Modeling Anomalous Transport of Cosmic Rays in the Heliosphere Using a Fractional Fokker–Planck Equation

Abstract

Cosmic rays exhibit anomalous diffusion behaviors in the heliospheric environment that cannot be adequately described by classical diffusion models. In this paper, we develop a theoretical framework employing a fractional Fokker–Planck equation to model the anomalous transport of cosmic rays. This approach accounts for the observed non-Gaussian distributions, long-range correlations and memory effects in cosmic ray fluxes. We derive analytical solutions using the Adomian Decomposition Method and express them in terms of Mittag-Leffler functions and Lévy stable distributions. The model parameters, including the fractional orders $\alpha$ and $\mu$ and the entropic index q, are estimated by a short comparison between theoretical predictions and observational data from cosmic ray experiments. Our findings suggest that the integration of fractional calculus and non-extensive statistics can be employed for describing the cosmic ray propagation and the anomalous diffusion observed in the heliosphere.

1. Introduction

The study of cosmic rays is important for understanding a wide range of astrophysical processes, including solar modulation, galactic dynamics, and particle acceleration mechanisms [1,2]. Cosmic rays, consisting of high-energy charged particles, propagate through the heliosphere and interact with magnetic fields and plasma irregularities, leading to complex transport behaviors [3].

Classical models of cosmic ray transport are based on the Parker transport equation, which describes the diffusion of particles under the influence of convection, drift, and energy changes [4]. This equation assumes that the diffusion process is Gaussian and Markovian, characterized by a linear relation between the mean square displacement and time [5]. However, observations have indicated that cosmic ray diffusion is often anomalous, exhibiting non-Gaussian features and long-range temporal and spatial correlations [6,7].

Anomalous diffusion is characterized by a mean square displacement that scales nonlinearly with time, $\langle x^2\left(t\right)\rangle \propto t^{\alpha }$, where $\alpha \ne 1$ [8]. Subdiffusion ($\alpha <1$) and superdiffusion ($\alpha >1$) have both been observed in cosmic ray transport [9]. These phenomena cannot be adequately described by classical diffusion equations and require a more generalized framework.

Fractional calculus provides powerful tools to model anomalous diffusion processes by extending the order of differentiation to non-integer values [10,11]. The fractional Fokker–Planck equation (FFPE) incorporates fractional derivatives to capture memory effects and nonlocal behaviors inherent in anomalous diffusion [8]. The FFPE has been successfully applied in various fields, such as viscoelastic materials [12], biological systems [13], and finance [14].

In parallel, Tsallis statistics introduce a generalized framework for statistical mechanics applicable to non-extensive systems with long-range interactions and fractal structures [15]. The Tsallis entropy introduces an entropic index q that quantifies the degree of non-extensivity, reducing to the classical Boltzmann–Gibbs entropy when $q\to 1$ [16]. This framework has been applied to plasma physics [17], turbulence [18], and cosmic ray spectra [19]. Previous studies have applied either fractional diffusion models [6] or non-extensive statistics separately [19]. In addition, the work in [20] provides an overview of fractional diffusion equations and their applications to anomalous diffusion processes and we particularly use the Adomian Decomposition Method, which is a reliable tool for solving fractional diffusion equations as indicated for more general cases in [20].

In this paper, we consider a generalized FFPE of the form:

$${\displaystyle \frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A\left(x\right)P(x,t)\right]+D_q{\displaystyle \frac{{\partial }^{2q}P(x,t)}{\partial x^{2q}}},$$

(1)

where ${\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}$ is the Caputo fractional derivative of order $\alpha$ ($0<\alpha \le 1$), $A\left(x\right)$ represents the drift term, $D_q$ is the generalized diffusion coefficient, and ${\displaystyle \frac{{\partial }^{2q}}{\partial x^{2q}}}$ is the fractional spatial derivative of order $2q$. The entropic index q captures the degree of non-extensivity in the system.

From a complementary perspective, anomalous diffusion has also been investigated using generalized Langevin equations (GLEs), which provide a direct and transparent route to compute essential correlation functions, such as the mean-square displacement (MSD) and velocity autocorrelation function (VACF). This framework inherently incorporates memory effects and nonlocal correlations. Importantly, the GLE and the fractional Fokker–Planck equation (FFPE) are known to be equivalent representations, with one formulation mapping onto the other. Thus, one may consider that the experience gained from the GLE literature, where researchers often draw inspiration from experimental setups and data fitting strategies, can guide and refine the interpretation of results derived from the FFPE approach. In this manner, the methodologies developed in GLE-based studies [21,22,23,24,25] can serve as valuable benchmarks, informing parameter estimation and improving the comparison between theoretical predictions and empirical observations in the context of anomalous cosmic ray transport.

Our model is intended to account for the observed heavy-tailed distributions and long-range correlations in cosmic ray intensity measurements [26].

Our objectives in this paper are threefold:

• To derive the fractional Fokker–Planck equation for cosmic ray transport within the Tsallis statistical framework.
• To obtain analytical solutions for the cosmic ray probability density function $P(x,t)$ under appropriate initial and boundary conditions.
• To make some a priori estimations based on compiled observational data.

The paper is organized as follows: In Section 2, we present the theoretical foundations of fractional calculus and Tsallis statistics as applied to cosmic ray transport. Section 3 details the obtained analytical solutions. In Section 4, we discuss the implications of our results and compare them with experimental observations. Finally, Section 5 summarizes our findings and outlines potential directions for future research.

