Persistent memory of diffusing particles

Nicolae Suciu
(original), ISI/JCR, Numerical Modeling, paper

Abstract

The variance of the advection-diffusion processes with variable coefficients is exactly decomposed as a sum of dispersion terms and memory terms consisting of correlations between velocity and initial positions. For random initial conditions, the memory terms quantify the departure of the preasymptotic variance from the time-linear diffusive behavior. For deterministic initial conditions, the memory terms account for the memory of the initial positions of the diffusing particles. Numerical simulations based on a global random walk algorithm show that the influence of the initial distribution of the cloud of particles is felt over hundreds of dimensionless times. In case of diffusion in random velocity fields with finite correlation range the particles forget the initial positions in the long-time limit and the variance is self-averaging, with clear tendency toward normal diffusion.

Authors

N. Suciu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

C. Vamoş
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

F.A. Radu
Computational Hydrosystems, Helmholtz Center for Environmental Research–UFZ, Leipzig, Germany 4
Institute of Geosciences, University of Jena, Jena, Germany

H. Vereecken
Research Center Jülich, Agrosphere Institute ICG-IV, Jülich, Germany

P. Knabner
Chair for Applied Mathematics I, Friedrich-Alexander University Erlangen-Nuremberg, Erlangen, Germany

Keywords

Cite this paper as:

Suciu, N., C. Vamoş, F.A. Radu, H. Vereecken, P. Knabner, Persistent memory of diffusing particles, Phys. Rev. E , 80 (2009), 061134,
doi: 10.1103/physreve.80.061134

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Physical Review E

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American Physical Society

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10.1103/physreve.80.061134

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Paper (preprint) in HTML form

