Equation (55) is calculated as

$$p_{ac\xi }(\xi ,t)\cong \mathit{exp}[{\displaystyle \frac{{\alpha }_2}{2}}{\int }_0^{t}dt[b_2(t)\xi \nu -a_2(t){\nu }^2]].$$

(56)

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

$$q_{ac\xi }^{st}(\xi ,t)=\mathit{exp}[-{\displaystyle \frac{{\alpha }_2}{2}}{\int }_0^{t}dt[b_2(t)\xi \nu -a_2(t){\nu }^2]].$$

(57)

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

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

(58)

where $r_{ac}(\xi ,t)=\Theta [t-\xi /{\gamma }_{ac}\nu D_{2-2h}]$ and $p_{ac}^{st}(\xi ,t)$ is equal to Equation (44). Applying Equation (43) for $\nu$ to a similar method from Equation (55) to Equation (58) of $p_{ac}(\xi ,t)$ derived, we also get the Fourier transforms of the probability density for velocity as

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

(59)

By calculating Equations (58) and (59), we have

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

(60)

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

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

(61)

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

(62)

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

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

(63)

2.2.3. $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ with Active Noise ${\zeta }_{ac}(t)$ for ${\tau }_{ac}=0$

In this subsection, we find $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ for the time domain ${\tau }_{ac}=0$. The approximate equation from Equation (42) for $\xi$ is written as

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac\xi }(\xi ,t)\cong -{\gamma }_{ac}\nu D_{2-2h}{\displaystyle \frac{\partial }{\partial \xi }}{p}_{ac\xi }(\xi ,t)-{\displaystyle \frac{{\alpha }_2}{2}}{a}_2(t){\nu }^2{p}_{ac\xi }(\xi ,t).$$

(64)

In the steady state, we can calculate $p_{ac}^{st}(\xi ,t)$ from Equation (64) as

$$p_{ac\xi }^{st}(\xi ,t)=\mathit{exp}[-{\displaystyle \frac{{\alpha }_2}{2{\gamma }_{ac}\nu D_{2-2H}}}{\nu }^2\xi ].$$

(65)

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

$$p_{ac}(\xi ,t)=\Theta [t-\xi /{\gamma }_{ac}\nu D_{2-2h}]p_{ac\varsigma }^{st}(\xi ,t).$$

(66)

Using a similar method to that of $p_{ac}(\xi ,t)$ derived, we get the Fourier transform of the probability density $p_{ac}(\nu ,t)$ as

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

(67)

We can calculate $p(\xi ,\nu ,t)$ by calculating Equations (66) and (67) as

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

(68)

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

$$p_{ac}(x_{ac},t)=[2\pi {\displaystyle \frac{{\alpha }_2{t}^3}{3}}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{3}{2{\alpha }_2{t}^3}}{x}_{ac}^2], p_{ac}(v,t)=[4\pi {\alpha }_{2}t{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{1}{4{\alpha }_{2}t}}{v}_{ac}^2].$$

(69)

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

$$<x_{ac}^2(t)>={\displaystyle \frac{1}{3}}{\alpha }_2{t}^3,<v_{ac}^2(t)>=2{\alpha }_{2}t.$$

(70)

3. Thermal and Active Fractional Fokker–Planck Equations

We consider a nonequilibrium dynamic model referred to as the active fractional Langevin equation, having a viscoelastic memory effect with a power-law kernel $k(t)=|{\displaystyle \frac{t-t^{\prime }}{\tau }}{|}^{2h-2}$ [25], with an optical trapped force $-\beta x$ and thermal and active noises. The two altered fractional generalized Langevin equations in our model are expressed in terms of

$${\displaystyle \frac{d}{dt}}{x}_{th}(t)=v_{th}(t), {\displaystyle \frac{d}{dt}}{v}_{th}(t)=-\beta x-{\gamma }_{th}{\int }_0^{t}d{t}^{\prime }|{\displaystyle \frac{t-t^{\prime }}{{\tau }_{th}}}{|}^{2h-2}{v}_{th}(t^{\prime })+{\zeta }_{th}(t),$$

(71)

$${\displaystyle \frac{d}{dt}}{x}_{ac}(t)=v_{ac}(t), {\displaystyle \frac{d}{dt}}{v}_{ac}(t)=-\beta x-{\gamma }_{ac}{\int }_0^{t}d{t}^{\prime }|{\displaystyle \frac{t-t^{\prime }}{{\tau }_{ac}}}{|}^{2h-2}{v}_{ac}(t^{\prime })+{\zeta }_{ac}(t),$$

(72)

where thermal equilibrium noise and active noise denote ${\zeta }_{\mathrm{th}}(t)$ and ${\zeta }_{\mathrm{ac}}(t)$, respectively. We introduce two noises, i.e., thermal noise and active noise, as

$$<{\zeta }_{th}(t){\zeta }_{th}(t^{\prime })>={\zeta }_{th}^2{\zeta }_{th}(t-t^{\prime })={\displaystyle \frac{{\zeta }_{th}^2}{2}}|{\displaystyle \frac{t-t^{\prime }}{{\tau }_{th}}}{|}^{2h-2},$$

(73)

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

(74)

where ${\zeta }_i^2=2{\gamma }_i{k}_{B}T$ for $i=th,ac$. The thermal energy is $k_{B}T,$ and the correlation times are ${\tau }_{th}$ and ${\tau }_{ac}$. Thermal equilibrium noise is fractional Gaussian noise coupled to the memory kernel via the fluctuation dissipation theorem, while active noise is responsible for both the self-propelled motion of an active particle and nonequilibrium motion from an active heat reservoir.

