PHYSICAL REVIEW E VOLUME 58, NUMBER 6 DECEMBER 1998 Generalization of the persistent ranom walk to imensions greater than 1 Marián Boguñá, Josep M. Porrà, an Jaume Masoliver Departament e Física Fonamental, Universitat e Barcelona, Diagonal, 647, Barcelona 08028, Spain Receive 16 July 1998 We propose a generalization of the persistent ranom walk for imensions greater than 1. Base on a cubic lattice, the moel is suitable for an arbitrary imension. We stuy the continuum limit an obtain the equation satisfie by the probability ensity function for the position of the ranom walker. An exact solution is obtaine for the projecte motion along an axis. This solution, which is written in terms of the free-space solution of the one-imensional telegrapher s equation, may open a new way to aress the problem of light propagation through thin slabs. S1063-651X 98 00312-2 PACS number s : 05.40. j, 05.60. w, 66.90. r I. INTRODUCTION The persistent ranom walk PRW, first introuce by Fürth 1 an shortly after by Taylor 2, is probably the earliest an simplest generalization of the orinary ranom walk that incorporates some form of momentum in aition to ranom motion. The persistent ranom walk iffers from the orinary ranom walk in that the probabilistic quantity use at each step is the probability of continually moving in a given irection rather than the probability of moving in a given irection regarless of the irection of the preceing step. In this way the PRW introuces some form of momentum, i.e., persistence, into the purely ranom motion. This remarkable feature of the moel is one of the reasons why the PRW has been recently applie to escribe scattering an iffusion in isorere meia 3. For one-imensional lattices an in the continuum or iffusive limit the probability ensity function for the isplacement at time t, p(x,t), satisfies the telegrapher s equation TE 4 : 2 p p 2 2 p v2 2 x 2. As is well known, Eq. 1 has solutions with a finite velocity of propagation given by v 6. This fact has justifie the extensive use of TE as a generalization of the mesoscopic iffusion equations in fiels such as heat propagation 5 an light ispersion in turbi meia 6,7. However, none of the generalizations explore in two an three imensions of persistent ranom walks obeys the TE in the continuum limit 8 11. On the other han, an besies the recent work of Gooy et al. 10 on two-imensional walks, there have been, to our knowlege, very few attempts to generalize the PRW to imensions higher than 1 in spite of its potential for moeling transport in isorere meia. One of the reasons for the lack of such a generalization is the nonexistence of a unique generalization of the PRW to imensions greater than 1 since several kins of lattices cubic, hexagonal, etc. can be use for the extension. Our main goal in this paper is to propose a generalization of the PRW to higher imensions assuming a 1 cubic lattice, an to obtain the governing equations for the probability ensity function of the process in the continuum limit. Unfortunately, in the continuum limit an for imensions greater than 1, the probability ensity function of the process oes not obey a higher-imensional telegrapher s equation. Nevertheless, in the context of transport in isorere meia, the partial ifferential equation escribing particle concentration i.e., the probability ensity function can suggest new approximations for the transport equation of more realistic moels. This is the case, for example, of light propagation in turbi meia where such an approach becomes extremely useful, especially when photons propagate in constraine geometries such as thin slabs where the Gaussian approximation becomes quite imprecise 12. In aition, the telegrapher s equation has been shown not to furnish better results than the iffusion approximation in two an three imensions 13. As we have mentione, in the one-imensional case the equation satisfie by p(x,t) is the telegrapher s equation. In two imensions, the moel consiere herein was partially analyze by Gooy et al. 10 but the equation for the probability ensity function, p(r,t), of the process was not obtaine. Another goal of this paper is to stuy the projecte motion, along a given irection, of the higher-imensional PRW. This projecte motion is relevant in the stuy of light propagation in turbi meia. Inee, when persistent ranom walks are use as moels for light propagation through slabs, the basic information is containe in the motion projecte