Persistent memory of diffusing particles N. Suciu, 1,2, * C. Vamoş, 2 F. A. Radu, 3,4 H. Vereecken, 5 and P. Knabner 1 1 Chair for Applied Mathematics I, Friedrich-Alexander University Erlangen-Nuremberg, Erlangen, Germany 2 Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj Napoca, Romania 3 Computational Hydrosystems, Helmholtz Center for Environmental Research–UFZ, Leipzig, Germany 4 Institute of Geosciences, University of Jena, Jena, Germany 5 Research Center Jülich, Agrosphere Institute ICG-IV, Jülich, Germany Received 30 September 2009; published 28 December 2009 The variance of the advection-diffusion processes with variable coefficients is exactly decomposed as a sum of dispersion terms and memory terms consisting of correlations between velocity and initial positions. For random initial conditions, the memory terms quantify the departure of the preasymptotic variance from the time-linear diffusive behavior. For deterministic initial conditions, the memory terms account for the memory of the initial positions of the diffusing particles. Numerical simulations based on a global random walk algorithm show that the influence of the initial distribution of the cloud of particles is felt over hundreds of dimensionless times. In case of diffusion in random velocity fields with finite correlation range the particles forget the initial positions in the long-time limit and the variance is self-averaging, with clear tendency toward normal diffusion. DOI: 10.1103/PhysRevE.80.061134 PACS numbers: 05.60.Cd, 02.50.Ey, 05.10.Gg, 05.10.Ln I. INTRODUCTION A stochastic process has diffusive behavior when its vari- ance is a linear-time function. The simplest example is the advection-diffusion process with constant coefficients de- scribed by a Gaussian normalized concentration one-time probability densitycx , t= 4Dt -1/2 exp-x - Vt 2 / 4Dt. The mean tand the variance stare both linear-time functions, t= xcx, tdx = Vt , st= x - t 2 cx, tdx =2Dt , and their time derivatives give the constant velocity and dif- fusion coefficients of the process 1, V = d dt t, D = d 2dt st. 1 In this case, the diffusion coefficient D describes both the shape of the Gaussian distribution cx , tand the width of the diffusion front st. For transport in systems with space-time- variable properties, the mean and the variance are no longer related with the variable velocity and diffusion coefficients by the simple relations Eq. 1. In Sec. II we derive general relations between coefficients and the covariance of the advection-diffusion process. These relations show that the variable diffusion coefficients con- tribute to the covariance of the process by the time integral of their expected values. The variability of the velocity in- stead yields two different contributions: dispersion terms, ex- pressed by Taylor-Kubo relations as time integrals of the Lagrangian velocity correlation, and memory terms, account- ing for correlations between initial positions of the diffusing particles and their Lagrangian velocity. In Sec. III we prove that the extinction of the memory terms in the long-time limit is a necessary condition for the occurrence of the nor- mal diffusion. Statistical physics approaches for spatially homogeneous systems 2,3or for transport in inhomogeneous and rapidly fluctuating velocity fields, as in plasma physics 4, are con- cerned with memory effects which characterize the departure of the process from diffusive behavior. Such effects are re- lated to the occurrence of the anomalous transport 4,5and are usually described by nonlocal equations, containing memory kernels which govern the velocity autocorrelation function or the ensemble average of a transported scalar 6–9. Our approach to investigate memory effects, though related to those cited above, is somehow simpler and straightforward. Instead of describing memory effects by convolution memory kernels 2,7,8, we investigate the time behavior of the memory terms. The latter can be expressed in general for either continuous time-space or discrete transport processes as correlations between displacements of the dif- fusing particles and their initial positions. Such correlations relate the linear-time diffusive behavior of the variance to the lose of memory of the past itinerary of the particles. This condition of diffusive behavior generalizes to the case of variable coefficients the independence of increments of the Wiener process Sec. III. Our approach allows one to treat in a unitary way memory effects manifested by departure from normal diffusive behav- ior and the memory of the initial positions of the particles. In Sec. IV we show, via a numerical experiment, that memory terms also quantify the persistent influence of the determin- istic initial conditions on the variance of the transport pro- cess and its departure from model statistical descriptions by ensemble averages. In particular, it is shown that the self- averaging behavior of the effective coefficients of diffusion in random velocity fields 10,11is clearly related to the * suciu@am.uni-erlangen.de PHYSICAL REVIEW E 80, 061134 2009 1539-3755/2009/806/06113412©2009 The American Physical Society 061134-1