We next derive the Fokker–Planck equation from the active fractional generalized Langevin equation. First, the joint probability density $p(x_i,v_i,t)$ for displacement $x_i$ and velocity $v_i$ is defined by

$$p\left(x_i,v_i,t\right)=<\delta \left(x_i-x_i\left(t\right)\right)\delta \left(v_i-v_i\left(t\right)\right)>,$$

(75)

where $i=th,ac$. Taking time derivatives in the joint probability density and inserting Equations (71) and (72) into Equation (75), we can write the time derivative of the joint probability equation for $p(x_i,v_i,t)\equiv p_i$ as follows:

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th}=-{\displaystyle \frac{\partial }{\partial x_{th}}}<{\displaystyle \frac{\partial x}{\partial t}}{\delta }_{x_{th}}{\delta }_{v_{th}}>-{\displaystyle \frac{\partial }{\partial v_{th}}}<[-{\gamma }_{th}{\int }_0^{t}d{t}^{\prime }(t-t^{\prime }{)}^{2h-2}{v}_{th}(t^{\prime })-\beta x+{\zeta }_{th}(t)]{\delta }_{x_{th}}{\delta }_{v_{th}}>,$$

(76)

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac}=-{\displaystyle \frac{\partial }{\partial x_{ac}}}<{\displaystyle \frac{\partial x_{ac}}{\partial t}}{\delta }_{x_{ac}}{\delta }_{v_{ac}}>-{\displaystyle \frac{\partial }{\partial v_{ac}}}<[-{\gamma }_{ac}{\int }_0^{t}d{t}^{\prime }(t-t^{\prime }{)}^{2h-2}{v}_{ac}(t^{\prime })-\beta x+{\zeta }_{ac}(t)]{\delta }_{x_{ac}}{\delta }_{v_{ac}}>,$$

(77)

where ${\delta }_{x_i}(x_i-x_i(t))\equiv {\delta }_{x_i}$, ${\delta }_{v_i}(v_i-v_i(t))\equiv {\delta }_{v_i}$, and the persistent Hurst exponent h ranges to $1/2<h<1$. In Equations (76) and (77), we have ${\int }_0^{t}d{t}^{\prime }|t-t^{\prime }{|}^{2h-2}v(t^{\prime })=[\Gamma (2h-1){]}^{-1}{\displaystyle \frac{d^{2-2h}}{d{t}^{2-2h}}}x(t)$. We assume from the joint probability density that the particle is initially at rest at time $t=0$. Then, the joint probability densities [34,35,36] satisfy the Fokker–Planck equations derived as

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th}=[-v{\displaystyle \frac{\partial }{\partial x_{th}}}+{\gamma }_{th}{D}_{2-2h}{\displaystyle \frac{\partial }{\partial v_{th}}}{x}_{th}+\beta {\displaystyle \frac{\partial }{\partial v_{th}}}{x}_{th}]p_{th}+{\alpha }_1[{\displaystyle \frac{t^{2h}}{2h{\tau }_{th}^{2h}}}{\displaystyle \frac{{\partial }^2}{\partial v_{th}\partial x_{th}}}+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}{\displaystyle \frac{{\partial }^2}{\partial v_{th}^2}}]p_{th},$$

(78)

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac}=[-v{\displaystyle \frac{\partial }{\partial x_{ac}}}+{\gamma }_{ac}{D}_{2-2h}{\displaystyle \frac{\partial }{\partial v_{ac}}}{x}_{ac}+\beta {\displaystyle \frac{\partial }{\partial v_{th}}}{x}_{th}]p_{ac}+{\alpha }_2[-b_2(t){\displaystyle \frac{{\partial }^2}{\partial x_{ac}\partial v_{ac}}}+a_2(t){\displaystyle \frac{{\partial }^2}{\partial v_{ac}^2}}]p_{ac},$$

(79)

where $D_{2-2h}=d^{2-2h}/d{t}^{2-2h}$. The parameters are

$${\alpha }_1={\displaystyle \frac{{\zeta }_{th}^2}{2}}, {\alpha }_2={\displaystyle \frac{{\zeta }_{ac}^2}{2}}, a_2(t)=1-\mathit{exp}(-{\displaystyle \frac{t}{{\tau }_{ac}}}), b_2(t)=(t+{\tau }_{ac})\mathit{exp}(-{\displaystyle \frac{t}{{\tau }_{ac}}})-{\tau }_{ac}.$$

(80)

Equations (78) and (79) are called the Fokker–Planck equations, as mentioned in the Introduction. Some derivations in relation to the correlated Gaussian force are given in Ref. [30]. We define the double Fourier transform of the joint probability density $p_i(\xi ,\nu ,t)$ by the equation

$$p_i(\xi ,\nu ,t)={\int }_{-\infty }^{+\infty }d{x}_i{\int }_{-\infty }^{+\infty }d{v}_i\mathit{exp}(-i\xi x_i-i\nu v_i)p_i(x_i,v_i,t),$$

(81)

where $i=th,ac$. The Fourier transforms of the Fokker–Planck equations, Equations (78) and (79), are, respectively, expressed in terms of