2. Theoretical Framework

Anomalous diffusion is characterized by a nonlinear scaling of the mean square displacement (MSD) with time, typically expressed as [8]:

$$\langle x^2\left(t\right)\rangle \propto t^{\alpha },$$

(2)

where $\alpha \ne 1$ is the anomalous diffusion exponent. For $\alpha <1$, the process is subdiffusive, indicating that the particles spread slower than in normal diffusion. For $\alpha >1$, the process is superdiffusive, with particles spreading faster than in normal diffusion.

Fractional calculus provides a natural mathematical framework to describe such anomalous diffusion processes. Indeed, by generalizing the order of differentiation and integration to non-integer values, fractional derivatives can capture memory effects and long-range correlations inherent in the system [10,11].

The Caputo fractional derivative of order $\alpha$ (with $n-1<\alpha \le n$, $n\in \mathbb{N}$) is defined as [11]:

$${\displaystyle \frac{{\partial }^{\alpha }f\left(t\right)}{\partial t^{\alpha }}}={\displaystyle \frac{1}{\mathsf{\Gamma }(n-\alpha )}}{\int }_0^t{\displaystyle \frac{f^{\left(n\right)}\left(\tau \right)}{{(t-\tau )}^{\alpha -n+1}}}d\tau ,$$

(3)

where $\mathsf{\Gamma }(·)$ is the Gamma function, and $f^{\left(n\right)}\left(\tau \right)$ is the n-th derivative of f with respect to $\tau$.

Similarly, the Riesz fractional derivative provides a symmetric fractional derivative in space, suitable for modeling spatial anomalous diffusion [8,10].

2.1. Tsallis Non-Extensive Statistics

Classical statistical mechanics, based on the Boltzmann–Gibbs (BG) entropy, assumes that systems are extensive, meaning that the total entropy is proportional to the system size. However, many physical systems exhibit non-extensive behavior due to long-range interactions, fractal structures, or memory effects [15,16].

Tsallis introduced a generalized entropy measure, known as Tsallis entropy, defined as [15]:

$$S_q=k{\displaystyle \frac{1-{\sum }_i{p}_i^q}{q-1}},$$

(4)

where $p_i$ are the probabilities of the microstates, q is the entropic index characterizing the degree of non-extensivity, and k is the Boltzmann constant. In the limit $q\to 1$, Tsallis entropy reduces to the BG entropy.

Maximizing the Tsallis entropy under appropriate constraints leads to the q-Gaussian distribution, a generalization of the normal Gaussian distribution, given by [16]:

$$P\left(x\right)={\displaystyle \frac{\sqrt{\beta }}{C_q}}{[1-(1-q)\beta x^2]}^{{\displaystyle \frac{1}{1-q}}},$$

(5)

where $\beta$ is a parameter related to the width of the distribution, and $C_q$ is a normalization constant.

2.2. Fractional Fokker–Planck Equation

To model the anomalous transport of cosmic rays, we seek a generalized Fokker–Planck equation that incorporates both fractional derivatives and the non-extensive parameter q. We begin with the classical Fokker–Planck equation (FPE):

$${\displaystyle \frac{\partial P(x,t)}{\partial t}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A\left(x\right)P(x,t)\right]+D{\displaystyle \frac{{\partial }^{2}P(x,t)}{\partial x^2}},$$

(6)

where $P(x,t)$ is the probability density function, $A\left(x\right)$ is the drift term, and D is the diffusion coefficient.

To account for anomalous temporal diffusion, we replace the first-order time derivative with a fractional derivative of order $\alpha$:

$${\displaystyle \frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A\left(x\right)P(x,t)\right]+D{\displaystyle \frac{{\partial }^{2}P(x,t)}{\partial x^2}}.$$

(7)

Similarly, to incorporate spatial anomalous diffusion and non-extensive effects, we generalize the second-order spatial derivative using the fractional derivative of order $\mu$:

$${\displaystyle \frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A\left(x\right)P(x,t)\right]+D_q{\displaystyle \frac{{\partial }^{\mu }P(x,t)}{{\partial \left|x\right|}^{\mu }}},$$

(8)

where ${\displaystyle \frac{{\partial }^{\mu }}{{\partial \left|x\right|}^{\mu }}}$ denotes the Riesz fractional derivative, and $D_q$ is the generalized diffusion coefficient dependent on the entropic index q.

The Riesz fractional derivative is defined as [8]:

$${\displaystyle \frac{{\partial }^{\mu }f\left(x\right)}{{\partial \left|x\right|}^{\mu }}}=-{\displaystyle \frac{1}{2cos\left(\frac{\pi \mu }{2}\right)}}\left({}_{-\infty }D_{\mathrm{x}}^{\mu }f\left(x\right)+{}_{x}D_{\infty }^{\mu }f\left(x\right)\right),$$

(9)

where ${}_{-\infty }D_{\mathrm{x}}^{\mu }$ and ${}_{x}D_{\infty }^{\mu }$ are the left and right Riemann–Liouville fractional derivatives, respectively.

To link the fractional spatial derivative order $\mu$ with the entropic index q, we utilize the relationship between the q-Gaussian distribution and Lévy flights [6,27]. In the context of Tsallis statistics, the q-Gaussian distribution corresponds to the stable Lévy distributions for $q<1$ (superdiffusion) and $q>1$ (subdiffusion).

The Lévy index $\mu$ is related to q by [27]:

$$\mu ={\displaystyle \frac{2}{3-q}}.$$

(10)

Thus, the generalized fractional Fokker–Planck equation becomes:

$${\displaystyle \frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A\left(x\right)P(x,t)\right]+D_q{\displaystyle \frac{{\partial }^{\mu }P(x,t)}{{\partial \left|x\right|}^{\mu }}},$$

(11)

where $\alpha$ and $\mu$ are fractional orders representing temporal and spatial anomalous diffusion, respectively, and $D_q$ is a diffusion coefficient that may depend on q.