destruction of the memory terms. Section V summarizes the results and outlines directions for further work. II. DISPERSION AND MEMORY TERMS We consider, for the beginning, the continuous diffusion process with space-time-variable diffusion coefficients D ij x , tand velocity components V i x , t, i, j =1,2,3. The density of the transition probability gx , t x 0 , t 0 is the solu- tion of the Fokker-Planck equation t g + x i V i g= x i x j D ij g2 for the initial condition gx , t 0 x 0 , t 0 = x - x 0 . The time evolution of the normalized concentration is given by cx, t= gx, tx 0 , t 0 cx 0 , t 0 dx 0 , 3 where cx 0 , t 0 is the initial concentration and the integral extends over the entire space. A diffusion process is defined by the following conditions uniformly satisfied in x and t for all 0 1,12,13: lim t→0 1 t x-x gx, t + tx, tdx=0, 4 V i x, t= lim t→0 1 t x-x x i - x i gx, t + tx, tdx, 5 D ij x, t= 1 2 lim t→0 1 t x-x x i - x i x j - x j gx, t + tx, tdx. 6 Condition 4ensures the continuity with probability 1 for the trajectories of the diffusion process. The velocity compo- nents V i and the diffusion coefficients D ij of the Fokker- Planck equation Eq. 2 are defined in Eqs. 5and 6by means of the first two local moments computed on spheres of radius of the transition probability of the process start- ing at x , t. Since the mean and the variance in Eq. 1are integrals over the entire space, it follows that for processes with constant coefficients two more conditions are fulfilled for every 0, lim t→0 1 t x-x x i gx, t + tx, tdx=0, 7 lim t→0 1 t x-x x i x j gx, t + tx, tdx= 0. 8 Conditions 7and 8correspond to the physically reason- able assumption that the first two moments are finite at finite times 12. In the following we restrict the solutions of the Fokker-Planck equation Eq. 2 to the class of transition probabilities obeying Eqs. 7and 8. In this case, the inte- grals in the definitions Eqs. 5and 6 of the coefficients can be extended over the entire space 12. Using the definitions Eqs. 5and 6 of the coefficients and the constraints on the transition probability Eqs. 4, 7, and 8, we computed in Appendix A the components of the mean i and of the covariance s ij , which have the following explicit dependence on the coefficients of the Fokker-Planck equation Eq. 2: i t, t 0 = x i cx, tdx = i t 0 + t 0 t V ¯ i tdt, 9 s ij t, t 0 = x i - i tx j - j tcx, tdx = s ij t 0 +2 t 0 t dt D ij x, tcx, tdx + s u,ij t, t 0 + m ij t, t 0 , 10 s u,ij t, t 0 = t 0 t dt t 0 t dt cx 0 , t 0 dx 0  u i x, tu j x, t+ u j x, tu i x, t gx, tx, tgx, tx 0 , t 0 dxdx, 11 m ij t, t 0 = t 0 t dt cx 0 , t 0 dx 0 x 0 j - j t 0 u i x, t + x 0i - i t 0 u j x, tgx, tx 0 , t 0 dx , 12 where V ¯ i t= V i x, tcx, tdx 13 is the mean velocity and u i x , t= V i x , t- V ¯ i tis the veloc- ity fluctuation. The physical meaning of the contributions Eqs. 11and 12 to the covariance Eq. 10 becomes clearer when we express these terms within the Lagrangian framework as av- erages over the trajectories of the diffusing particles 9. To this end, we note that for a given diffusion process with sufficiently smooth coefficients Eqs. 5and 6, it is al- ways possible to construct weak solutions of the associated Itô equation with transition probability densities described by the Fokker-Planck equation Eq. 2 12,13. Let us consider a diffusion process X i t, t t 0 , i =1,2,3with stationary velocity Vxand isotropic constant diffusion coefficients D ij = D and the Itô equation X i t= X 0i + t 0 t V i Xtdt+ W i t - t 0 , 14 where W i t - t 0 is a Brownian motion modeled as a Wiener process with mean zero and variance 2Dt - t 0 13. The Itô equation, which when written in differential form dX i t = V i Xtdt + dW i tis also referred to as Langevin equation, is often used as model for the movement of diffusing par- ticles in physical systems 1,14 –16. SUCIU et al. PHYSICAL REVIEW E 80, 061134 2009 061134-2
In the following we denote averages over trajectories of particles starting at given initial positions and averages with respect to initial distributions of particles by angular brackets with subscripts D and X 0 , respectively, and we pass to a Lagrangian description using the expressions of the probabil- ity densities derived in Appendix B. From Eq. B1it be- comes evident that the mean velocity Eq. 13 is the aver- age of the velocity field viewed by particles moving on trajectories, V ¯ i t= V i X i t DX 0 , i.e., it is the average of the Lagrangian velocity 17. In the same way, using Eq. B2, the integrands in Eqs. 11and 12are expressed in terms of autocorrelation and correlation with initial positions of the Lagrangian velocity. The diagonal components of the cova- riance Eq. 10 become s ii t, t 0 = s ii t 0 +2Dt - t 0 +2 t 0 t dt t 0 t u i Xt, tu i Xt, t DX 0 dt +2 t 0 t X i t 0 - X i t 0  DX 0 u i Xt DX 0 dt. 15 The relation Eq. 15 has also been derived directly from Eq. 14by using the Itô formalism 18. As follows from the comparison with the third term in Eq. 15, the contribution s u,ij Eq. 11 to the covariance Eq. 10 is of the Taylor-Kubo type 9,19, i.e., given by time integrals of the Lagrangian velocity correlation. The novelty of this result is that Eq. 11is a Taylor-Kubo relation valid for a deterministic velocity field or for a fixed realization of a random field. The contribution Eq. 11 of the variable velocity field is an equivalent expression of the usual term, given by a time integral of the correlation position velocity see Eq. A3, derived from computations of the spatial mo- ments of the concentration field 20or within the Lagrang- ian approach used in this paper e.g., 21. Note that in previous works Taylor-Kubo contributions to the covariance of the advection-diffusion process or to the effective diffu- sion coefficients were obtained after averaging over en- sembles of velocity realizations 9,19,22,24or as first-order approximations in velocity variance 20,21,23,24. The last term of Eq. 15shows that Eq. 12is the cor- relation of the Lagrangian velocity with the initial position cumulated in time. Hence m ii describes the memory of the diffusing particles. Further, using Eq. 14, we rewrite Eq. 15as s ii t, t 0 = s ii t 0 + s ˜ ii t, t 0 + m ii t, t 0 , 16 where the sum of second and third terms of Eq. 15is ex- pressed in terms of displacements X ˜ i t= X i t- X i t 0 , s ˜ ii t, t 0 = ŠX ˜ i t- X ˜ i t DX 0 2 ‹ DX 0 . 