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th}(\xi ,\nu ,t)=[-[\beta \nu +\gamma D_{2-2h}\nu ]{\displaystyle \frac{\partial }{\partial \xi }}+\xi {\displaystyle \frac{\partial }{\partial \nu }}]p_{th}(\xi ,\nu ,t)-{\alpha }_1[{\displaystyle \frac{1}{2h{\tau }^{2h}}}{t}^{2h}\xi \nu +{\displaystyle \frac{1}{(2h-1){\tau }^{2h-1}}}{t}^{2h-1}{\nu }^2]p_{th}(\xi ,\nu ,t),$$

(82)

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac}(\xi ,\nu ,t)=-\beta \nu {\displaystyle \frac{\partial }{\partial \xi }}{p}_{ac}(\xi ,\nu ,t)+[\xi {\displaystyle \frac{\partial }{\partial \nu }}-\gamma \nu D_{2-2h}{\displaystyle \frac{\partial }{\partial \xi }}]p_{ac}(\xi ,\nu ,t)+{\alpha }_2[b_2(t)\xi \nu -a_2(t){\nu }^2]p_{ac}(\xi ,\nu ,t).$$

(83)

3.1. $p_{th}(x_{th},t)$ and $p_{th}(v_{th},t)$ with Thermal Noise ${\zeta }_{th}(t)$

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

In this subsection, we obtain the solutions of the probability densities $p_{th}(x_{th},t)$ and $p_{th}(v_{th},t)$ in the short-time domain $t<<{\tau }_{th}$. To find the special solutions for $\xi ,\nu$ by the variable separation from Equation (82), the two equations for the displacement and the velocity are given by

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th}(\xi ,t)=[-\beta \nu {\displaystyle \frac{\partial }{\partial \xi }}-\gamma \nu D_{2-2h}{\displaystyle \frac{\partial }{\partial \xi }}-{\displaystyle \frac{{\alpha }_1}{2}}[{\displaystyle \frac{1}{2h{\tau }_{th}^{2h}}}{t}^{2h}\xi \nu +{\displaystyle \frac{1}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{\nu }^2]+C]p_{th}(\xi ,t),$$

(84)

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th}(\nu ,t)=[\xi {\displaystyle \frac{\partial }{\partial \nu }}+{\displaystyle \frac{{\alpha }_1}{2}}[{\displaystyle \frac{1}{2h{\tau }_{th}^{2h}}}{t}^{2h}\xi \nu +{\displaystyle \frac{1}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{\nu }^2]-C]p_{th}(\nu ,t),$$

(85)

where $C$ denotes the separation constant.

In the steady state, assuming ${\displaystyle \frac{\partial }{\partial t}}{p}_{th}(\xi ,t)=0$, we obtain

$$p_{th}^{st}(\xi ,t)=\mathit{exp}[{\displaystyle \frac{{\alpha }_1}{2\beta \nu }}[1-\gamma D_{2-2h}/\beta ][{\displaystyle \frac{1}{2h{\tau }_{th}^{2h}}}{t}^{2h}{\displaystyle \frac{{\xi }^2}{2}}\nu +{\displaystyle \frac{1}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}\xi {\nu }^2+C\xi ]].$$

(86)

We assume that $[\beta \nu +2\gamma D_{2-2h}{]}^{-1}\cong (\beta \nu {)}^{-1}[1-2\gamma D_{2-2h}/\beta ]$. To obtain the probability density for $\xi$ from $q_{th}(\xi ,t)\equiv r_{th}(\xi ,t)q_{th}^{st}(\xi ,t)$, we calculate the Fourier transform of the probability density after including terms up to order $1/{\tau }_{th}^2$:

$$p_{th}(\xi ,t)=q(\xi ,t)\mathit{exp}[{\displaystyle \frac{{\alpha }_1}{2\beta \nu }}[1-\gamma D_{2-2h}/\beta ][{\displaystyle \frac{1}{2h{\tau }_{th}^{2h}}}{t}^{2h}{\displaystyle \frac{{\xi }^2}{2}}\nu +{\displaystyle \frac{1}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}\xi {\nu }^2+CA\xi ]],$$

(87)

$$q_{th}(\xi ,t)=r_{th}(\xi ,t)\mathit{exp}[{\displaystyle \frac{{\alpha }_1}{2(\beta \nu {)}^2}}[1-2\gamma D_{2-2h}/\beta ][{\displaystyle \frac{1}{(2h{)}^2{\tau }_{th}^{2h}}}{t}^{2h-1}{\displaystyle \frac{{\xi }^3}{6}}\nu +{\displaystyle \frac{1}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{\displaystyle \frac{{\xi }^2}{2}}{\nu }^2]].$$

(88)

Considering the solutions as arbitrary functions of variable $t-\xi /(\beta \nu +2\gamma \nu D_{2-2h})$, the arbitrary function $r_{th}(\xi ,t)$ becomes $\Theta [t-\xi /(\beta \nu +2\gamma \nu D_{2-2h})]$. Thus,

$$p_{th}^{st}(\xi ,t)=r_{th}^{st}(\xi ,t)q_{th}^{st}(\xi ,t)p_{th}^{st}(\xi ,t)=\Theta [t-\xi /(\beta \nu +2\gamma \nu D_{2-2h})]q_{th}^{st}(\xi ,t)p_{th}^{st}(\xi ,t).$$

(89)

Using a similar method from Equation (86) to Equation (89) for $\nu$, we also get the Fourier transform of the probability density for the velocity as