along the coorinate orthogonal to the faces of the slab 14. We thus obtain the equation that governs the evolution of the projecte motion an write its solution in terms of the freespace solution of the one-imensional telegrapher s equation. The paper is organize as follows. In Sec. II we set the general analysis for the PRW in a cubic lattice of imension. In Sec. III we obtain the continuum limit an fin the governing equation for the probability ensity function of the resulting process. Section IV is evote to the projecte motion on a given coorinate an we obtain an exact expression of the ensity. Conclusions are rawn in Sec. V. II. GENERAL ANALYSIS We consier a ranom walk in a cubic lattice of arbitrary imension. The istance between nearest lattice points is l. 1063-651X/98/58 6 /6992 7 /$15.00 PRE 58 6992 1998 The American Physical Society
PRE 58 GENERALIZATION OF THE PERSISTENT RANDOM... 6993 Jumps to another lattice point occur after a time interval. At each point, the ranom walker can take 2 ifferent irections. Among all possible events, we are only intereste in three of them: a The forwar scattering, where the ranom walker moves in the same irection as the previous jump, b the backwar scattering, where the walker reverses its previous irection, an c the scattering in the 2( 1) remaining irections. Let us enote by,, an the probabilities of events a, b, an c, respectively. As absorption will not be consiere here, the scattering probabilities satisfy the normalization conition 2 1 1. 2 This kin of ranom walk is terme persistent because it introuces a persistent probability an generalizes the oneimensional PRW to arbitrary imensions 3. Let us now set the general equations of this moel. We efine a set of auxiliary probabilities P n ( k) (i 1,...,i ), k 1,...,, where P n ( k) (i 1,...,i ) is the probability that the walker reaches the lattice point (i 1,...,i ) at step n moving along irection k. A similar efinition applies to P n ( k) (i 1,...,i ). Following an analogous reasoning to that of the one-imensional PRW 3, we can see that P n ( k) (i 1,...,i ) obeys the following set of recursive equations: P k n 1 i 1,...,i P k n i 1,...,i k 1,...,i P k n i 1,...,i k 1,...,i P j n i 1,...,i k 1,...,i P n j i 1,...,i k 1,...,i, 3 P k n 1 i 1,...,i P k n i 1,...,i k 1,...,i P k n i 1,...,i k 1,...,i P j n i 1,...,i k 1,...,i P n j i 1,...,i k 1,...,i. 4 Let us briefly explain how these equations can be obtaine. Consier an the origin locate at (0,...,0). Suppose the ranom walker has reache the origin at step n 1 moving along irection 1. Then, it necessarily was at the lattice point ( 1,0,...,0) at step n. The probability that the ranom walker jumps from this point to the origin epens on the arrival irection to the point ( 1,0,...,0). Each irection an there are 2 irections contributes to the total probability P ( 1) n 1 (0,...,0) with a ifferent weight. Inee, the term P ( 1) n ( 1,0,...,0) gives the probability that the ranom walker arrives at ( 1,0,...,0) along irection 1 an keeps going on the same irection. The probability that the ranom walker reverses its arrival irection after reaching ( 1,0,...,0) is P ( 1) n ( 1,0,...,0). The rest of the 2( 1) irections contributes with the terms P ( k) n ( 1,0,...,0), with. Equations 3 an 4 are easily obtaine after generalizing this reasoning to arbitrary points (i 1,...,i ) an irections k. These equations completely characterize our extension of the PRW an they are a convenient starting point for numerical analysis when no further analytical treatment can be mae. III. THE CONTINUUM LIMIT We now procee to the continuum limit. In this situation the length of each step, l, an the time interval between jumps,, both go to zero in such a way that the ranom walker moves at finite velocity v, l v lim. l, 0 5 As in the case of the one-imensional PRW 3,4, we also have to scale the scattering probabilities in the form 1, c, a /2, 6 where is the interval between jumps an is a parameter whose units are T 1. Let us now see what the physical meaning is of the parameters appearing in Eq. 6. We first observe that, as a irect consequence of the scaling 6, the occurrence of collision events is governe by the Poisson law. Inee, in the iscrete case the probability that the ranom walker jumps k times in the same irection is k. Therefore, in the continuum limit the probability that the particle keeps moving in the same irection for a time t k is lim k lim 1 t/ e. 0 