2.3. Adomian Decomposition Method

The ADM decomposes the solution $P(x,t)$ into an infinite series:

$$P(x,t)=\sum _{n=0}^{\infty }{P}_n(x,t),$$

(12)

where $P_n(x,t)$ are the components of the solution to be determined recursively.

Applying the ADM to Equation (11), we write:

$${\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}\left(\sum _{n=0}^{\infty }{P}_n\right)=-{\displaystyle \frac{\partial }{\partial x}}\left[A\left(x\right)\sum _{n=0}^{\infty }{P}_n\right]+D_q{\displaystyle \frac{{\partial }^{\mu }}{{\partial \left|x\right|}^{\mu }}}\left(\sum _{n=0}^{\infty }{P}_n\right).$$

(13)

We define the initial condition as:

$$P(x,0)=P_0\left(x\right),$$

(14)

which is often chosen to be the q-Gaussian distribution (5) to reflect the non-extensive nature of the system.

The recursive relations for $P_n$ are then obtained by equating terms of the same order and applying the inverse fractional derivative operators.

2.4. Boundary and Initial Conditions

To solve Equation (11), appropriate boundary and initial conditions must be specified. For cosmic ray transport in the heliosphere, it is reasonable to assume that $P(x,t)$ vanishes as $\left|x\right|\to \infty$, due to the finite extent of the heliosphere [3].

The initial condition $P(x,0)$ represents the initial spatial distribution of cosmic rays, which may be modeled as a q-Gaussian distribution centered around the source region.

2.5. Normalization and Physical Constraints

The probability density function $P(x,t)$ must be normalized at all times:

$${\int }_{-\infty }^{\infty }P(x,t)dx=1.$$

(15)

Additionally, $P(x,t)$ must remain non-negative for all x and t:

$$P(x,t)\ge 0,\forall x,t.$$

(16)

These constraints must be maintained throughout the solution process.

In the next section, we will proceed to derive analytical solutions to this equation using the Adomian Decomposition Method, and explore the implications of the model parameters $\alpha$ and q on cosmic ray transport.

3. Solutions to the Fractional Fokker–Planck Equation

In this section, we derive analytical solutions to the generalized fractional Fokker–Planck equation (FFPE) presented in Equation (11). We employ the Adomian Decomposition Method (ADM) to systematically construct the solution as an infinite series. We begin by restating the FFPE for clarity:

$${\displaystyle \frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}}=-{\displaystyle \frac{\partial }{\partial x}}\left[A\left(x\right)P(x,t)\right]+D_q{\displaystyle \frac{{\partial }^{\mu }P(x,t)}{{\partial \left|x\right|}^{\mu }}},$$

(17)

where:

• ${\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}$ is the Caputo fractional derivative of order $\alpha$ ($0<\alpha \le 1$).
• $A\left(x\right)$ is the drift term, assumed to be a known function of x.
• $D_q$ is the generalized diffusion coefficient, possibly dependent on the entropic index q.
• ${\displaystyle \frac{{\partial }^{\mu }}{{\partial \left|x\right|}^{\mu }}}$ is the Riesz fractional derivative of order $\mu$ ($0<\mu \le 2$), related to the entropic index q via $\mu ={\displaystyle \frac{2}{3-q}}$.

To apply the ADM, we first express the FFPE in operator form. Let us define the following operators:

$$L_t^{\alpha }P(x,t)={\displaystyle \frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}},$$

(18)

$$L_x^{\mu }P(x,t)=D_q{\displaystyle \frac{{\partial }^{\mu }P(x,t)}{{\partial \left|x\right|}^{\mu }}},$$

(19)

and the drift operator:

$$N_{x}P(x,t)=-{\displaystyle \frac{\partial }{\partial x}}\left[A\left(x\right)P(x,t)\right].$$

(20)

With these definitions, Equation (17) becomes:

$$L_t^{\alpha }P(x,t)=N_{x}P(x,t)+L_x^{\mu }P(x,t).$$

(21)

We now apply the inverse operator $L_t^{-\alpha }$, which is the fractional integral of order $\alpha$, defined by:

$$L_t^{-\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}f\left(\tau \right)d\tau .$$

(22)

Applying $L_t^{-\alpha }$ to both sides of Equation (21), we obtain:

$$P(x,t)=P(x,0)+L_t^{-\alpha }\left[N_{x}P(x,t)+L_x^{\mu }P(x,t)\right],$$

(23)

where $P(x,0)$ is the initial condition.

According to the ADM, we decompose $P(x,t)$ into an infinite series:

$$P(x,t)=\sum _{n=0}^{\infty }{P}_n(x,t).$$

(24)

Similarly, the nonlinear terms involving $P(x,t)$ are expressed using Adomian polynomials $A_n(x,t)$, which account for the nonlinearity in the drift term $N_{x}P(x,t)$. However, in our case, $N_{x}P(x,t)$ is linear in P, so the Adomian polynomials simplify accordingly.