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

(90)

Hence, by calculating Equations (89) and (90), we get the Fourier transform of the joint probability density as

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

(91)

Using the inverse Fourier transform, we get

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

(92)

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

(93)

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

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

(94)

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

Now, we derive the probability densities $p_{th}(x_{th},t)$ and $p_{th}(v_{th},t)$ in the long-time domain ($t>>{\tau }_{th}$). An approximate equation from Equation (84) can be written as

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th\xi }(\xi ,t)\cong -{\displaystyle \frac{{\alpha }_1}{2h{\tau }_{th}^{2h}}}{t}^{2h}\xi \nu p_{th\xi }(\xi ,t)-{\displaystyle \frac{{\alpha }_1}{(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{\nu }^2{p}_{th\xi }(\xi ,t).$$

(95)

The Fourier transform of the probability density $p_{th\xi }(\xi ,t)$ from Equation (95) is obtained as

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

(96)

We find the Fourier transform of the steady probability density $q_{th}^{st}(\xi ,t)$ for $\xi$, defined by $p_{th}(\xi ,t)\equiv q_{th\xi }(\xi ,t)p_{th\xi }(\xi ,t)$, as

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

(97)

Since the Fourier transform of probability density $q(\xi ,t)$ in the short-time domain is given by $q_{th}(\xi ,t)=r_{th}(\xi ,t)q_{th}^{st}(\xi ,t)$, we can write $p_{th}(\xi ,t)$ as

$$p_{th}(\xi ,t)=\Theta [t-\xi /(\beta \nu +\gamma \nu D_{2-2h})]q_{th\xi }^{st}(\xi ,t)p_{th}^{st}(\xi ,t),$$

(98)

where $r_{th}(\xi ,t)=\Theta [t-\xi /(\beta \nu +\gamma \nu D_{2-2h})]$. Applying Equation (85) for $\nu$ in the same manner as in Equations (95)–(98) of $p(\xi ,t)$ derived, we also obtain the Fourier transforms of the probability density for velocity $\nu$ as

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

(99)

By calculating Equations (98) and (99), we have

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

(100)

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

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

(101)

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

(102)

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

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

(103)

3.1.3. $p_{th}(x_{th},t)$ and $p_{th}(v_{th},t)$ with Thermal Noise ${\zeta }_{th}(t)$ for ${\tau }_{th}=0$

In this subsection, we derive $p_{th}(x_{th},t)$ and $p_{th}(v_{th},t)$ for the case ${\tau }_{th}=0$. The approximate equation from Equation (84) for $\xi$ is written as

$${\displaystyle \frac{\partial }{\partial t}}{p}_{th\xi }(\xi ,t)\cong -\beta \nu {\displaystyle \frac{\partial }{\partial \xi }}{p}_{th\xi }(\xi ,t)-\gamma \nu D_{2-2h}{\displaystyle \frac{\partial }{\partial \xi }}{p}_{th\xi }(\xi ,t)-{\displaystyle \frac{{\alpha }_1}{2(2h-1){\tau }_{th}^{2h-1}}}{t}^{2h-1}{\nu }^2{p}_{th\xi }(\xi ,t).$$

(104)

In the steady state, $p_{th}^{st}(\xi ,t)$ can be obtained from Equation (104) as

$$p_{th\xi }^{st}(\xi ,t)=\mathit{exp}[-{\displaystyle \frac{{\alpha }_1}{2\beta \nu }}[1-2\gamma D_{2-2h}/\beta ][{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{th}^{2h-1}}}\xi {\nu }^2]].$$

(105)

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

$$p_{th}(\xi ,t)=\Theta [t-\xi /(\beta \nu +\gamma \nu D_{2-2h})]p_{th\xi }^{st}(\xi ,t).$$

(106)

Using a similar procedure to that for $p_{th}(\xi ,t)$, the Fourier transform of the probability density $p_{th}(\nu ,t)$ becomes

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

(107)

We can get $p_{th}(\xi ,\nu ,t)$ by calculating Equations (106) and (107) as

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

(108)

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

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

(109)

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

(110)

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

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

(111)

3.2. $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ with Active Noise ${\zeta }_{ac}(t)$

3.2.1. $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ with Active Noise ${\zeta }_{ac}(t)$ in the Short-Time Domain

In this subsection, we obtain the solutions of the probability densities $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ in the short-time domain $t<<{\tau }_A$. To find the particular solutions for $\xi$ and $\nu$ by separating variables from Equation (83), the two equations for displacement and velocity are given by

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac}(\xi ,t)=[-\beta \nu {\displaystyle \frac{\partial }{\partial \xi }}-\gamma \nu D_{2-2h}{\displaystyle \frac{\partial }{\partial \xi }}]p_{ac}(\xi ,t)+{\displaystyle \frac{{\alpha }_2}{2}}[b_2(t)\xi \nu -a_2(t){\nu }^2+E]p_{ac}(\xi ,t),$$

(112)

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac}(\nu ,t)=[\xi {\displaystyle \frac{\partial }{\partial \nu }}+{\displaystyle \frac{{\alpha }_2}{2}}[b_2(t)\xi \nu -a_2(t)]{\nu }^2-E]p_{ac}(\nu ,t),$$