0 We thus see the physical meaning of the parameter efine in Eq. 6, since it represents the mean frequency at which the ranom walker changes its irection, that is, is the mean number of scattering events per unit time, or equivalently 1 is the mean time between collision events. As a consequence, since is the probability of reversing irection, we see from Eq. 6 that c/ is the conitional probability of reversing the irection of motion. Analogously a/(2 ) is the probability that there is a turn to an orthogonal irection. Finally, the normalization conition requires that c 1 a. We also note that the angle between irections of motion before an after a collision is necessarily a multiple of /2 7
6994 BOGUÑÁ, PORRÀ, AND MASOLIVER PRE 58 p r,t p k r,t p k r,t. 10 For 1, we alreay explaine that p(x,t) evolves accoring to a TE, Eq. 1. We now present the partial ifferential equation escribing the evolution of this ensity when 2. We show in Appenix A that this equation reas t t b t a 1 t b 1 t a 2 t a v 2 2 p r,t 0, 11 FIG. 1. A realization of the continuum moel in three imensions escribe in the paper. Note that the ranom walker moves along irecctions that are parallel to the axes. recall that the iscrete moel has been built on a cubic lattice. Therefore, the only possible velocities of the ranom walker are the 2 values vî k (,...,), where î k is the unit vector in the x k coorinate irection. Figure 1 shows a realization of the moel in three imensions. In orer to get the iffusive limit of Eqs. 3 an 4, we efine t n an x k i k l,,...,, an let l 0 an 0 with the conition that v efine by Eq. 5 is constant. In this limit, probabilities P ( ) n (i 1,...,i ) become P k n 1 i 1,...,i j,...,i p k r,t p k r,t O 2, P n k i 1,...,i j 1,...,i p k r,t l p k r,t x j O l 2, 8 where p ( k) (r,t)r is the joint probability ensity function for the position r (x 1,...,x ) an the velocity of the ranom walker at time t. All the information about the continuous moel is containe in the following set of equations: p k r,t p k r,t v p k r,t p k r,t cp k r,t 1 2 a p j r,t p j r,t, v p k r,t p k r,t cp k r,t 1 2 a p j r,t p j r,t, which are the result of applying the continuum limit 8 to the general recursive equations 3 an 4. In many applications the most interesting quantity is the probability ensity p(r,t) for the position inepenent of the velocity, that is, 9 where 2 is the Laplacian of the spatial coorinates, b c, an is an operator incluing all spatial partial erivatives of fourth orer or greater. For 2 an 3, the operator has the following expressions: 3 v 6 6 x 2 y 2 z 2 2 v 4 4 x 2 y 2, 12 v 4 t b t 2a x 2 y 2 x 2 z 2 y 2 z 2. 13 We have thus obtaine the 2 orer partial ifferential equation that satisfies the probability ensity function of the persistent ranom walk in higher imensions. This equation oes not show spherical symmetry because of the velocities allowe by the moel. Nevertheless, using the so-calle ominant balance technique 18 one can easily see that the behavior of the probability ensity function at long times, an for positions r sufficiently far away from the moving bounary, is given by the lowest-orer partial erivatives. This central limit approximation transforms Eq. 11 into the iffusion equation t p r,t D 2 p r,t, where the iffusion constant D is D v2 b. 4 4 4 14 15 Note that the iffusion constant is in agreement with the result that follows from transport theory D vl t /, where l t v/b is the transport mean free path 15. IV. THE PROJECTED MOTION Let us now stuy the projecte motion of the PRW on a given axis. The probability ensity function of this motion is given by p x,t p r,t x2 x. This marginal ensity obeys a simpler equation than Eq. 11 mainly because the integration of p(r,t) over the coorinates x 2 x is zero. We show in Appenix B that the equation for p(x,t) reas t t b t a p x,t v 2 t a 2 x 2p x,t. 16