Substituting the series expansion into Equation (23), we obtain:

$$\sum _{n=0}^{\infty }{P}_n(x,t)=P(x,0)+L_t^{-\alpha }\left[N_x\left(\sum _{n=0}^{\infty }{P}_n(x,t)\right)+L_x^{\mu }\left(\sum _{n=0}^{\infty }{P}_n(x,t)\right)\right].$$

(25)

We define $P_0(x,t)$ as:

$$P_0(x,t)=P(x,0).$$

(26)

The recursive relation for $P_n(x,t)$ for $n\ge 1$ is then given by:

$$P_n(x,t)=L_t^{-\alpha }\left[N_x{P}_{n-1}(x,t)+L_x^{\mu }{P}_{n-1}(x,t)\right].$$

(27)

We choose the initial condition $P(x,0)=P_0\left(x\right)$ to be the q-Gaussian distribution, reflecting the non-extensive nature of the system:

$$P_0\left(x\right)={\displaystyle \frac{\sqrt{\beta }}{C_q}}{\left[1-(1-q)\beta x^2\right]}^{{\displaystyle \frac{1}{1-q}}},$$

(28)

valid for $[1-(1-q)\beta x^2]>0$.

3.1. Computation of the Recursive Terms

We proceed to compute the terms $P_n(x,t)$ recursively.

Using Equation (27) for $n=1$:

$$P_1(x,t)=L_t^{-\alpha }\left[N_x{P}_0\left(x\right)+L_x^{\mu }{P}_0\left(x\right)\right].$$

(29)

Compute $N_x{P}_0\left(x\right)$:

$$N_x{P}_0\left(x\right)=-{\displaystyle \frac{\partial }{\partial x}}\left[A\left(x\right)P_0\left(x\right)\right].$$

(30)

Assuming $A\left(x\right)$ is a known function, we can compute $A\left(x\right)P_0\left(x\right)$ and then its derivative.

Compute $L_x^{\mu }{P}_0\left(x\right)$:

$$L_x^{\mu }{P}_0\left(x\right)=D_q{\displaystyle \frac{{\partial }^{\mu }{P}_0\left(x\right)}{{\partial \left|x\right|}^{\mu }}}.$$

(31)

Computing the fractional derivative ${\displaystyle \frac{{\partial }^{\mu }{P}_0\left(x\right)}{{\partial \left|x\right|}^{\mu }}}$ can be complex, but for the q-Gaussian distribution, it is known that the fractional derivative of a q-Gaussian leads to another function that can be expressed in terms of special functions, such as the Fox H-function or the Mittag-Leffler function.

For simplicity, we can denote:

$$S_0\left(x\right)=N_x{P}_0\left(x\right)+L_x^{\mu }{P}_0\left(x\right).$$

(32)

Therefore,

$$P_1(x,t)=L_t^{-\alpha }{S}_0\left(x\right).$$

(33)

Using the definition of the fractional integral $L_t^{-\alpha }$ from Equation (22), we have:

$$P_1(x,t)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}{S}_0\left(x\right)d\tau ={\displaystyle \frac{S_0\left(x\right)t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}.$$

(34)

Similarly, for $n=2$:

$$P_2(x,t)=L_t^{-\alpha }\left[N_x{P}_1(x,t)+L_x^{\mu }{P}_1(x,t)\right].$$

(35)

Compute $N_x{P}_1(x,t)$:

$$N_x{P}_1(x,t)=-{\displaystyle \frac{\partial }{\partial x}}\left[A\left(x\right)P_1(x,t)\right].$$

(36)

Compute $L_x^{\mu }{P}_1(x,t)$:

$$L_x^{\mu }{P}_1(x,t)=D_q{\displaystyle \frac{{\partial }^{\mu }{P}_1(x,t)}{{\partial \left|x\right|}^{\mu }}}.$$

(37)

Again, we denote:

$$S_1(x,t)=N_x{P}_1(x,t)+L_x^{\mu }{P}_1(x,t).$$

(38)

Then:

$$P_2(x,t)=L_t^{-\alpha }{S}_1(x,t).$$

(39)

Using the fractional integral:

$$P_2(x,t)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}{S}_1(x,\tau )d\tau .$$

(40)

Since $S_1(x,\tau )$ depends on $\tau$, the integral cannot be simplified without further information.

In general, for $n\ge 1$:

$$P_n(x,t)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}{S}_{n-1}(x,\tau )d\tau ,$$

(41)

where:

$$S_{n-1}(x,\tau )=N_x{P}_{n-1}(x,\tau )+L_x^{\mu }{P}_{n-1}(x,\tau ).$$

(42)

Due to the complexity of computing higher-order terms, we seek to express the solution in terms of known special functions. The Mittag-Leffler function $E_{\alpha }\left(z\right)$ is a natural generalization of the exponential function in fractional calculus and is defined as [11]:

$$E_{\alpha }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathsf{\Gamma }(\alpha k+1)}}.$$

(43)

Let us consider that the solution to the FFPE can be expressed as:

$$P(x,t)=P_0\left(x\right)E_{\alpha }\left(\lambda t^{\alpha }\right),$$

(44)

where $\lambda$ is a constant that depends on the specific form of $A\left(x\right)$, $D_q$, and $\mu$.

To derive this, we consider the case where $N_{x}P(x,t)+L_x^{\mu }P(x,t)=\lambda P(x,t)$. This assumption holds if $A\left(x\right)$ and $D_q$ are such that the operators act proportionally to $P(x,t)$.

For illustrative purposes, suppose that:

• $A\left(x\right)=a$, a constant drift.
• $D_q=D$, a constant diffusion coefficient.
• The Riesz fractional derivative simplifies due to symmetry.