(113)

where E denotes the separation constant. By setting ${\displaystyle \frac{\partial }{\partial t}}{p}_{ac}(\xi ,t)=0$ in the steady state, we get $p_{ac}^{st}(\xi ,t)$ as

$$p_{ac}^{st}(\xi ,t)=\mathit{exp}[{\displaystyle \frac{{\alpha }_2}{2\beta \nu }}[1-2\gamma D_{2-2h}/\beta ][{\displaystyle \frac{b_2(t)}{2}}\nu {\xi }^2-a_2(t){\nu }^2\xi +E\xi ]].$$

(114)

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

$$p_{ac}(\xi ,t)=q_{ac}(\xi ,t)\mathit{exp}[{\displaystyle \frac{{\alpha }_2}{2\beta \nu }}[1-2\gamma D_{2-2h}/\beta ][{\displaystyle \frac{b_2(t)}{2}}\nu {\xi }^2-a_2(t){\nu }^2\xi +E\xi ]].$$

(115)

$$q_{ac}(\xi ,t)=r_{ac}(\xi ,t)\mathit{exp}[-{\displaystyle \frac{{\alpha }_2}{2(\beta \nu {)}^2}}[1-2\gamma D_{2-2h}/\beta ][{\displaystyle \frac{b_2^{\prime }(t)}{6}}\nu {\xi }^3-{\displaystyle \frac{a_2^{\prime }(t)}{2}}{\nu }^2{\xi }^2]].$$

(116)

$$r_{ac}(\xi ,t)=s_{ac}(\xi ,t)\mathit{exp}[{\displaystyle \frac{{\alpha }_2}{2(\beta \nu {)}^3}}[1-2\gamma D_{2-2h}/\beta ][{\displaystyle \frac{b_2^{″}(t)}{24}}\nu {\xi }^4-{\displaystyle \frac{a_2^{″}(t)}{6}}{\nu }^2{\xi }^3]].$$

(117)

$$s_{ac}\left(\xi ,t\right)=t_{ac}(\xi ,t)\mathit{exp}[-{\displaystyle \frac{{\alpha }_2}{2(\beta \nu {)}^4}}[1-2\gamma D_{2-2H}/\beta ]{\displaystyle \frac{b_2^{‴}(t)}{120}}\nu {\xi }^5].$$

(118)

Taking the solutions as arbitrary functions of variable $t-\xi /(\beta \nu +\gamma \nu D_{2-2h})$, the arbitrary function $t_{ac}(\xi ,t)$ becomes $\Theta [t-\xi /(\beta \nu +\gamma \nu D_{2-2h})]$. As a result, we find that

$$p_{ac}(\xi ,t)=t_{ac}(\xi ,t)s_{ac}^{st}(\xi ,t)r_{ac}^{st}(\xi ,t)q_{ac}^{st}(\xi ,t)p_{ac}^{st}(\xi ,t)=\Theta [t-\xi /(\beta \nu +\gamma \nu D_{2-2h})]s_{ac}^{st}(\xi ,t)r_{ac}^{st}(\xi ,t)q_{ac}^{st}(\xi ,t)p_{ac}^{st}(\xi ,t).$$

(119)

Using a similar method as Equations (114)–(119), we also get the Fourier transform of the probability density for the velocity as

$$p_{ac}(\nu ,t)=t_{ac}(\nu ,t)s_{ac}^{st}(\nu ,t)r_{ac}^{st}(\nu ,t)q_{ac}^{st}(\nu ,t)p_{ac}^{st}(\nu ,t)=\Theta [t+\nu /\xi ]s_{ac}^{st}(\nu ,t)r_{ac}^{st}(\nu ,t){q^{st}}_{ac}^{st}(\nu ,t)p_{ac}^{st}(\nu ,t).$$

(120)

Therefore, by calculating Equations (119) and (120), we get the Fourier transform of the joint probability density as

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

(121)

Using the inverse Fourier transform, we get

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

(122)

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

(123)

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

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

(124)

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

We now find the probability densities $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ in the long-time domain $t>>{\tau }_{ac}$. An approximate equation from Equation (112) can be written as

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac\xi }(\xi ,t)\cong {\displaystyle \frac{{\alpha }_2}{2}}[b_2(t)\xi \nu -a_2(t){\nu }^2]p_{ac\xi }(\xi ,t).$$

(125)

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

$$p_{ac\xi }(\xi ,t)\cong \mathit{exp}[{\displaystyle \frac{{\alpha }_2}{2}}{\int }_0^{t}dt[b_2(t)\xi \nu -a_2(t){\nu }^2]].$$

(126)

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

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

(127)

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

$$p_{ac}(\xi ,t)=r_{ac}(\xi ,t)q_{ac\xi }^{st}(\xi ,t)p_{ac}^{st}(\xi ,t)=\Theta [t-\xi /(\beta \nu +\gamma \nu D_{2-2h})]q_{ac\xi }^{st}(\xi ,t)p_{ac}^{st}(\xi ,t),$$

(128)

where $r_{ac}(\xi ,t)=\Theta [t-\xi /(\beta \nu +\gamma \nu D_{2-2H})]$ and $p_{ac}^{st}(\xi ,t)$ is equal to Equation (114). Applying Equation (113) for $\nu$ to a similar method from Equation (125) to Equation (128) for $p_{ac}(\xi ,t)$ derived, we also get the Fourier transforms of the probability density for velocity $\nu$ as

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

(129)

By calculating Equations (128) and (129), we have

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

(130)