17 Hence, Eq. 17is the dispersion of the random variable X ˜ i . The memory terms m ii can be equivalently expressed as cor- relations between displacements and initial positions of par- ticles, m ii t, t 0 =2ŠX i t 0 - X i t 0  DX 0 X ˜ i t- X ˜ i t DX 0 ‹ DX 0 . 18 Thus the variance Eq. 16 consists of a sum of two disper- sion terms, s ii t 0 + s ˜ ii t , t 0 , and a memory term, m ii t , t 0 . Though Eqs. 16–18were introduced here by using the equivalence of the Fokker-Plank and Itô descriptions for dif- fusion processes, it is noteworthy to mention that they are general relations that hold true for any continuous or discrete time-space process 25. In fact, Eq. 16is strictly equiva- lent to the usual definition of the variance, s ii = ŠX i - X i DX 0 2 ‹ DX 0 . For deterministic processes, Eq. 16without averages with subscript D is a decomposition of the moment of inertia s ii of the cloud of particles moving on known tra- jectories X i t, for instance, the trajectories of the dynamical system generated by a nonsingular velocity field 26. III. MEMORY TERMS FOR RANDOM INITIAL CONDITIONS Let us consider a process Xt, starting at t 0 from Xt 0 = 0. Since s ii t 0 and m ii t , t 0 Eq. 18 vanish, Eq. 16 reduces to s ii t , t 0 = s ˜ ii t , t 0 , which is the dispersion of the random variable X ˜ i t= X i t. If one observes the same pro- cess from time t 0 on, then the initial positions are random variables X i , outcome of the evolution of the process for t . In these conditions, the dispersion of the total displace- ments X i tcan be written according to Eq. 16as s ˜ ii t, t 0 = s ˜ ii , t 0 + s ˜ ii t, + m ii t, , 19 where the terms s ˜ ii are dispersions given by Eq. 17for X ˜ i equal to X i t, X i , and X i t- X i , respectively, and the memory term describes the correlation of the successive dis- placements, m ii t, =2ŠX ˜ i - X ˜ i DX 0 X ˜ i t- X ˜ i DX 0 t‹ DX 0 . 20 According to Eq. 19, cancellation of memory terms m ii t , is equivalent to additivity of the dispersion s ˜ ii with respect to nonoverlapping time intervals, s ˜ ii t , t 0 = s ˜ ii , t 0 + s ˜ ii t , . In particular, if the dispersion of the particle dis- placements is a linear function of time, then the memory terms necessarily vanish. This is obviously the case of the Wiener process starting at 0,0, for which the memory term Eq. 20 vanishes because the increments of the process are independent, W i - W i 0W i t- W i  D =0. A. Discrete diffusion processes with finite memory The following example illustrates the case of processes with finite memory which after a transient time reaches a diffusive regime characterized by linearity of the dispersion with respect to time. The trajectory of the Wiener process can be simulated numerically by summing up Gaussian random variables, Z = Z n , n =0, 1, 2,..., of mean zero and unit variance. This is the particular case = 1 of the more general algo- PERSISTENT MEMORY OF DIFFUSING PARTICLES PHYSICAL REVIEW E 80, 061134 2009 061134-3
rithm for autoregressive processes of order 1 generated by the recursive relation X n = X n-1 + Z n . For 0 1, the dis- crete process X = X n is stationary, with mean zero, constant variance s X =1 / 1- 2 , and autocovariance X n X n+r = s X r . Finite sequences are also stationary, with mean zero and with the same covariance, if the first term X 0 is chosen as a ran- dom variable with the same variance s X as the infinite autore- gressive processes 27. Summing up realizations of X n is a simple way to obtain diffusion processes with memory. The process Y n , n 0starting from Y 0 = 0 generated by Y n = Y n-1 + X n = s=1 n X s has the expectation Y n = 0 and, because X n is stationary, the variance s Y n = Y n 2 can be expressed by a discrete Taylor- Kubo relation, s Y n = s=1 n X s 2 +2 r=1 n-1 s=1 n-r X s X s+r = ns X +2s X r=1 n-1 n - r r . 21 Einstein formula gives a finite diffusion coefficient 28, D = lim n→ s Y n 2n = s X 2 1+2 1- . 22 The discrete form of Eq. 19is s ˜ Y n = s ˜ Y l + s ˜ Y n,l + m Y n,l , where s ˜ Y n = s Y n whereas s ˜ Y l and s ˜ Y n,l are expressed analogous to Eq. 21after replacing the upper summation limit by l and the lower limit by l +1, respectively. It is ready to check that, for fixed l, lim n→ s ˜ Y l + s ˜ Y n,l / 2n= D; that is, the memory term m Y n,l has no contribution to the diffusion coef- ficient Eq. 22. This can also be checked directly by com- puting the memory term m Y n,l =2Y l - Y 0 Y n - Y l  =2ls X r=l n-1 r =2ls X 1- n-l 1- l . 