On a Generalization of One-Dimensional Kinetics

: One-dimensional random walks with a constant velocity between scattering are considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several special cases are considered when the convolutions are expressed in explicit form. As a particular case, the solution of A. S. Monin for a symmetric random walk with exponential path distribution and its generalization to the asymmetric case are obtained. Solution of fractional telegraph equation with the fractional material derivative is presented. Asymptotic behavior of its solution for an asymmetric case is provided.


Introduction
The main difference between the Boltzmann kinetic equation and widely used diffusion approximation is the velocity loss. Whereas the Boltzmann trajectory is a broken line consisting of rectilinear segments, each of which is characterized by a certain velocity, and any part of the trajectory has a length and travel time, a Brownian particle in the Wiener representation has a continuous everywhere but nowhere differentiable trajectory. The length of its section between any two points is infinite. From a physical point of view, some of the consequences of this approximation are unacceptable. Particularly, a small diffusion packet placed in an infinite medium will instantly spread to all space that conflicts with special relativity. This is the price to pay for the ease of a mathematical solution.
To the best of our knowledge, the first who undertook an attempt to involve a finite velocity into this model was Soviet physicist-theorist V. A. Fock [1]. His aim was to describe the diffusion of light in a medium with small transparent reflecting particles suspended in it. Beginning his article with consideration of the random hopping process on a discrete system of nodes, he formulated the mathematical problem by means of algebraic equations and gave its solution in terms of finite differences, and only to the end of his article, Fock introduced a finite constant velocity c (he considered photons) and derived the system of differential interrelations under the Markovian assumption that the probability for the particle changing its motion direction into the opposite one during interval (t, t + dt) is equal to µdt: This pair of equalities is a one-dimensional case of the linear Boltzmann equation expressed through differentials. Each of these two functions obeys an equation of the wave-type where is the unscattered part, and p (s) (x, t) is the scattered one, expressed through the modified Bessel functions with argument µt 1 − (x/vt) 2 , where −vt < x < vt.
A fundamentally important circumstance accompanying the derivation of the telegraph equation is the proportionality of the scattering probability on an elementary path to the length of this path, the result of which is the exponential type of the free path distribution. In a review [6], a detailed analysis of Einstein's derivation of diffusion equation was done and revealed an important circumstance taking place in the case with the telegraph equation as well: random positions of scatterers acting on the tracer are considered to be statistically independent. In other words, these equations model passing a tracer through an ideal gas, molecules of which do not interact with each other, so there are no correlations between consecutive collisions. Certainly, the inertia of the tracer possessing some mass produces some small correlations, especially in a rarified matter, but much stronger correlations arise in a turbulent media, when the tracer is picked up by the random flow (having the form of a jet or vortex) and carried to large distances. A similar situation takes place in the case of charged particle diffusion in random magnetic fields, correlations of which are produced by a random system of magnetic force lines. Some of these cases allow the broken-line approximation of certain parts of the trajectory, but the lengths of the random segments have distributions that differ from the classical exponential form.
We consider in this paper other examples of one-dimensional random motion also admitting a rather easy solution but being beyond the Markovian paradigm. In a general case, we should be ready to describe a process by means of using integral operators with an arbitrary kernel and find a way to its solution. Such an approach is known as the nonlocal transport theory and find a more and more broadening field of applications.

The Generalized Process
One-dimensional walking of a particle under consideration starts from the point x = 0 at the moment t = 0 with probability ε 1 on the left (towards negative x) and with probability ε 2 = 1 − ε 1 in the opposite direction. Velocities of free motion in different directions are also different and are of v 1 and v 2 correspondingly. The probability density for a random free path (to the point of changing the direction of the motion) depends on direction as well and is denoted by p 1 (ξ) for the left direction and p 2 (ξ) for the right (ξ > 0). The problem is to determine the probability density ρ(x, t) of the particle coordinate at the moment t > 0.
A further modification of this scheme is possible by introduction of a square matrix ε ij , determining the transition probability from the state of motion j to the state i (i, j = 1, 2). Let us denote the path-distribution in this modifications q i (ξ), so free path distributions become where q (n) i (ξ) denotes multiple convolutions of the densities q i (ξ): The sought density consists of two parts corresponding to possible directions (states) of particle motion. Denote by ξ the distance between points of observation and the appearance of the particle in some state via transition from another state or emitted by a source. The minimum value of ξ is zero, and the maximum value depends on x, t and a considered state. For state 1, it satisfies the system of equations where t * is the moment of time when the particle performs a jump from a corresponding point of the cone (x = −vt, x = vt) to the point x. Excluding t * , we find Similarly, for state 2, we find Let f 1 (x , t )dx dt be the probability of particle appearance in state 1 in the element dx during dt and let P 1 (x , t → x, t) be the probability that it will pass a segment [x, x ] without collisions and at time t turns out to be at point x: Here stands for probability that the random path of the particle will exceed ξ ≥ 0. The first density component is expressed through the integral having the sense of the absolute value of the current particle j 1 in the direction 1, so j 1 = vρ 1 . Inserting (10) into (12) and using the δ-function property, we arrive at expression Considering in a similar way the motion in the opposite direction, we find the second relation: The collision densities f i (x, t) satisfy the system of integral equations, derived from the same reasoning as (13) and (14), and have the form The free terms f describe the emission of particles by the source.