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

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

(131)

$$p_{ac}(v,t)=[\pi \alpha \gamma t^{2h+1}{]}^{-1/2}\mathit{exp}[-{\displaystyle \frac{1}{\alpha \gamma t^{2h+1}}}{v}_{ac}^2].$$

(132)

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

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

(133)

3.2.3. $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ with Active Noise ${\zeta }_{ac}(t)$ for ${\tau }_{ac}=0$

In this subsection, we derive $p_{ac}(x_{ac},t)$ and $p_{ac}(v_{ac},t)$ in the time domain ${\tau }_{ac}=0$. The approximate equation from Equation (112) for $\xi$ is written as

$${\displaystyle \frac{\partial }{\partial t}}{p}_{ac\xi }(\xi ,t)\cong -\beta \nu {\displaystyle \frac{\partial }{\partial \xi }}{p}_{ac\xi }(\xi ,t)-\gamma \nu D_{2-2h}{\displaystyle \frac{\partial }{\partial \xi }}{p}_{ac\xi }(\xi ,t)-{\displaystyle \frac{{\alpha }_2}{2}}{a}_2(t){\nu }^2{p}_{ac\xi }(\xi ,t).$$

(134)

In the steady state, we can calculate $p_{ac}^{st}(\xi ,t)$ from Equation (134) as

$$p_{ac\xi }^{st}(\xi ,t)=\mathit{exp}[-{\displaystyle \frac{{\alpha }_2}{2\beta \nu }}[1-\gamma D_{2-2h}/\beta ]{\nu }^2\xi ].$$

(135)

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

$$p_{ac}(\xi ,t)=\Theta [t-\xi /(\beta \nu +\gamma \nu D_{2-2h})]p_{ac\varsigma }^{st}(\xi ,t).$$

(136)

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

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

(137)

We can calculate $p(\xi ,\nu ,t)$ by calculating Equations (136) and (137) as

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

(138)

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

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

(139)

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

(140)

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

$$<x_{ac}^2(t)>={\displaystyle \frac{1}{3}}{\alpha }_2{t}^3,<v_{ac}^2(t)>=2{\alpha }_{2}t.$$

(141)

4. Statistical Quantities

In this section, we calculate statistical quantities, including the non-Gaussian parameter for displacement and velocity, the correlation coefficient, the entropy, the combined entropy, and the moments from the moment equations. The moment equations for an active particle with a viscoelastic memory effect, derived from Equations (8) and (9), are expressed as

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

(142)

$${\displaystyle \frac{d{\mu }_{m,n}}{dt}}=-m{\mu }_{m-1,n+1}+n{\gamma }_{ac}{D}_{2-2h}{\mu }_{m+1,n-1}-mn{\alpha }_2{b}_2(t){\mu }_{m-1,n-1}+n(n-1){\alpha }_2{a}_2(t){\mu }_{m,n-2}.$$

(143)

where ${\mu }_{m,n}={\int }_{-\infty }^{+\infty }dx{\int }_{-\infty }^{+\infty }dv{x}^m{v}^{n}P(x,v,t)$. For an active particle in an optical trap, the moment equations from Equations (78) and (79) are

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

(144)

$${\displaystyle \frac{d{\mu }_{m,n}}{dt}}=-m{\mu }_{m-1,n+1}+(n{\gamma }_{ac}{D}_{2-2h}+\beta n){\mu }_{m+1,n-1}+-mn{\alpha }_2{b}_2(t){\mu }_{m-1,n-1}+n(n-1){\alpha }_2{a}_2(t){\mu }_{m,n-2}.$$

(145)

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

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

(146)

The combined entropy is defined by

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

(147)

Entropy provides a numerical measure of the uncertainty or information content associated with a probability distribution; as the probability of a particular value increases and that of other values decreases, the entropy decreases.

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

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

(148)

The non-Gaussian parameter quantifies the degree of heavy tails in the probability distribution of a real-valued variable, providing insight into the deviation from Gaussian behavior. The correlation coefficient is defined as

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

(149)

The correlation coefficient describes the strength and direction of the linear relationship between two variables, $x_i(t)$ and $v_i(t)$. Here, we assume that an active 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 1 and Table 2, respectively, summarize the values of the entropy, non-Gaussian parameter, and correlation coefficient for the joint probability density with thermal equilibrium noise ${\zeta }_{th}(t)$ and active noise ${\zeta }_{ac}(t)$ in the three-time domains. Table 3 and Table 4 show the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density for an active Brownian particle with a harmonic trap, ${\zeta }_{th}(t)$, and ${\zeta }_{ac}(t)$ in the three-time domains.

Table 1. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density with ${\zeta }_{th}(t)$ in the three-time domains.

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

Table 2. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density with ${\zeta }_{ac}(t)$ in the three-time domains.

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

Table 3. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density with optical trap and ${\zeta }_{th}(t)$ in the three-time domains.

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

Table 4. Values of the non-Gaussian parameter, the correlation coefficient, the moment, the entropy, and the combined entropy for the joint probability density with an optical trap and ${\zeta }_{ac}(t)$ in the three-time domains.

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

5. Conclusions

In conclusion, we studied the nonequilibrium effects of an active Brownian particle in a viscoelastic medium, subjected to thermal noise ${\zeta }_{th}(t)$ and active noise ${\zeta }_{ac}(t)$, using the fractional generalized Langevin equation. Thermal noise satisfies the fluctuation–dissipation theorem, whereas active noise is modeled as an active Ornstein–Uhlenbeck process. We derived the Fokker–Planck equation for the joint probability density and obtained its solution via double Fourier transforms across three-time regimes.