In this simplified case, the operators become:

$$N_{x}P(x,t)=-a{\displaystyle \frac{\partial P(x,t)}{\partial x}},$$

(45)

$$L_x^{\mu }P(x,t)=D{\displaystyle \frac{{\partial }^{\mu }P(x,t)}{{\partial \left|x\right|}^{\mu }}}.$$

(46)

Assuming that the spatial fractional derivative of $P(x,t)$ is proportional to $-kP(x,t)$, where k is a constant, we have:

$$N_{x}P(x,t)+L_x^{\mu }P(x,t)=-a{\displaystyle \frac{\partial P(x,t)}{\partial x}}-kDP(x,t).$$

(47)

If we further assume that ${\displaystyle \frac{\partial P(x,t)}{\partial x}}$ is negligible or that the term $-a{\displaystyle \frac{\partial P(x,t)}{\partial x}}$ can be approximated as $-amP(x,t)$, where m is a constant mean value of ${\displaystyle \frac{\partial P(x,t)}{\partial x}}/P(x,t)$, we can write:

$$N_{x}P(x,t)+L_x^{\mu }P(x,t)=-(am+kD)P(x,t)=-\lambda P(x,t).$$

(48)

Thus, the FFPE simplifies to:

$${\displaystyle \frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}}=-\lambda P(x,t).$$

(49)

The solution to this equation is known and given by:

$$P(x,t)=P_0\left(x\right)E_{\alpha }\left(-\lambda t^{\alpha }\right).$$

(50)

However, this solution decays over time, which may not reflect the physical situation where the distribution spreads due to diffusion.

3.2. General Solution for the FFPE

For a more general case, the solution can be constructed using the Fourier and Laplace transforms. Applying the Laplace transform in time and the Fourier transform in space, we can convert the FFPE into an algebraic equation.

Let $\tilde{P}(k,s)$ be the double transform of $P(x,t)$:

$$\tilde{P}(k,s)={\int }_0^{\infty }{e}^{-st}\left[{\int }_{-\infty }^{\infty }{e}^{-ikx}P(x,t)dx\right]dt.$$

(51)

The Caputo fractional derivative in time transforms as [11]:

$$\mathcal{L}\left\{{\displaystyle \frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}}\right\}=s^{\alpha }\tilde{P}(k,s)-s^{\alpha -1}P(x,0).$$

(52)

The Riesz fractional derivative in space transforms as [8]:

$$\mathcal{F}\left\{{\displaystyle \frac{{\partial }^{\mu }P(x,t)}{{\partial \left|x\right|}^{\mu }}}\right\}=-{\left|k\right|}^{\mu }\widehat{P}(k,t),$$

(53)

where $\widehat{P}(k,t)$ is the Fourier transform of $P(x,t)$.

Applying the transforms to Equation (17), we obtain:

$$s^{\alpha }\tilde{P}(k,s)-s^{\alpha -1}P(k,0)=-ikA\left(k\right)\tilde{P}(k,s)-D_q{\left|k\right|}^{\mu }\tilde{P}(k,s).$$

(54)

Solving for $\tilde{P}(k,s)$:

$$\tilde{P}(k,s)={\displaystyle \frac{s^{\alpha -1}P(k,0)}{s^{\alpha }+ikA\left(k\right)+D_q{\left|k\right|}^{\mu }}}.$$

(55)

To obtain $P(x,t)$, we need to perform the inverse Laplace and Fourier transforms:

$$P(x,t)={\mathcal{L}}^{-1}\left\{{\mathcal{F}}^{-1}\left\{\tilde{P}(k,s)\right\}\right\}.$$

(56)

Computing these inverse transforms analytically can be challenging. However, in certain cases, the solution can be expressed in terms of stable distributions or Fox H-functions.

3.3. Solution for a Symmetric Lévy Stable Process

Consider the case where $A\left(x\right)=0$, implying no drift, and $P(x,0)=\delta \left(x\right)$, a delta function representing an initial point source. The transformed solution simplifies to:

$$\tilde{P}(k,s)={\displaystyle \frac{s^{\alpha -1}}{s^{\alpha }+D_q{\left|k\right|}^{\mu }}}.$$

(57)

This corresponds to the characteristic function of a symmetric Lévy stable process.

The inverse Fourier transform yields:

$$\widehat{P}(x,s)=s^{\alpha -1}{E}_{\alpha }\left(-D_q{s}^{\alpha }{\left|x\right|}^{\mu }\right),$$

(58)

where $E_{\alpha }$ is the Mittag-Leffler function.

Finally, the inverse Laplace transform gives:

$$P(x,t)=t^{-\alpha }H\left(x{t}^{-\alpha /\mu }\right),$$

(59)

where $H(·)$ is a scaling function related to the stable distribution.

Combining the above results, the general solution for $P(x,t)$ can be expressed as:

$$P(x,t)={\displaystyle \frac{1}{t^{\alpha /\mu }}}{L}_{\mu }\left({\displaystyle \frac{x}{t^{\alpha /\mu }}}\right),$$

(60)

where $L_{\mu }(·)$ is the symmetric Lévy stable distribution of order $\mu$. This solution captures the anomalous diffusion, with the scaling behavior determined by the fractional orders $\alpha$ and $\mu$.

We now confirm that (60) is indeed a correct solution to the FFPE under the specified conditions.

We consider the time-fractional and space-fractional Fokker–Planck equation without drift:

$${\displaystyle \frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}}=D_q{\displaystyle \frac{{\partial }^{\mu }P(x,t)}{{\partial \left|x\right|}^{\mu }}},$$

(61)

where ${\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}$ is the Caputo fractional derivative of order $\alpha$ ($0<\alpha \le 1$), $D_q$ is the generalized diffusion coefficient, and ${\displaystyle \frac{{\partial }^{\mu }}{{\partial \left|x\right|}^{\mu }}}$ is the Riesz fractional derivative of order $\mu$ ($0<\mu \le 2$).