PRE 58 GENERALIZATION OF THE PERSISTENT RANDOM... 6995 This equation is a thir-orer hyperbolic partial ifferential equation with finite velocity of propagation v. The reuction in the orer from 2 for Eq. 11 to 3 for Eq. 16 is ue to the fact that the projecte motion is couple to the motion along any other irection by a first-orer equation see Appenix B. Equation 16 may provie a better approximation to transport problems than the iffusion equation, especially in thin slabs where the effect of taking into account the spee of propagation is crucial 16. Note that Eq. 16 possesses a richer structure than the telegrapher s equation. Moreover, we have been able to erive Eq. 16 from a microscopic moel we recall that any attempt to erive TE from a microscopic moel of transport fails in higher imensions. Nevertheless, Eq. 16 shares with TE at least two important features: i a finite velocity of propagation, an ii a similar asymptotic behavior of the moments see below. An equation similar to Eq. 16 was use several years ago in the context of heat propagation in rigi solis 17. It is possible to exactly solve Eq. 16 in the Fourier- Laplace space for the isotropic initial conitions p k r,t p k r,t 1 2 r, 17 which provie the three initial conitions require for solving Eq. 16, p x,t 0 x, 2 p x,t 2 p x,t 0, 0 0 v2 x. 18 In Appenix C we show how to erive these conitions from Eq. 17. It is possible to solve Eq. 16 in the Fourier- Laplace space. To this en we efine the joint transform pˆ,s e st e i x p x,t x t. 0 Then the transformation of Eq. 16 an the use of Eq. 18 lea to the solution pˆ,s s a s b v 2 2 1 1/ s s a s b v 2 2 s a. 19 We observe that this case is equivalent to a three-state continuous-time ranom walk where the particle is moving to the right or left with velocity v ( v) or is at rest, which correspons to the motion in an orthogonal irection. The transition matrix of this three-state walk is 3 c/ 1 a/ p i j 0 c/ 0 1 a/, 20 a/ 2 a/ 2 1 a/ where i 1 correspons to the state with velocity v, i 2 with velocity v, an i 3 to the state without isplacement along the x irection. Let us now stuy the moments m n (t), n 1,2,3,..., of the istribution p(x,t). In terms of the characteristic function, p (,t), the moments are given by m n (t) i n pˆ (,t) 0. Due to the isotropic initial conitions, Eq. 18, all o moments vanish, m 2n 1 (t) 0. Then it follows from Eq. 19 that the Laplace transform of m 2n (t) is given by mˆ 2n s 2n! s a s s a The leaing behavior for small s is v n 1 s s b 2 n. 21 mˆ 2n s v2n 2n! b n n s 1 n s 0 an the leaing behavior for large s is mˆ 2n s v2n 2n! s 1 2n s. Then, by the Tauberian theorems 3,18, we get the asymptotic behaviors an m 2n 2n! n! v2 n b t 22 m 2n v2n t2n 0. 23 Note that the moments as t are ientical to those of orinary iffusion with D v 2 /b in agreement with the result of the preceing section, which means that p(x,t) behaves in this limit as p x,t 1 4 Dt exp x2 /4Dt. 24 On the other han, when t 0, the motion behaves as if it were eterministic. In fact, taking into account the isotropic initial conitions, we can easily show that the ensity p(x,t) has the following expansion at short times: p x,t 1 2 x vt 1 2 1 x vt x 0, 25 which immeiately leas to the result 23 for the moments. Therefore, the behavior of moments is similar to that of the solution of the telegrapher s equation. Inee, we see from Eqs. 22 an 23 that if t, we have orinary iffusion where, for instance, the secon moment goes as t), while if t 0 the behavior is eterministic where the secon moment goes as t 2 ). We have not been able to invert Eq. 19 in general. However, the solution in real space can be written for the two-
6996 BOGUÑÁ, PORRÀ, AND MASOLIVER PRE 58 imensional case 2 an when the backscattering probability vanishes, c 0. In effect, now a b, an the expression for pˆ (,s) reas pˆ,s 1 2 s 1 2 pˆ te,s 1 2 s where s 2 pˆ te,s s s 2 v 2 2 pˆ te,s, 26 27 is the Fourier-Laplace transform of the free-space solution of the one-imensional telegrapher s equation for isotropic initial conitions. Inee, the Fourier-Laplace transform of Eq. 1 along with the isotropic initial conitions p te (x,0) (x) an t p te (x,0) 0 leas to Eq. 27 6,19. Therefore, we see from Eq. 26 that the sought-after expression reas p x,t 1 2 e x 1 2 p te x,t 2 e 0 t e p te x,, where p te (x,t) is 4,19 p te x,t e 2v e x /v 2v x /v I 0 2 x 2 /v 2 t 2 x 2 /v I 1 2 x 2 /v 2, 2 28 29 where I n (z) are moifie Bessel functions. Surprisingly, p te (x, ) is also the solution of a three-state ranom walk with transition matrix 20 when the initial conitions have an equal probability of moving to the left or to the right, an zero probability to be at rest. This fact clarifies the origin of solution 28. Inee, the initial conitions for that solution were 1/2 probability to be at rest an 1/4 to move in either x irection. The term in p(x,t) accounts for the probability that the ranom walker keeps moving along the y axis for all perio t. The secon term is the contribution of ranom walkers that are not at rest initially, accoring to the