The main work and conclusions are as follows:

(1)

In the fractional generalized Langevin equation with a viscoelastic memory effect and thermal noise for $1/2<h<1$ (where $h$ is the Hurst exponent), the mean squared displacement $⟨x_{\mathrm{t}\mathrm{h}}^2(t)⟩$ scales with $t^{2h+1}$ in the limits of $t\ll {\tau }_{\mathrm{a}\mathrm{c}}$ and $t\gg {\tau }_{\mathrm{a}\mathrm{c}}$ and for ${\tau }_{\mathrm{a}\mathrm{c}}=0$, exhibiting super-diffusive behavior, while the mean squared velocity $<v_{th}^2(t)>$ scales with $t^{2h}$. In particular, for $h\to 1/2$, the mean squared displacement scales with $t^2$ in the limit of $t<<\tau$, while the mean squared velocity $<v_{th}^2(t)>$ behaves as $t$, consistent with simulated results [33,34]. The mean squared displacement $⟨x_{\mathrm{a}\mathrm{c}}^2\left(t\right)⟩$ for the fractional generalized Langevin equation scales with $\sim t^4$ in the short-time limit $t\ll {\tau }_{\mathrm{a}\mathrm{c}}$, consistent with the simulation result reported in Ref. [26].

(2)

As we solve the fractional Fokker–Planck equation with a harmonic force and thermal noise, the mean squared displacement $<x_{ac}^2(t)>$ behaves as $t^{2h+2}$ in $t<<\tau$ and $t>>\tau$. The mean squared velocity $<v_{ac}^2(t)>$ for $\tau =0$ exhibits normal diffusion from the fractional generalized Langevin equation with a harmonic force and active noise.

(3)

From statistical quantities, the entropy for the joint probability density with ${\zeta }_{th}(t)$ as $h\to 1/2$ is the same value as that for the joint probability density with ${\zeta }_{ac}(t)$ in $t>>\tau$ and $\tau =0$.

(4)

The combined entropy for the joint probability density with ${\zeta }_{th}(t)$ is larger than that for the joint probability density with ${\zeta }_{ac}(t)$ in each time domain. In Table 3, the combined entropy of displacement and velocity follows a scaling behavior of $\ln t^{(4h+4)}$ in the time regime $t\gg \tau$, where the scaling exponent $h$ lies in the range $1/2<h<1$. Since the combined entropy maintains functional dependence, $\ln t^{(4h+2)}$, in the other regimes ($t\gg \tau$ and $\tau =0$), comparison among these regimes indicates that the system in the $t\gg \tau$ region reaches equilibrium more rapidly. This result suggests that the long-time dynamics are dominated by a faster relaxation toward equilibrium despite the identical entropy scaling form.

(5)

For $h\to 1$, the displacement non-Gaussian parameter $K_x$ for the joint probability density with ${\zeta }_{ac}(t)$ is the same value proportional to time $t^{-8}$ as that for the joint probability density with ${\zeta }_{ac}(t)$ in $t<<\tau$. The velocity non-Gaussian parameter $K_v$ for the joint probability density with ${\zeta }_{ac}(t)$ is also the same value proportional to time $t^{-3}$ as that for the joint probability density with ${\zeta }_{ac}(t)$ for $h\to 1/2$ in $\tau =0$. In Table 1, Table 2, Table 3 and Table 4, as the non-Gaussian parameter increases, the distribution exhibits heavier tails and stronger deviations from Gaussian behavior. Such behavior typically appears in the time regime $t\ll \tau$ for $x_{th}(t)$, $v_{th}(t)$ and $x_{ac}(t)$, $v_{ac}(t)$. In contrast, for $t\gg \tau$ and at $\tau =0$, the non-Gaussian parameters of $x_{th}(t)$, $v_{th}\left(t\right),$ and $x_{ac}(t)$, $v_{ac}(t)$ correspond to flatter central regions and shorter tails, indicating light-tailed distributions.

(6)

Moments for an active Brownian particle in a harmonic trap with thermal noise in the limits of $t>>\tau$ and for $\tau =0$ scale with $t^{4h+2}$, consistent with our calculation result.

The formal approach of our study shares similarities with analytical frameworks, but extends them in several important aspects. We particularly addressed the roles of active noise, the non-Gaussian parameter, and entropy aspects that have not been sufficiently analyzed in previous studies. Moreover, we hope that the present results provide new physical insights into the mechanisms of transition kinetics, fractional diffusion, and the suppression of rare events. Future work combining theoretical [36,37,38,39], computational [40,41,42,43], and experimental approaches can clarify how inertial effects, finite correlation times, and non-Gaussian statistics influence transport and stability in active matter. The methodology developed here offers a versatile framework applicable to generalized stochastic systems, anomalous diffusion, and self-propelled particle dynamics in complex environments, such as shear flows [44,45]. Interdisciplinary extensions of this work are expected to deepen our understanding of noise-driven processes in fractional active and nonequilibrium systems [46,47,48,49].

