Generalization of the persistent random walk to dimensions greater than 1

We propose a generalization of the persistent random walk for dimensions greater than 1. Based on a cubic lattice, the model is suitable for an arbitrary dimension d. We study the continuum limit and obtain the equation satisfied by the probability density function for the position of the random walker. An exact solution is obtained for the projected motion along an axis. This solution, which is written in terms of the free-space solution of the one-dimensional telegrapher’s equation, may open a new way to address the problem of light propagation through thin slabs. @S1063-651X~98!00312-2#


I. INTRODUCTION
The persistent random walk ͑PRW͒, first introduced by Fu ¨rth ͓1͔ and shortly after by Taylor ͓2͔, is probably the earliest and simplest generalization of the ordinary random walk that incorporates some form of momentum in addition to random motion.The persistent random walk differs from the ordinary random walk in that the probabilistic quantity used at each step is the probability of continually moving in a given direction rather than the probability of moving in a given direction regardless of the direction of the preceding step.In this way the PRW introduces some form of momentum, i.e., persistence, into the purely random motion.This remarkable feature of the model is one of the reasons why the PRW has been recently applied to describe scattering and diffusion in disordered media ͓3͔.
For one-dimensional lattices and in the continuum or diffusive limit the probability density function for the displacement at time t, p(x,t), satisfies the telegrapher's equation ͑TE͒ ͓4͔: As is well known, Eq. ͑1͒ has solutions with a finite velocity of propagation given by v ͓6͔.This fact has justified the extensive use of TE as a generalization of the mesoscopic diffusion equations in fields such as heat propagation ͓5͔ and light dispersion in turbid media ͓6,7͔.However, none of the generalizations explored in two and three dimensions of persistent random walks obeys the TE in the continuum limit ͓8-11͔.
On the other hand, and besides the recent work of Godoy et al. ͓10͔ on two-dimensional walks, there have been, to our knowledge, very few attempts to generalize the PRW to dimensions higher than 1 in spite of its potential for modeling transport in disordered media.One of the reasons for the lack of such a generalization is the nonexistence of a unique generalization of the PRW to dimensions greater than 1 since several kinds of lattices ͑cubic, hexagonal, etc.͒ can be used for the extension.Our main goal in this paper is to propose a generalization of the PRW to higher dimensions assuming a cubic lattice, and to obtain the governing equations for the probability density function of the process in the continuum limit.
Unfortunately, in the continuum limit and for dimensions greater than 1, the probability density function of the process does not obey a higher-dimensional telegrapher's equation.Nevertheless, in the context of transport in disordered media, the partial differential equation describing particle concentration ͑i.e., the probability density function͒ can suggest new approximations for the transport equation of more realistic models.This is the case, for example, of light propagation in turbid media where such an approach becomes extremely useful, especially when photons propagate in constrained geometries such as thin slabs where the Gaussian approximation becomes quite imprecise ͓12͔.In addition, the telegrapher's equation has been shown not to furnish better results than the diffusion approximation in two and three dimensions ͓13͔.
As we have mentioned, in the one-dimensional case the equation satisfied by p(x,t) is the telegrapher's equation.In two dimensions, the model considered herein was partially analyzed by Godoy et al. ͓10͔ but the equation for the probability density function, p(r,t), of the process was not obtained.Another goal of this paper is to study the projected motion, along a given direction, of the higher-dimensional PRW.This projected motion is relevant in the study of light propagation in turbid media.Indeed, when persistent random walks are used as models for light propagation through slabs, the basic information is contained in the motion projected along the coordinate orthogonal to the faces of the slab ͓14͔.
We thus obtain the equation that governs the evolution of the projected motion and write its solution in terms of the freespace solution of the one-dimensional telegrapher's equation.
The paper is organized as follows.In Sec.II we set the general analysis for the PRW in a cubic lattice of dimension d.In Sec.III we obtain the continuum limit and find the governing equation for the probability density function of the resulting process.Section IV is devoted to the projected motion on a given coordinate and we obtain an exact expression of the density.Conclusions are drawn in Sec.V.