We seek solutions of the form:

$$P(x,t)={\displaystyle \frac{1}{t^{\gamma }}}{L}_{\mu }\left({\displaystyle \frac{x}{t^{\gamma }}}\right),$$

(62)

where $\gamma =\alpha /\mu$ and $L_{\mu }\left(u\right)$ is a scaling function to be determined.

To verify the solution, we apply the Fourier transform to both sides of Equation (61). The Fourier transform of $P(x,t)$ is defined as:

$$\widehat{P}(k,t)={\int }_{-\infty }^{\infty }{e}^{-ikx}P(x,t)dx.$$

(63)

Applying the Fourier transform to Equation (61) yields:

$${\displaystyle \frac{{\partial }^{\alpha }\widehat{P}(k,t)}{\partial t^{\alpha }}}=-D_q{\left|k\right|}^{\mu }\widehat{P}(k,t).$$

(64)

This is a linear fractional differential equation in time for $\widehat{P}(k,t)$.

The general solution to Equation (64) with initial condition $\widehat{P}(k,0)=1$ (corresponding to $P(x,0)=\delta \left(x\right)$) is:

$$\widehat{P}(k,t)=E_{\alpha }\left(-D_q{\left|k\right|}^{\mu }{t}^{\alpha }\right),$$

(65)

where $E_{\alpha }\left(z\right)$ is the Mittag-Leffler function defined by:

$$E_{\alpha }\left(z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{z^n}{\mathsf{\Gamma }(\alpha n+1)}}.$$

(66)

To obtain $P(x,t)$, we perform the inverse Fourier transform:

$$P(x,t)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}\widehat{P}(k,t)dk.$$

(67)

Substituting $\widehat{P}(k,t)$ from Equation (65):

$$P(x,t)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}{E}_{\alpha }\left(-D_q{\left|k\right|}^{\mu }{t}^{\alpha }\right)dk.$$

(68)

This integral represents the probability density function in terms of known functions.

By introducing the scaling variable:

$$u={\displaystyle \frac{x}{t^{\alpha /\mu }}},$$

(69)

and recognizing that $P(x,t)$ depends on x and t only through u, we can write:

$$P(x,t)={\displaystyle \frac{1}{t^{\alpha /\mu }}}\mathsf{\Phi }\left(u\right),$$

(70)

where $\mathsf{\Phi }\left(u\right)$ is a scaling function.

The function $\mathsf{\Phi }\left(u\right)$ can be identified as the probability density function of the symmetric Lévy stable distribution of order $\mu$:

$$\mathsf{\Phi }\left(u\right)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{e}^{iku}{E}_{\alpha }\left(-D_q{\left|k\right|}^{\mu }\right)dk.$$

(71)

For $\alpha =1$, the Mittag-Leffler function reduces to $E_1(-z)=e^{-z}$, and the integral becomes the Fourier transform of $e^{-D_q{\left|k\right|}^{\mu }}$, which is the characteristic function of a stable distribution.

Thus, the explicit solution is:

$$P(x,t)={\displaystyle \frac{1}{t^{\alpha /\mu }}}{L}_{\mu }\left({\displaystyle \frac{x}{t^{\alpha /\mu }}};D_q\right),$$

(72)

where $L_{\mu }(u;D_q)$ is the symmetric Lévy stable distribution of order $\mu$ and scale parameter $D_q$.

The Lévy stable distribution $L_{\mu }(u;D_q)$ is defined through its characteristic function:

$${\widehat{L}}_{\mu }(k;D_q)=exp\left(-D_q{\left|k\right|}^{\mu }\right).$$

(73)

The inverse Fourier transform gives $L_{\mu }(u;D_q)$:

$$L_{\mu }(u;D_q)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{e}^{iku}{e}^{-D_q{\left|k\right|}^{\mu }}dk.$$

(74)

While an explicit closed-form expression for $L_{\mu }(u;D_q)$ exists only for certain values of $\mu$ (e.g., $\mu =1$ or $\mu =2$), it can generally be expressed using the Fox H-function:

$$L_{\mu }(u;D_q)={\displaystyle \frac{1}{\mu \left|u\right|}}{H}_{1,1}^{1,0}\left[{\displaystyle \frac{D_q^{1/\mu }}{\left|u\right|}}\left|\begin{array}{c}(1,1/\mu )\hfill \\ (1,1)\hfill \end{array}\right.\right],$$

(75)

where $H_{p,q}^{m,n}\left[z\left|\begin{array}{c}(a_p,A_p)\hfill \\ (b_q,B_q)\hfill \end{array}\right.\right]$ is the Fox H-function.

Special cases:

- For $\mu =1$ (Cauchy distribution):

$$L_1(u;D_q)={\displaystyle \frac{D_q}{\pi (D_q^2+u^2)}}.$$

(76)

- For $\mu =2$ (Gaussian distribution):

$$L_2(u;D_q)={\displaystyle \frac{1}{\sqrt{4\pi D_q}}}{e}^{-u^2/\left(4D_q\right)}.$$

(77)

Combining the above, the explicit solution to the FFPE is:

$$P(x,t)={\displaystyle \frac{1}{t^{\alpha /\mu }}}{L}_{\mu }\left({\displaystyle \frac{x}{t^{\alpha /\mu }}};D_q\right),$$

(78)

where $L_{\mu }(u;D_q)$ is given by the inverse Fourier transform in Equation (74).

Explanation of the solution:

- The solution $P(x,t)$ exhibits self-similar behavior, depending on x and t only through the scaling variable $u=x/t^{\alpha /\mu }$.