Appendix A.1. Joint Probability Equation

The joint probability density function is

$$p_i(x_i,v_i,t)=⟨\delta (x_i-x_i(t)) \delta (v_i-v_i(t))⟩.$$

(A3)

Taking the time derivative of Equation (A3) and using Equation (A1), we obtain

$${\displaystyle \frac{\partial p_i}{\partial t}}=-{\displaystyle \frac{\partial }{\partial x_i}}⟨{\dot{x}}_i \delta (x_i-x_i(t)) \delta (v_i-v_i(t))⟩-{\displaystyle \frac{\partial }{\partial v_i}}⟨{\dot{v}}_i \delta (x_i-x_i(t)) \delta (v_i-v_i(t))⟩.$$

(A4)

Substituting the Langevin equations gives

$${\displaystyle \frac{\partial p_i}{\partial t}}=-v_i{\displaystyle \frac{\partial p_i}{\partial x_i}}+{\gamma }_i{D}_{(2-2h_i)}{\displaystyle \frac{\partial }{\partial v_i}} [x_i{p}_i]-{\displaystyle \frac{\partial }{\partial v_i}}⟨{\zeta }_i(t) \delta (x_i-x_i(t)) \delta (v_i-v_i(t))⟩,$$

(A5)

where $D_{(2-2h_i)}=d^{2-2h}/d{t}^{2-2h}$.

Appendix A.2. Evaluation of the Noise Term

To evaluate the last term in Equation (A5), we use Novikov’s theorem for Gaussian colored noise:

$$⟨{\zeta }_i(t)F[{\zeta }_i]⟩={\int }_0^{t}ds ⟨{\zeta }_i(t){\zeta }_i(s)⟩⟨{\displaystyle \frac{\delta F[{\zeta }_i]}{\delta {\zeta }_i(s)}}⟩.$$

(A6)

Choosing $F=\delta (x_i-x_i(t))\delta (v_i-v_i(t))$ and differentiating functionally yields two contributions: one proportional to ${\partial }^2/(\partial v_i \partial x_i)$ and another to ${\partial }^2/(\partial v_i^2)$. Carrying out the integration with the correlation Equation (A2) gives

$${\displaystyle \frac{\partial }{\partial v_i}}⟨{\zeta }_i(t) \delta (x_i)\delta (v_i)⟩=-{\alpha }_i[c_{1i}(t){\displaystyle \frac{{\partial }^2{p}_i}{\partial v_i \partial x_i}}+c_{2i}(t){\displaystyle \frac{{\partial }^2{p}_i}{\partial v_i^2}}],$$

(A7)

where the time-dependent coefficients are

$$c_{1i}(t)={\displaystyle \frac{t^{2h_i}}{2h_i {\tau }_i^{2h_i}}},c_{2i}(t)={\displaystyle \frac{t^{2h_i-1}}{(2h_i-1){\tau }_i^{2h_i-1}}}.$$

(A8)

Equation (A7) corresponds to Equation (A1) in the original text for $i=th$ and to Equation (A2) for $i=ac$.

Appendix A.3. Final Form of the Fokker–Planck Equation

Substituting Equation (A7) into Equation (A5) yields the fractional Fokker–Planck equation for the thermal case:

$${\displaystyle \frac{\partial p_{\mathrm{t}\mathrm{h}}}{\partial t}}=[-v_{\mathrm{t}\mathrm{h}}{\displaystyle \frac{\partial }{\partial x_{\mathrm{t}\mathrm{h}}}}+{\gamma }_{\mathrm{t}\mathrm{h}}{D}_{(2-2h)}{\displaystyle \frac{\partial }{\partial v_{\mathrm{t}\mathrm{h}}}}{x}_{\mathrm{t}\mathrm{h}}]p_{\mathrm{t}\mathrm{h}}+{\alpha }_1[{\displaystyle \frac{t^{2h}}{2h {\tau }_{\mathrm{t}\mathrm{h}}^{2h}}}{\displaystyle \frac{{\partial }^2}{\partial v_{\mathrm{t}\mathrm{h}}\partial x_{\mathrm{t}\mathrm{h}}}}+{\displaystyle \frac{t^{2h-1}}{(2h-1){\tau }_{\mathrm{t}\mathrm{h}}^{2h-1}}}{\displaystyle \frac{{\partial }^2}{\partial v_{\mathrm{t}\mathrm{h}}^2}}]p_{\mathrm{t}\mathrm{h}},$$

(A9)

This is identical to Equation (8) in the main text. By applying the same procedure to the active case, we have the following:

$${\displaystyle \frac{\partial p_{\mathrm{a}\mathrm{c}}}{\partial t}}=[-v_{\mathrm{a}\mathrm{c}}{\displaystyle \frac{\partial }{\partial x_{\mathrm{a}\mathrm{c}}}}+{\gamma }_{\mathrm{a}\mathrm{c}}{D}_{(2-2h)}{\displaystyle \frac{\partial }{\partial v_{\mathrm{a}\mathrm{c}}}}{x}_{\mathrm{a}\mathrm{c}}]p_{\mathrm{a}\mathrm{c}}+{\alpha }_2 [-b_2(t){\displaystyle \frac{{\partial }^2}{\partial x_{\mathrm{a}\mathrm{c}}\partial v_{\mathrm{a}\mathrm{c}}}}+a_2(t){\displaystyle \frac{{\partial }^2}{\partial v_{\mathrm{a}\mathrm{c}}^2}}]p_{\mathrm{a}\mathrm{c}},$$

(A10)

This reproduces Equation (9). In the limit $h\to 1/2$, $a_2,b_2$ and the fractional coefficients reduce to constants, and Equations (A9) and (A10) recover the ordinary (Markovian) Fokker–Planck equation for white noise.