II. GENERAL ANALYSIS
We consider a random walk in a cubic lattice of arbitrary dimension d.The distance between nearest lattice points is l.
Jumps to another lattice point occur after a time interval .At each point, the random walker can take 2d different directions.Among all possible events, we are only interested in three of them: ͑a͒ The ''forward scattering,'' where the random walker moves in the same direction as the previous jump, ͑b͒ the ''backward scattering,'' where the walker reverses its previous direction, and ͑c͒ the scattering in the 2(dϪ1) remaining directions.Let us denote by ␣, ␤, and ␥ the probabilities of events ͑a͒, ͑b͒, and ͑c͒, respectively.As absorption will not be considered here, the scattering probabilities satisfy the normalization condition
Let us now set the general equations of this model.We define a set of auxiliary probabilities P n (Ϯk) (i 1 , . . .,i d ), k ϭ1, . . .,d, where P n (ϩk) (i 1 , . . .,i d ) is the probability that the walker reaches the lattice point (i 1 , . . .,i d ) at step n moving along direction ϩk.A similar definition applies to P n (Ϫk) (i 1 , . . .,i d ).Following an analogous reasoning to that of the one-dimensional PRW ͓3͔, we can see that P n (Ϯk) (i 1 , . . .,i d ) obeys the following set of recursive equations: Let us briefly explain how these equations can be obtained.Consider kϭ1 and the origin located at (0, . . .,0).Suppose the random walker has reached the origin at step nϩ1 moving along direction ϩ1.Then, it necessarily was at the lattice point (Ϫ1,0, . . .,0) at step n.The probability that the random walker jumps from this point to the origin depends on the arrival direction to the point (Ϫ1,0, . . .,0).Each direction ͑and there are 2d directions͒ contributes to the total probability P nϩ1 (ϩ1) (0, . . .,0) with a different weight.Indeed, the term ␣ P n (ϩ1) (Ϫ1,0, . . .,0) gives the probability that the random walker arrives at (Ϫ1,0, . . .,0) along direction ϩ1 and keeps going on the same direction.The probability that the random walker reverses its arrival direction after reaching (Ϫ1,0, . . .,0) is ␤P n (Ϫ1) (Ϫ1,0, . . .,0).The rest of the 2(dϪ1) directions contributes with the terms ␥ P n (Ϯk) (Ϫ1,0, . . .,0), with k 1. Equations ͑3͒ and ͑4͒ are easily obtained after generalizing this reasoning to arbitrary points (i 1 , . . .,i d ) and directions Ϯk.These equations completely characterize our extension of the PRW and they are a convenient starting point for numerical analysis when no further analytical treatment can be made.

III. THE CONTINUUM LIMIT
We now proceed to the continuum limit.In this situation the length of each step, l, and the time interval between jumps, , both go to zero in such a way that the random walker moves at finite velocity v, As in the case of the one-dimensional PRW ͓3,4͔, we also have to scale the scattering probabilities in the form ␣ϭ1Ϫ, ␤ϭc, ␥ϭa/2, ͑6͒ where is the interval between jumps and is a parameter whose units are ͓T Ϫ1 ͔.Let us now see what the physical meaning is of the parameters appearing in Eq. ͑6͒.We first observe that, as a direct consequence of the scaling ͑6͒, the occurrence of collision events is governed by the Poisson law.Indeed, in the discrete case the probability that the random walker jumps k times in the same direction is ␣ k .Therefore, in the continuum limit the probability that the particle keeps moving in the same direction for a time t уk is We thus see the physical meaning of the parameter defined in Eq. ͑6͒, since it represents the mean frequency at which the random walker changes its direction, that is, is the mean number of scattering events per unit time, or equivalently Ϫ1 is the mean time between collision events.As a consequence, since ␤ is the probability of reversing direction, we see from Eq. ͑6͒ that c/ is the conditional probability of reversing the direction of motion.Analogously a/(2) is the probability that there is a turn to an orthogonal direction.Finally, the normalization condition requires that cϩ͑dϪ1 ͒aϭ.
We also note that the angle between directions of motion before and after a collision is necessarily a multiple of /2 ͑recall that the discrete model has been built on a cubic lat-tice͒.Therefore, the only possible velocities of the random walker are the 2d values Ϯv i ˆk (kϭ1, . . .,d), where i ˆk is the unit vector in the x k coordinate direction.Figure 1 shows a realization of the model in three dimensions.
In order to get the diffusive limit of Eqs.͑3͒ and ͑4͒, we define tϭn and x k ϭi k l, kϭ1, . . .,d, and let l→0 and →0 with the condition that v defined by Eq. ͑5͒ is constant.In this limit, probabilities P n (Ϯ) (i 1 , . . .,i d ) become ͓ p ͑ϩ j͒ ͑r,t͒ϩp ͑Ϫ j͒ ͑r,t͔͒, ͑9͒ which are the result of applying the continuum limit ͑8͒ to the general recursive equations ͑3͒ and ͑4͒.
In many applications the most interesting quantity is the probability density p(r,t) for the position independent of the velocity, that is, For dϭ1, we already explained that p(x,t) evolves according to a TE, Eq. ͑1͒.We now present the partial differential equation describing the evolution of this density when d у2.We show in Appendix A that this equation reads where ٌ 2 is the Laplacian of the spatial coordinates, bϭ ϩc, and ⌽ d is an operator including all spatial partial derivatives of fourth order or greater.For dϭ2 and dϭ3, the operator ⌽ d has the following expressions:

͒. ͑13͒
We have thus obtained the 2d order partial differential equation that satisfies the probability density function of the persistent random walk in higher dimensions.This equation does not show spherical symmetry because of the velocities allowed by the model.Nevertheless, using the so-called ''dominant balance technique'' ͓18͔ one can easily see that the behavior of the probability density function at long times, and for positions r sufficiently far away from the moving boundary, is given by the lowest-order partial derivatives.This ''central limit approximation'' transforms Eq. ͑11͒ into the diffusion equation ‫ץ‬ t p͑r,t ͒ϭDٌ 2 p͑r,t ͒, ͑14͒ where the diffusion constant D is Note that the diffusion constant is in agreement with the result that follows from transport theory Dϭvl t /d, where l t ϭv/b is the transport mean free path ͓15͔.

IV. THE PROJECTED MOTION
Let us now study the projected motion of the PRW on a given axis.The probability density function of this motion is given by This marginal density obeys a simpler equation than Eq.͑11͒ mainly because the integration of ⌽ d p(r,t) over the coordinates x 2 •••x d is zero.We show in Appendix B that the equation for p(x,t) reads 2 p͑x,t͒.͑16͒ This equation is a third-order hyperbolic partial differential equation with finite velocity of propagation v.The reduction in the order from 2d for Eq.͑11͒ to 3 for Eq.͑16͒ is due to the fact that the projected motion is coupled to the motion along any other direction by a first-order equation ͑see Appendix B͒.Equation ͑16͒ may provide a better approximation to transport problems than the diffusion equation, especially in thin slabs where the effect of taking into account the speed of propagation is crucial ͓16͔.Note that Eq. ͑16͒ possesses a richer structure than the telegrapher's equation.Moreover, we have been able to derive Eq. ͑16͒ from a microscopic model ͑we recall that any attempt to derive TE from a microscopic model of transport fails in higher dimensions͒.Nevertheless, Eq. ͑16͒ shares with TE at least two important features: ͑i͒ a finite velocity of propagation, and ͑ii͒ a similar asymptotic behavior of the moments ͑see below͒.An equation similar to Eq. ͑16͒ was used several years ago in the context of heat propagation in rigid solids ͓17͔.
It is possible to exactly solve Eq. ͑16͒ in the Fourier-Laplace space for the isotropic initial conditions ␦͑r͒, ͑17͒ which provide the three initial conditions required for solving Eq. ͑16͒, ␦Љ͑x͒.
In Appendix C we show how to derive these conditions from Eq. ͑17͒.It is possible to solve Eq. ͑16͒ in the Fourier-Laplace space.To this end we define the joint transform e Ϫst e Ϫix p͑x,t͒dx dt.
Then the transformation of Eq. ͑16͒ and the use of Eq. ͑18͒ lead to the solution We observe that this case is equivalent to a three-state continuous-time random walk where the particle is moving to the right ͑or left͒ with velocity v (Ϫv) or is at rest, which corresponds to the motion in an orthogonal direction.The transition matrix of this three-state walk is ͓3͔ where iϭ1 corresponds to the state with velocity ϩv, i ϭ2 with velocity Ϫv, and iϭ3 to the state without displacement along the x direction.
Let us now study the moments m n (t), nϭ1,2,3, . . ., of the distribution p(x,t).In terms of the characteristic function, p ˜(,t), the moments are given by m n (t) ϭi n ‫ץ‬ p ˆ(,t)͉ ϭ0 .Due to the isotropic initial conditions, Eq. ͑18͒, all odd moments vanish, m 2nϪ1 (t)ϭ0.Then it follows from Eq. ͑19͒ that the Laplace transform of m 2n (t) is given by The leading behavior for small s is and the leading behavior for large s is Then, by the Tauberian theorems ͓3,18͔, we get the asymptotic behaviors and Note that the moments as t→ϱ are identical to those of ordinary diffusion with Dϭv 2 /bd in agreement with the result of the preceding section, which means that p(x,t) behaves in this limit as p͑x,t ͒ϳ 1 ͱ4Dt exp͓Ϫx 2 /4Dt͔ ͑t→ϱ͒.͑24͒ On the other hand, when t→0, the motion behaves as if it were deterministic.In fact, taking into account the isotropic initial conditions, we can easily show that the density p(x,t) has the following expansion at short times:

͑25͒
which immediately leads to the result ͑23͒ for the moments.Therefore, the behavior of moments is similar to that of the solution of the telegrapher's equation.Indeed, we see from Eqs. ͑22͒ and ͑23͒ that if t→ϱ, we have ordinary diffusion ͑where, for instance, the second moment goes as t), while if t→0 the behavior is deterministic ͑where the second moment goes as t 2 ).
We have not been able to invert Eq. ͑19͒ in general.However, the solution in real space can be written for the two- is the Fourier-Laplace transform of the free-space solution of the one-dimensional telegrapher's equation for isotropic initial conditions.Indeed, the Fourier-Laplace transform of Eq. ͑1͒ along with the isotropic initial conditions p te (x,0) ϭ␦(x) and ‫ץ‬ t p te (x,0)ϭ0 leads to Eq. ͑27͒ ͓6,19͔.Therefore, we see from Eq. ͑26͒ that the sought-after expression reads where I n (z) are modified Bessel functions.Surprisingly, p te (x,) is also the solution of a three-state random walk with transition matrix ͑20͒ when the initial conditions have an equal probability of moving to the left or to the right, and zero probability to be at rest.This fact clarifies the origin of solution ͑28͒.Indeed, the initial conditions for that solution were 1/2 probability to be at rest and 1/4 to move in either x direction.The ␦ term in p(x,t) accounts for the probability that the random walker keeps moving along the y axis for all period t.The second term is the contribution of random walkers that are not at rest initially, according to the interpretation for p te (x,) given above.Finally, particles at rest at tϭ0 that begin to move at some time between (0,t) can be considered as a source of particles that evolve as p te (x,).The third term is therefore the convolution of the source function (e Ϫt ) with p te (x,).In Fig. 2, we plot the solution for different times ͑without ␦ terms͒.When t is less than Ϫ1 , the ␦-function terms account for most of p(x,t).As time grows, the contribution of these terms decays exponentially and the solution converges to the Gaussian distribution, Eq. ͑24͒.

V. CONCLUSIONS
The generalization of the persistent random walk to dimensions higher than 1 does not evolve according to a tele-grapher's equation as it does in one dimension.We have explicitly obtained the 2d order partial differential equation governing the evolution of the probability density function.Therefore, higher-dimensional persistent random walks in cubic lattices cannot be considered as microscopic models to derive TE in dimensions greater than 1.
Moreover, even the motion projected along an axis, besides being one-dimensional, does not evolve according to a TE.We have found that the probability density of the projection obeys a third-order partial differential equation.The order of the equation is completely consistent with the fact that the projected motion is equivalent to a three-state random walk.
The evolution equation for the projected motion may be very useful when considered as an approximation to the full transport equation for models of light propagation because Eq. ͑16͒ may overcome the problems of the diffusive approximation when light propagates through thin slabs.In this sense we have shown that the evolution equation for the projected motion correctly links the expected short-time behavior ͑deterministic͒ with the long-time behavior ͑diffusive͒ of these models.Finally, applying ͟ iϭ1 d S i to both sides of Eq. ͑A2͒ and using Eq.͑A5͒, we obtain the equation for p(r,t), The expansion of the product ͟ iϭ1 d S i in powers of D b D ad leads to Eq. ͑11͒.

FIG. 1 .
FIG.1.A realization of the continuum model in three dimensions described in the paper.Note that the random walker moves along direcctions that are parallel to the axes.
dimensional case dϭ2 and when the backscattering probability vanishes, cϭ0.In effect, now aϭbϭ, and the expression for p ˆ(,s) reads p i j , . . .,i d ͒ All the information about the continuous model is contained in the following set of equations: where p(Ϯk)(r,t)dr is the joint probability density function for the position rϭ(x 1 , . . .,x d ) and the velocity of the random walker at time t.