Solving the Generalized Equation
Let us add to ρ ij and f ij the second index, indicating a fixed initial state of the particle, so After one iteration, the system of equations for f ij (x, t) (15)-(16) become a pair of independent equations (20) with free terms Let us consider the equation for f 11 (x, t) in detail, omitting lower indexes to be short. Representing its solution in the form of von Neumann series, we obtain where For the farther exposition, it is convenient to extend the notion of multiple convolution (24) to the case n = 0: p A simple check confirms that Substituting (26) into (23) and finding a few first iterations, we see that each term has the form of a product Substituting (27) into (23) and taking into account that and and

Solution of the Generalized Equation for the Process with an Arbitrary Transition Density
The latter functions are expressed in terms of multiple convolutions of distributions p 1 (ξ) and p 2 (ξ), so we have: The meaning of the summation index n is clear: it is the number of particle steps after which it appears in state 1 at point x at moment t. Since the initial state is also 1, the probability that after some (any) number of steps n the particle will be at state 1 without reference to x and t, is equal to 1. It is easy to verify that this normalization condition is satisfied. Expressing ξ 1 through ξ 2 using relation ξ 1 Similarly, we get function f 21 (x, t), but since the right side of the corresponding equation has the form the result will be We also give the expressions for the other two functions: and However, it is clear from obvious reasons that f 12 can be obtained from f 21 , and f 22 from f 11 , by replacing all lower indexes (1 → 2, 2 → 1) in the right side of expressions (35) and (32), respectively. Substituting the expressions found in the right sides of (13), (14) and considering that and is a distribution function, we shall get As in the case of f ij (x, t), expressions for densities ρ 22 (x, t) and ρ 12 (x, t) are obtained from (41)-(42) by permutation of the lower indexes. Note that the distribution presented here includes δ-features, given by the term with n = 0, so the normalization condition is valid: The representation (41)-(42) of the walking problem reduces it to a calculation of multiple convolutions p (n) 1 (x) and p (n) 2 (x) of path-densities. We divide the set of such distributions into four classes: A, whose convolutions are expressed in terms of elementary or special functions; B, whose convolutions are expressed in terms of the original one with changed parameters; and C, whose convolutions are expressed through the initial one up to the linear scale transform (Lévy-stable densities). Below, we consider some of them. Taking any of these distributions allows us to get the solution in a closed form. Finally, the last (D) class includes densities, producing solutions with self-similar (with respect to space-time variables) long-time asymptotics.

Solution for the Process with Exponential and Gamma-Distributions
We begin the search for explicit solutions with the simplest term of this family In this case and ∆F Substituting these expressions into (41), we obtain for the regular component (n = 0) The last sum is an expansion to the power series of (2/z)I 1 (z) function, where As a result, we havẽ Similary, from (42) we find The componentsρ 22 (x, t) andρ 12 (x, t) are obtained by changing indices of the coefficients µ and changing the sign before x. The total distributioñ reads for x ∈ (−vt, vt): Solutions for exponential and gamma distribution of path lengths are demonstrated in Figure 1. All results were confirmed using Monte Carlo simulations; for example, the result of the modeling for exponential and gamma walks are shown in Figure 2.
The result for gamma distribution (only regular parts) for ν 1 = ν 2 = 2. The other parameters are the same as in (a). Equation (37) is a particular case of the gamma distribution.
The family of gamma distributions is closed under convolution [7]. Therefore, choosing we immediately get Using (46), we get where Γ(β, x) is the upper incomplete gamma function. Thus, for regular components of densities ρ 11 and ρ 21 , we get for integer ν i , this sum can be expressed through integrals of generalized hypergeometric functions. For ν 1 = ν 2 = 2, these expressions become especially simplẽ

Solutions for the Process with Some Other Transition Densities
There are some more distributions in which multiple convolutions can be expressed in elementary form, but sums (41) and (42) can not be expressed in closed form (as (57) and (58) with arbitrary ν i ). One example is the Bessel density Using (46), we get where F p;q (a 1 , .., a p ; b 1 , .., b q , x) is a generalized hypergeometric function [8]. The sums (41) and (42) can be calculated numerically, and the result are shown in Figure 3. Another distribution is uniform distribution [7] where θ(x) is a Heaviside function. For this distribution, we have and ∆F (ξ) A wide class of distributions whose convolutions lie in the same class are one-sided stable densities with 0 < α < 1, concentrated on the positive semiaxis [9] Using a cumulative distribution function of stable densities G (α) (x), we can get For a special case α = 1/2 we have Some distributions with different α are shown in Figure 4a. As in the previous section, all results were proved by means of the Monte Carlo simulations. One example is shown in Figure 4b.