23 Since the limit n → of Eq. 23is a constant, 2ls X l / 1- , the term m Y n,l / 2nhas no contribution to D. More- over, the limit n, l → , n l of Eq. 23vanishes, hence the increments of the diffusion process with memory Y n , n 0become independent in the long-time limit. B. Diffusion in random velocity fields If the velocity is a realization of a random space function, a key issue is whether the average over the ensemble of velocity realizations of the diffusion process described by Eqs. 2and 14behaves diffusively at some large time scale 19,22,29. The relevant variance is now ŠX i - X i DX 0 V 2 ‹ DX 0 V , which, as follows from Eq. B1, is the second, central, spatial moment of the ensemble average concentration cx , t V = x - Xt DX 0 V . Here and in the following, angular brackets with subscript V denote en- semble averages.The identity Eq. 19 can be used in investigations on the second moment of the mean concentra- tion when averages ¯ DX 0 are replaced by ¯ DX 0 V . Fur- ther, let us consider the Itô process, X i t= X i + t V i Xtdt+ W i t - , 24 with the same coefficients and with initial position given by the solution X i of the Eq. 14at the moment . Assuming that the Itô equation has unique solutions, from Eqs. 14and 24one obtains the following explicit dependence of the memory term Eq. 20 on the Lagrangian correlation func- tion C L : m ii t, =2 t 0 C ii L t, tdtdt, 25 where C ii L t, t= V i Xt, tV i Xt, t DX 0 V - V i Xt, t DX 0 V V i Xt, t DX 0 V . 26 While the question whether the Lagrangian correlation in- herits properties of the Eulerian correlation of the random space function Vxhas no simple answer and requires prov- ing specific limit theorems 19,22, some insight can how- ever be provided by formal asymptotic expansions 29. The simplest approach is that considering statistically homoge- neous velocity fields and first-order approximations of the transport equations. For advection dominated transport prob- lems, a consistent first-order approximation to the solutions of Itô equation Eq. 14 with initial condition Xt 0 = 0 is obtained by the first iteration about the unperturbed solution X i 0 t= i,1 Ut, where U is the constant mean of the velocity field Vx V , assumed to be oriented along the one axis of the coordinate system: X i 1 t= t 0 t V i Utdt+ W i t - t 0 18,23,24. In this approximation the arguments of V i in Eq. 26have to be replaced by Utand Utand the Lagrangian correlation will be C ii L t, t= C ii E Ut- t, where C ii E r = V i xV i x + r V - V i x V V i x + r V is the homogeneous Eulerian correlation. The first-order approximation of the dispersion s ˜ ii t , t 0 , associated to the ensemble average con- centration, becomes s ˜ ii t, t 0 =2Dt +2 t 0 t dt t 0 t C ii E Ut- tdt. 27 If the Eulerian field has finite correlation lengths ii =1 / C ii E 0C ii E rdr, then the first-order approximation of the Lagrangian field also has finite correlation times ii / U 24. Heuristically, because the velocity field has finite cor- relation scale, one assumes that the displacements over times larger than the correlation time will be uncorrelated and by the central limit theorem the process of diffusion in random fields behaves like a Gaussian diffusion e.g., the macrodis- persion model for transport of solutes in geological porous formations 30. Nevertheless, rigorous proofs of this state- ment are only obtained under stronger assumptions on the velocity field 19,22. Following a common approach in physics literature 14,31, the behavior of the process at large distances can be SUCIU et al. PHYSICAL REVIEW E 80, 061134 2009 061134-4
described by considering white noise correlations C ii E = ii r. Then, the memory term Eq. 25 can be com- puted exactly and yields m ii t , =0. Similarly, from Eq. 27 one obtains the linear dispersion s ˜ ii t , t 0 =2D + ii / Ut - t 0 . Thus, for white noise correlations the memory terms vanish at all times, the displacement increments from Eq. 20are uncorrelated, like in the case of a Gaussian diffu- sion, and the dispersion Eq. 19 is a linear-time function. In case of finite correlation lengths, as for instance the often used isotropic exponential C ii L = C ii E 0e -t-tU/ and Gaussian C ii L = C ii E 0e -t- t 2 U/ correlations, information on the asymptotic behavior can be obtained by considering the long-time limit of the time derivatives of the dispersion and memory terms. From Eq. 25one obtains lim t→ d dt m ii t, = 2 lim t→ 0 C ii L t - tdt=0 because for t - tU  the Lagrangian correlation C L will be very close to zero. The derivative of the dispersion Eq. 27 instead approaches a finite limit and defines upscaled diffusion coefficients D ii = 1 2 lim t→ d dt s ˜ ii t, t 0 = D + lim t→ t 0 t C ii E Ut - tdt = D + C E 0 U . 28 Since, according to Eq. 28, the dispersion s ˜ ii has a long- time-linear behavior, Eq. 19implies the cancellation of the memory terms m ii . It follows that for sufficiently fast decay of the Lagrangian correlations, the diffusing particles forget the memory and behave diffusively like Brownian particles. A special situation, for which no approximations are re- quired, is that of a system of particles undergoing diffusion in a stratified velocity field. If the advective velocities of the particles have only longitudinal components which depend randomly on the transverse coordinate alone and a white noise correlation, then the Lagrangian correlation Eq. 26 can be computed exactly: C 11 L = C 11 E 0e t- t -1/2 / 2 D 14,31. Since the integral of C 11 L diverges, the longitudinal Lagrangian velocity has an infinite correlation time and the decorrelation of the displacements from Eq. 20cannot be expected. The longitudinal memory term computed from Eq. 25, m 11 t, = 2C 11 E 0 3 D t - t 0 3/2 - - t 0 3/2 - t - 3/2 , 29 expresses, according to Eq. 19, the nonlinearity of the dis- persion terms with a t 3/2 time behavior obtained explicitly by replacing C 11 L in Eq. 27, m 11 t , = s ˜ ii t , t 0 - s ˜ ii , t 0 + s ˜ ii t , . The memory term Eq. 29 is nonvanishing at all times t and tends to infinity for t → . This shows that, according to Eq. 20, the particles undergoing diffusion in perfectly stratified velocity fields never forget the initial ran- dom position X 1 . Or