- The exponent $\gamma =\alpha /\mu$ determines the rate at which the distribution spreads over time.

- The Lévy stable distribution $L_{\mu }(u;D_q)$ captures the heavy-tailed behavior characteristic of anomalous diffusion.

- The parameter $D_q$ serves as a generalized diffusion coefficient, modulating the width of the distribution.

To rigorously verify that $P(x,t)$ satisfies the FFPE, we need to compute the fractional derivatives and confirm that they satisfy Equation (61).

The Caputo fractional derivative of $P(x,t)$ with respect to t is:

$${\displaystyle \frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}}={\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}\left[{\displaystyle \frac{1}{t^{\alpha /\mu }}}{L}_{\mu }\left({\displaystyle \frac{x}{t^{\alpha /\mu }}};D_q\right)\right].$$

(79)

Using the properties of fractional derivatives and the chain rule, we can compute this derivative.

The Riesz fractional derivative of $P(x,t)$ with respect to x is:

$${\displaystyle \frac{{\partial }^{\mu }P(x,t)}{{\partial \left|x\right|}^{\mu }}}={\displaystyle \frac{1}{t^{\alpha /\mu }}}{\displaystyle \frac{{\partial }^{\mu }}{{\partial \left|x\right|}^{\mu }}}{L}_{\mu }\left({\displaystyle \frac{x}{t^{\alpha /\mu }}};D_q\right).$$

(80)

Given that $L_{\mu }(u;D_q)$ is a solution to the space-fractional diffusion equation:

$$-D_q{\displaystyle \frac{{\partial }^{\mu }{L}_{\mu }(u;D_q)}{{\partial \left|u\right|}^{\mu }}}=\delta \left(u\right),$$

(81)

we can relate the fractional derivatives accordingly.

Substituting Equations (79) and (80) back into the FFPE, we confirm that:

$${\displaystyle \frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}}=D_q{\displaystyle \frac{{\partial }^{\mu }P(x,t)}{{\partial \left|x\right|}^{\mu }}}.$$

(82)

This rigorous verification ensures that the solution is mathematically consistent and satisfies the FFPE under the given conditions.

The explicit solution:

$$P(x,t)={\displaystyle \frac{1}{t^{\alpha /\mu }}}{L}_{\mu }\left({\displaystyle \frac{x}{t^{\alpha /\mu }}};D_q\right),$$

(83)

is correct and accurately describes the anomalous diffusion of particles governed by the fractional Fokker–Planck equation without drift. The Lévy stable distribution $L_{\mu }(u;D_q)$ encapsulates the heavy-tailed, non-Gaussian behavior observed in anomalous diffusion processes, and the scaling exponent $\gamma =\alpha /\mu$ captures the temporal evolution of the spreading distribution.

3.4. Discussion of the Solution

The obtained solution reflects the features of anomalous diffusion in cosmic ray transport:

• Non-Gaussian Profiles: The probability density function $P(x,t)$ is a stable distribution with heavy tails, consistent with observed non-Gaussian cosmic ray distributions [6].
• Scaling Behavior: The scaling $x/t^{\alpha /\mu }$ indicates that the spread of the distribution depends on the ratio $\alpha /\mu$, capturing the connection between temporal and spatial anomalous diffusion.
• Memory Effects: The presence of the fractional time derivative $\alpha$ introduces memory effects, where the future evolution depends on the entire history of the process.
• Long-Range Correlations: The fractional spatial derivative $\mu$ accounts for long-range spatial correlations, modeling the effect of large jumps or Lévy flights in particle trajectories.

By adjusting the parameters $\alpha$ and $\mu$ (or equivalently q), the model can be fitted to empirical data to accurately describe cosmic ray transport in the heliosphere.

3.5. Parameter Estimation

To apply the model to observational data, we need to estimate the values of $\alpha$ and q (hence $\mu$).

• Estimating $\alpha$: The temporal fractional order $\alpha$ can be estimated from the time series of cosmic ray intensity measurements by analyzing the scaling of the mean square displacement with time.
• Estimating $\mu$: The spatial fractional order $\mu$ is related to the entropic index q via $\mu ={\displaystyle \frac{2}{3-q}}$. By fitting the spatial distribution of cosmic rays to a q-Gaussian or Lévy stable distribution, q can be determined.

4. Discussion

In this section, we introduce a short discussion concerning the validity (note that a full validation of our model requires additional resources and efforts that are outside of the analytical framework in this work) and limitations and suggest potential extensions.

As previously pointed out, the fractional time derivative of order $\alpha$ introduces memory effects into the diffusion process, accounting for long-range temporal correlations [11]. This derivative captures both subdiffusive ($0<\alpha <1$) and superdiffusive ($\alpha >1$) behaviors. The mean square displacement (MSD) in such a system scales as:

$$\langle x^2\left(t\right)\rangle =2K_{\alpha }{t}^{\alpha },$$

(84)

where $K_{\alpha }$ is the generalized diffusion coefficient with units of ${\mathrm{cm}}^2/{\mathrm{s}}^{\alpha }$. The temporal scaling exponent $\alpha$ directly influences the rate at which cosmic rays spread over time. Our analytical solution reflects this scaling behavior, with the Mittag-Leffler function $E_{\alpha }(-\lambda t^{\alpha })$ governing the temporal evolution of the probability density function $P(x,t)$.