interpretation for p te (x, ) given above. Finally, particles at rest at t 0 that begin to move at some time between (0,t) can be consiere as a source of particles that evolve as p te (x, ). The thir term is therefore the convolution of the source function ( e ) with p te (x, ). In Fig. 2, we plot the solution for ifferent times without terms. When t is less than 1, the -function terms account for most of p(x,t). As time grows, the contribution of these terms ecays exponentially an the solution converges to the Gaussian istribution, Eq. 24. V. CONCLUSIONS FIG. 2. Probability ensity function p(x,t) for the motion projecte along the x irection at ifferent times as a function of the position x. Soli line: 2xt 1; ashe line: 2 5; ot-ashe line: 2 10. The generalization of the persistent ranom walk to imensions higher than 1 oes not evolve accoring to a telegrapher s equation as it oes in one imension. We have explicitly obtaine the 2 orer partial ifferential equation governing the evolution of the probability ensity function. Therefore, higher-imensional persistent ranom walks in cubic lattices cannot be consiere as microscopic moels to erive TE in imensions greater than 1. Moreover, even the motion projecte along an axis, besies being one-imensional, oes not evolve accoring to a TE. We have foun that the probability ensity of the projection obeys a thir-orer partial ifferential equation. The orer of the equation is completely consistent with the fact that the projecte motion is equivalent to a three-state ranom walk. The evolution equation for the projecte motion may be very useful when consiere as an approximation to the full transport equation for moels of light propagation because Eq. 16 may overcome the problems of the iffusive approximation when light propagates through thin slabs. In this sense we have shown that the evolution equation for the projecte motion correctly links the expecte short-time behavior eterministic with the long-time behavior iffusive of these moels. ACKNOWLEDGMENTS This work has been supporte in part by Dirección General e Investigación CientíficayTécnica uner Contract No. PB96-0188, an by Societat Catalana e Física Institut Estuis Catalans. APPENDIX A: DERIVATION OF EQ. 11 Let us efine the following functions: U k r,t p k r,t p k r,t, V k r,t p k r,t p k r,t, A1 for,...,. The probability ensity of the position becomes
PRE 58 GENERALIZATION OF THE PERSISTENT RANDOM... 6997 an the set of Eqs. 9 reas U k r,t p r,t U k r,t, v V k r,t au r,t ap r,t, k A2 where x 1 was enote by x an U k (x,t) an V k (x,t) correspon to the functions U k (r,t) an V k (r,t) integrate over the coorinates x 2 x. The first equation of system B1 implies that the quantity q x,t U k x,t k 2 V k r,t v U k r,t bv r,t, A3 k where b c. If we efine the operator D t, where is a constant, then the equation for U k (r,t) becomes D b ap r,t D a U k r,t v 2 2 U k r,t,,...,. 2 A4 For 1, this equation coincies with the telegrapher s equation because, in this case, a 0 an U 1 (x,t) p(x,t). For 2, the efinition of the operator S k D b D a v 2 2 xk further simplifies Eq. A4 into S k U k r,t ad b p r,t. A5 Finally, applying i 1 S i to both sies of Eq. A2 an using Eq. A5, we obtain the equation for p(r,t), i 1 S i p r,t ad b i k S i p r,t. A6 The expansion of the prouct i 1 S i in powers of D b D a leas to Eq. 11. APPENDIX B: DERIVATION OF EQ. 16 The integration of the system A3 over the coorinates x 2 x yiels satisfies the first-orer equation q x,t aq x,t a 1 U 1 x,t q x,t. B3 This equation along with system B2 leas to a thir-orer partial ifferential equation for p x,t U 1 x,t q x,t. Inee, from Eqs. B1 an B2 we get an t D b p x,t v 2 x 2 U 1 x,t D a U 1 x,t D a p x,t. B4 B5 Finally, the combination of Eqs. B4 an B5 immeiately leas to Eq. 16. APPENDIX C: INITIAL CONDITIONS The integration of the isotropic initial conitions 17 over the coorinates x 2 x gives the initial conitions for the marginal probability p k x,t p k x,t 1 2 x. The first initial conition follows irectly from them, p x,t 0 p k x,t 0 p k x,t 0 x. U k x,t au k x,t ap x,t, B1 Taking into account that V k (x,t 0) 0, we see from Eq. A3 that for k 2,..., an U 1 x,t V k x,t bv k x,t, v V 1 x,t au x 1 x,t ap x,t, V 1 x,t v U 1 x,t bv x 1 x,t, B2 U k x,t a 1 0 x a x 0, an therefore the secon initial conition reas p x,t 0. 0 C1 C2 After taking the erivative with respect to time in the equa-
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