equivalently, according to Eq. 25, the particles always remember the past Lagrangian velocity they had before the initial observation time . IV. MEMORY EFFECTS FOR DETERMINISTIC INITIAL CONDITIONS Now, we consider the memory terms Eq. 18 for initial positions X i t 0 that are no longer the outcome of the evolu- tion of the same process but arbitrary deterministic quanti- ties. Since in this case the correlation of increments in Eq. 18cannot be expressed by correlations of the Lagrangian velocity, the issue was investigated through numerical ex- periments. Simulations of diffusion of large collections of particles in realizations of a random velocity field were carried out with the global random walk GRWalgorithm 32. Though equivalent with a superposition of many particle tracking procedures Euler schemes for the Itô equation Eq. 14, GRW is rather a cellular automaton: at given time step all the particles located at grid points are simultaneously advected with the local velocity and spread according to the random walk rule. This allows global simulations of diffusion for huge numbers of particles which render the statistical fluc- tuations of the estimated expectations ¯ DX 0 e.g., concen- tration momentssmaller than the limit of double precision of the computing platform. For the simulations presented in the following this precision was ensured by using 10 10 par- ticles. We considered a hydrological problem of contaminant transport through an isotropic two-dimensional aquifer sys- tem, characterized by logarithmic-normal distributed hydrau- lic conductivity K with small variance ln K 2 = 0.1 and expo- nentially decaying isotropic correlation with correlation length = 1 m. Darcy velocity fields were approximated nu- merically by Gaussian fields 11,23. For fixed mean flow velocity U =1 m / d and isotropic local dispersion with con- stant coefficient D =0.01 m 2 / d, the Péclet number got a typical value Pe= U/ D = 100. Details on algorithm and nu- merical setup can be found in Ref. 33. A. Memory terms for fixed velocity realizations To estimate memory terms Eq. 18 for a single realiza- tion we considered a number of 121 initial positions Xt 0 uniformly distributed in rectangular domains with dimen- sions L 1  L 2 . By releasing 10 10 particles from each initial position, we performed GRW simulations to compute the displacements along the trajectories of the diffusion pro- cesses starting from these positions. Finally, the correlations between displacements and initial positions from Eq. 18 were computed by averages over the initial positions Xt 0 . The results presented in Fig. 1 show that at early times the memory terms m ii increase with the dimension of the source in the i direction and are mainly significant for asymmetric sources. The overall decay of m ii in all cases indicates that the averages u i Xt D of the fluctuations of the Lagrangian velocity averaged over realizations of the diffusion process for fixed initial positions X i t 0 become independent of X i t 0 see Eq. 15. This can be the case if u i Xt D tends to the PERSISTENT MEMORY OF DIFFUSING PARTICLES PHYSICAL REVIEW E 80, 061134 2009 061134-5
constant ensemble averages of the Eulerian field. Although such self-averaging properties can only be proved in particu- lar cases e.g., 10, they are often found in numerical mod- eling of transport in random fields with finite correlation scales 11,25. In our case, the self-averaging is indicated by the decay in time of the fluctuations of u i Xt D and the good agreement between the space-averaged Lagrangian ve- locity and the ensemble mean Eulerian velocity shown in Fig. 2. The two panels of Fig. 2 also show that the shape of the initial distribution longitudinal and transverse slabs of thickness has little influence on the self-averaging behav- ior, provided that the support of the initial concentration ex- tends over at least one correlation length in all directions. Intriguingly, in the case of diffusion in perfectly stratified flows with infinite correlation range and infinitely persistent memory of the random initial conditions, analyzed in Sec. III B, the dependence on deterministic initial conditions in- duces only transitory effects. The transverse dispersion is that of a memory-free Brownian motion with m 22 =0 and, because the longitudinal velocity does not depend on the longitudinal coordinates, m 11 Eq. 18 identically vanishes at all times. Equation 12shows that the off-diagonal term m 21 , corresponding to the correlation between longitudinal velocities and transverse initial positions of particles, ŠX 2 t 0 u 1 X 2 t D ‹ DX 0 , is nonvanishing at finite times. How- ever, since u 1 has a finite correlation length in the transverse direction it has a self-averaging behavior similar to that in Fig. 1 and m 21 vanishes in the long-time limit. B. Persistent memory of the initial conditions When diffusion takes place in random environments, the ensemble averaged variance of the process and the second moment of the ensemble averaged concentration have differ- ent behaviors and describe different features of the physical process 14,15,34. Their difference is the variance of the center of mass, which is also strongly influenced by the ini- tial conditions of the transport problem 25,33. To account for the randomness of the center of mass, we define dispersion terms ii = ŠX i - X i DX 0 V 2 ‹ DX 0 , 30 r ii = X i DX 0 - X i DX 0 V 2 31 and we write the variance Eq. 16 in the equivalent form s ii = ii - r ii . 