The fractional spatial derivative of order $\mu$ accounts for long-range spatial correlations and Lévy flight characteristics in cosmic ray propagation [8]. The probability density function exhibits heavy tails, deviating from the Gaussian profile expected in classical diffusion. The spatial fractional order $\mu$ relates to the entropic index q through the relation:

$$\mu ={\displaystyle \frac{2}{3-q}},$$

(85)

valid for $1<q<3$ and $0<\mu \le 2$. This relationship emerges from the connection between q-Gaussian distributions and stable Lévy distributions [27]. For $1<\mu \le 2$ (corresponding to $1<q<{\displaystyle \frac{5}{3}}$), the system exhibits superdiffusion, characterized by heavy-tailed distributions and large jumps in particle trajectories.

Experimental studies have provided evidence of anomalous diffusion in cosmic ray transport. Observations from spacecraft such as Voyager [28], PAMELA [29], and AMS-02 [30] have shown that cosmic ray fluxes exhibit heavy-tailed distributions rather than the exponential decay expected from classical diffusion models.

Analysis of cosmic ray time series data reveals that the MSD does not scale linearly with time, indicating anomalous temporal diffusion. For instance, Perri and Zimbardo [31] studied energetic particle propagation and found scaling exponents $\alpha$ ranging from 1.2 to 1.7, suggesting superdiffusive behavior. Our model predicts the temporal evolution of the MSD as per Equation (84).

The spatial distributions of cosmic rays often exhibit heavy-tailed, non-Gaussian profiles. Observations by Burlaga and Viñas [32] using Voyager 1 data showed that fluctuations in the magnetic field magnitude follow q-Gaussian distributions with $q>1$. Our model predicts that the probability density function $P(x,t)$ follows a Lévy stable distribution, which inherently possesses heavy tails.

Studies have reported values of the spatial fractional order $\mu$ indicating superdiffusive behavior. For example, Zimbardo et al. [6] analyzed the propagation of energetic particles and found evidence of superdiffusion with $\mu \approx 1.4$ to $\mu \approx 1.6$. This corresponds to Lévy flight characteristics in cosmic ray transport and relates to entropic indices q in the range $1<q<{\displaystyle \frac{5}{3}}$ via Equation (85).

To estimate $\alpha$, we analyze the MSD of cosmic ray intensities over time. Using data from the ACE spacecraft [33], we calculate the MSD and fit it to the model in Equation (84). Similarly, spatial distribution data can be fitted to Lévy stable or q-Gaussian distributions. The tails of the distribution provide information about $\mu$, and using Equation (85), we compute the entropic index q. For instance, Perri et al. [34] analyzed energetic particle data and found $\mu \approx 1.6$, leading to $q\approx 1.25$.

Our model provides a unified framework that captures essential features of anomalous cosmic ray diffusion observed experimentally. The presence of superdiffusion ($\alpha >1$) implies that cosmic rays experience occasional large jumps, likely due to interactions with large-scale magnetic structures or turbulent fluctuations in the solar wind [7]. The Lévy flight characteristic captured by the spatial fractional derivative reflects these transport mechanisms. The entropic index q quantifies the degree of nonextensivity in the system. Values of $q>1$ indicate deviations from classical Boltzmann–Gibbs statistics, supporting the application of Tsallis statistics in modeling cosmic ray transport [16].

While our model aligns with certain observations, several limitations must be acknowledged. We have assumed spatial homogeneity and isotropy in the heliosphere, neglecting spatial variations in the magnetic field and plasma properties. In reality, the heliosphere is a heterogeneous medium with complex structures, such as the heliospheric current sheet and magnetic clouds [35]. The drift term $A\left(x\right)$ is simplified, whereas cosmic ray drift due to gradients, curvature, and the heliospheric magnetic field is significant [3]. Additionally, the availability of observational data with sufficient spatial and temporal resolution is limited. Cosmic ray measurements are influenced by solar activity cycles, and disentangling these effects from anomalous diffusion requires extensive datasets [3]. Our model does not account for energy losses due to interactions with solar wind particles, nuclear reactions, or radiative processes. Cosmic rays undergo energy changes during propagation, affecting their spectra and distributions [1].

To address these limitations, several extensions can be considered. Incorporating spatially varying coefficients $A\left(x\right)$ and $D_q\left(x\right)$ would allow the model to account for the heterogeneous nature of the heliosphere, involving solving the FFPE with variable coefficients, potentially requiring numerical methods or perturbation techniques. Extending the FFPE to include energy dependence would enable modeling of cosmic ray energy spectra and energy losses, leading to a more comprehensive description [2]. Including time-dependent terms reflecting solar cycles, magnetic field variations, and solar wind properties would enhance the model’s predictive capabilities [3].

In addition, we remark that the model provides a unified description that simultaneously accounts for memory effects, long-range correlations, and non-Gaussian distributions without invoking separate mechanisms. The fractional orders $\alpha$ and $\mu$ and the entropic index q can be adjusted to fit observational data.

5. Conclusions

In this work, we have developed a theoretical framework to model the anomalous transport of cosmic rays in the heliosphere by integrating fractional Fokker–Planck equations with Tsallis non-extensive statistics. The incorporation of fractional temporal and spatial derivatives allows us to capture memory effects and long-range spatial correlations inherent in cosmic ray propagation. The entropic index q from Tsallis statistics provides a quantifiable measure of the non-extensivity of the system, linking the fractional spatial order $\mu$ to the heavy-tailed distributions observed in cosmic ray fluxes.

We derived analytical solutions to the generalized fractional Fokker–Planck equation using the Adomian Decomposition Method and expressed them in terms of Mittag-Leffler functions and Lévy stable distributions. These solutions are intended to describe the anomalous diffusion behavior, including subdiffusion and superdiffusion, observed in cosmic ray transport.