Integral Transforms and Asymptotic Results
In the case of an arbitrary density p(z) with a finite second moment, the asymptotic part of the solution satisfies the telegraph equation. In References [10][11][12], the authors investigated one-dimensional random walks with the asymptotically power-law distribu-tion p(ξ) ∝ ξ −α−1 , 0 < α < 2. Such a random process is sometimes called a fractal walk. Recently, Lévy flights with a bi-modal distribution of jumps were studied in Reference [13] and the interesting effect of loss of self-similarity was predicted for certain parameters.
We rewrite Equations (13)-(17) as the following system, After the Fourier-Laplace transformation, the solution can be represented as the following transform, Consider two cases of fractal walk. Let an asymptotic expansion (λ → 0, i.e., t → ∞) of the Laplace transformp be of the form, This case corresponds to the heavy-tailed distribution, p(z) ∼ αz 0 α z −α−1 , 0 < α < 1, and all moments of natural order diverge.
Substituting Equation (71) into expression (70), we obtain for the first casẽ In Reference [12], we inverted this transform in the symmetric case and expressed the result through elementary functions for all values of α of the indicated interval. In the asymmetric case, we have Here, C = γ 1 /γ 2 . Plots of these distributions for several values of α are presented in Figure 5. The pdf of such type was obtained by Lamperti in frames of the mathematical theory of occupation times [14] and used in statistical physics of weakly non-ergodic systems [15]. Evolution of pdf for instantaneous point source and its tendency to the asymptotic Lamperti distribution (α = 0.75) is demonstrated in Figure 6. Path lengths are distributed according to the Pareto distribution with the same α.
Rewriting relation (72) in the form and performing the inverse Fourier-Laplace transformation, we arrive at the equation with material derivatives of fractional order The multiplier (λ ± ivk) α in the formulas derived above presents the Fourier-Laplace transform of the fractional material derivative [11]: that can be verified by rewriting the operator in the Riemann-Liouville form: and applying the Fourier-Laplace transformation. The second case is characterized by a finite first moment (1 < α < 2), For asymptotic solution, we can put λ → 0, k → 0, |λ/vk| → 0. Thus, the transform takes the following form: The inverse Laplace transformation leads to the characteristic functioñ related to the characteristic function of the Lévy stable density g(x; α, β), by the following expressioñ p(k, t) = exp(iβkvt)g((Kt) 1/α k; α; β).
The passage to the original leads to the solution: where The inverse Fourier-Laplace transformation of the latter expression corresponds to the superdiffusion equation with a fractional operator.

One-Sided Fractal Walks with Traps
In some applications, the models of a random walk with localization events can be useful. Particularly, traps characterized by random waiting times play a major role in the kinetics of dispersive transport of charge carriers in disordered semiconductors [16]. Let us call the localized state off-state, and the state of motion is on-state.
Let p(x, t) be the distribution density of total residence time in on-state. For the double Laplace transform of the density p(x, t), the following expression is derived [17]: Here, v denotes some proportional coefficient responsible for time compression in on-state, v = 1 if time measures in on-and off-states coincide. Under the assumption of power-law distributions of waiting times in off-and on-states,

Generalized Cases
In this section, we shortly discuss some generalized cases of one-dimensional random walks, where the obtained solutions can be applied.

Two-Sided Lévy Walks with Traps
The asymptotic (t → ∞) solution for the pdf of a walking particle coordinate can be written in terms of an integral with the subordinating function Here, p 1 (x, τ) is the solution for a random walk without traps, and w(τ, t) is the distribution density for operational time. For the case of Lévy walks, expressions for p 1 (x, τ) are given in and Sections 8 and 9. For w(τ, t), we can use the asymptotic expression (83) with τ instead of x/v, when the distribution of waiting times and motion times are distributed according to (81) and (82). The comparison of a numerically computed integral (84) with results of the Monte Carlo simulation for the case α = 0.6, β = 0.7 and asymmetry parameter θ = 0.5 is shown in Figure 8. The agreement is satisfactory.

Conclusions
One-dimensional non-Markovian random walk models with a finite velocity between scattering events have been considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several particular cases have been considered when the convolutions can be expressed in explicit form. A solution of the fractional telegraph equation has been investigated, and asymptotic behavior of its solution has been provided.