32 For processes governed by the Fokker-Planck equation Eq. 2 and Itô equation Eq. 14, Eq. 32is obtained by replacing in the variance Eq. 10 the center of mass for a single realization X i DX 0 = i t , t 0 = i t 0 + t 0 t V ¯ i tdtwith its ensemble average i t 0 + t 0 t V ¯ j t V dtand by subtracting the correction Eq. 31. The latter is expressed by using Eq. 9as -150 -100 -50 0 50 100 150 200 250 300 350 400 1 10 100 1000 m 11 / (2Dt) Ut / λ 100λ x λ 10λ x 10λ λ x 100λ -200 -150 -100 -50 0 50 100 150 200 1 10 100 1000 m 22 / (2Dt) Ut / λ 100λ x λ 10λ x 10λ λ x 100λ FIG. 1. Memory terms in longitudinal leftand transverse rightdirections with respect to the mean flow obtained from GRW simulations for a given velocity field and for sources with different dimensions, shapes, and orientations. -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 10 100 1000 Ut / λ Longitudinal slab 100λ x λ 1+<u 1 > D /U <u 2 > D /U -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 10 100 1000 Ut / λ Transverse slab λ x 100λ 1+<u 1 > D /U <u 2 > D /U FIG. 2. Averages over 121 initial positions Xt 0 thick linesand standard deviations thin linesof the mean Lagrangian velocity u i Xt D for longitudinal and transverse slab sources. SUCIU et al. PHYSICAL REVIEW E 80, 061134 2009 061134-6
r ii t, t 0 = t 0 t dt uˆ i x, tcx, tdx t 0 t dt uˆ i x, tcx, tdx, where the velocity fluctuation is now defined with respect to the ensemble averaged velocity, uˆ i x , t= V i x , t- V ¯ i t V . Expression 10can thus be rewritten in the equivalent form, s ii t, t 0 = s ii t 0 +2Dt - t 0 + s uˆ ,ii t, t 0 + m ii t, t 0 - r ii t, t 0 , 33 where the memory term m ii t , t 0 is given by Eq. 12for i = j and the contribution of the velocity fluctuations uˆ i has a form similar to Eq. 11, s uˆ ,ii t, t 0 = t 0 t dt t 0 t dt cx 0 , t 0 dx 0  uˆ i x, tuˆ i x, tgx, tx, t gx, tx 0 , t 0 dxdx. 34 The ensemble average of the velocity correlation under the time integrals in Eq. 34is the mean Lagrangian correlation Eq. 26 analyzed through first-order approximations in Sec. III B. The ensemble average of Eq. 32is a well known iden- tity 15,20,33,34which relates the expected second moment S ii = s ii V to the second moment of the mean concentration ii = ii V and the variance of the center of mass R ii = r ii V , S ii = ii - R ii . 35 Assuming all necessary joint measurability conditions which allow permutations of averages 17leads to ii = s ii t 0 + X ii X 0 + M ii + Q ii , 36 where X ii = ŠX ˜ i - X ˜ i DV 2 ‹ DV is the one-particle dispersion defined by averaging with respect to D and V for a fixed initial position, M ii = m ii V is the mean memory term, and Q ii = ŠX ˜ i DV - X ˜ i DX 0 V 2 ‹ X 0 is the spatial variance of the one- particle center of mass X ˜ i DV computed by averages over initial positions35. The terms of Eq. 36depend, via the trajectory Eq. 14, on the Lagrangian velocity field. If the Lagrangian field is statistically homogeneous the one-particle center of mass X ˜ i DV and dispersion X ii are independent of X 0 17,24; hence they are memory-free quantities. Then M ii and Q ii vanish and Eq. 36takes on the simpler form, ii = s ii t 0 + X ii . 37 Assuming Lagrangian homogeneity leads, according to Eqs. 33and 35, to ii = ii V = s ii t 0 +2Dt - t 0 + s uˆ ,ii V . Then, the upscaled diffusion coefficients describing the behavior of the ensemble mean concentration, D ii = lim t→ 1 2t ii = 1 2 lim t→ d dt ii , are the sum between the local coefficient D and the long-time limit of the half derivative of the ensemble averages s uˆ ,ii V of the velocity fluctuation contributions Eq. 34. Their first-order approximations Eq. 28 are related to the statistics of the hydraulic conductivity field by D ii = D + 1i ln K 2 U, 38 which for the parameters of the numerical experiment pre- sented here take the values D 11 =0.11 m 2 / d and D 22 =0.01 m 2 / d 33. The dispersion terms Eqs. 30–32 were estimated from GRW simulations by using for every velocity realiza- tion 10 10 particles that were initially uniformly distributed in rectangular domains L 1  L 2 or released from the origin of the computational grid. For each initial condition, 1024 real- izations of the velocity field were used to asses ensemble averages ¯ V expectations and standard deviations of vari- ous concentration moments. Preliminary tests and compari- son with reference simulations using an algorithm free of overshooting errors 23showed that the overall precision of this Monte Carlo approach was of the order of the local dispersion 2Dt at early times and smaller than Dt after simu- lation times of about 30 dimensionless times Ut / . Figure 3 shows that for large dimensions of the support of the initial concentration the source of particlesS ii ii . Hence, the variance of the center of mass R ii 0, a property which is quite insensitive to the shape and orientation of the source. This is a somewhat expected result because for trans- port in velocity fields with finite correlation range, which are ergodic 36, the center of mass for point sources X i D is ergodic too, i.e., the space average approximates the en- semble average, X i DX 0 X i DV , and according to Eq. 31 R ii = r ii DV 0. Another information provided by Fig. 3 is that for large slab sources transverse to the i direction, for which the memory terms Eq. 18 are small Fig. 1, ii - s ii t 0 is independent of the source dimension. Therefore, we approxi- mated the memory-free one-particle dispersion X ii by using in Eq. 37the second moment of the mean concentration ii estimated from GRW simulations done for the largest slab source L = 100oriented perpendicular to the i axis. Since it has been found that the term Q ii is much smaller than the local dispersion 2Dt 35, the irregular behavior at early times of ii for different initial conditions can be attributed, according to Eq. 36, to the mean memory terms M ii = ii - s ii t 0 - X ii . The mean memory terms are a consequence of the statis- tical inhomogeneity of the Lagrangian velocity field, for which the ensemble average of Eq. 12is nonvanishing. Such memory effects on the ensemble averaged dispersion can be tracked back to the lack of smoothness of the velocity samples of the random field with exponential correlation used in simulations 24. The issue is somewhat similar to that of memory-induced oscillations of the diffusion coeffi- cient observed in the case of charged particles driven by a uniform magnetic field and a stationary Gaussian stochastic force with exponential time correlation 8, as well as in the case of particles driven by an anticorrelated autoregressive noise 28. However, in the two latter examples the memory mechanism is no longer the dependence of the Lagrangian statistics on deterministic initial conditions but an intrinsic PERSISTENT MEMORY OF DIFFUSING PARTICLES PHYSICAL REVIEW E 80, 061134 2009 061134-7
interdependence of the increments of the process due to the colored noise 28. The smallness of the standard deviation SD ii for large slab sources perpendicular to the i direction shown in Fig. 4 indicates that ii - s ii t 0 also approximates X ii for such initial conditions. The large values of SD ii for large slab sources parallel to the i direction are due, according to Eq. 33, to the large memory terms m ii which, as shown by Eq. 18, increase with the source dimension. We have seen that for large sources, irrespective of their shape and orientation, r ii 0. Thus, according to Eq. 32 ii s ii and we can adopt the following estimation of the memory terms, m ii t - t 0 = s ii t - t 0 - s ii t 0 - X ii t - t 0 . 39 As seen in Fig. 3, the long-time limit of the effective diffusion coefficients, defined by the half slope of the mean dispersion S ii - s ii t 0  / 2t - t 0 , approaches the upscaled coefficients Eq. 38. The relevance of the upscaled coeffi- cients for single realizations of the transport is an ergodicity issue. The overall trend of fluctuations shown in Fig. 4 indi- cates that single-realization dispersion coefficients are self- averaging. Together, the results from Figs. 3 and 4 indicate an ergodic behavior in the sense that the mean square dis- tance between single-realization and upscaled coefficients decreases in time 33. Another ergodic behavior of interest in practical applica- tions is that with respect to the memory-free dispersion X ii , which is often available from estimations of the dispersion terms Eq. 34 for given correlations of the Eulerian veloc- 0 2 4 6 8 10 12 1 10 100 1000 [Σ 11 (t) - s 11 (0)] / (2Dt) Transverse slab λ xLλ 0 2 4 6 8 10 12 14 1 10 100 1000 Longitudinal slab Lλ x λ 0 2 4 6 8 10 12 1 10 100 1000 Square Lλ xLλ -0.5 0 0.5 1 1.5 2 1 10 100 1000 [Σ 22 (t) - s 22 (0)] / (2Dt) Ut / λ 1 1.2 1.4 1.6 1.8 2 2.2 1 10 100 1000 Ut / λ point L = 10 L = 100 1 1.2 1.4 1.6 1.8 2 2.2 1 10 100 1000 Ut / λ FIG. 3. Second moments of the ensemble average concentration ii and ensemble averaged second moments S ii thin linesfor different shapes and extensions of the source. 0.1 1 10 1 10 100 1000 SD[σ 11 (t)] / (2Dt) Transverse slab λ xLλ 0.1 1 10 100 1 10 100 1000 Longitudinal slab Lλ x λ 0.1 1 10 1 10 100 1000 Square Lλ xLλ point L = 10 L = 100 0.01 0.1 1 10 1 10 100 1000 SD[σ 22 (t)] / (2Dt) Ut / λ 0.01 0.1 1 1 10 100 1000 Ut / λ 0.01 0.1 1 10 1 10 100 1000 Ut / λ FIG. 4. Standard deviation of the second moment with respect to ensemble averaged center of mass ii and standard deviation of the second moment of the actual center of mass s ii thin linesfor different shapes and extensions of the source. SUCIU et al. PHYSICAL REVIEW E 80, 061134 2009 061134-8
ity field 11,23. Using Eq. 39, we estimated the mean and the standard deviations of the largest memory terms m ii cor- responding to sources with the largest extension in the i di- rection. The results presented in Fig. 5 show that the mean memory terms are negligible small as compared to the stan- dard deviations. So, practically the mean square distance be- tween s ii and X ii is given by SDm ii . In Fig. 5 we also represented the memory terms for pure advective transport with Pe= , simulated by dropping the diffusion step in the GRW algorithm. It can be seen that the standard deviations are almost the same as in the case with Pe=100. According to Eq. 39, the extinction of the memory terms is equivalent to the self-averaging of the single-realization variance s ii and of the effective diffusion coefficients. At finite times, memory effects manifest mainly through deviations of single-realization dispersion from the ideal model-behavior described by the memory-free dispersion X ii . Such effects strongly depend on the shape, the orientation, and the spatial extension of the source of particles. V. CONCLUSIONS We decomposed the variance of the transport processes in dispersion and memory terms and for continuous diffusion process we derived explicit relations between these terms and the coefficients of the Fokker-Planck equation. Never- theless, this decomposition is a general property of the vari- ance and allows investigations on discrete processes as well. The memory terms govern the preasymptotic behavior of the transport process. We have shown that normal diffusion occurs only if the memory terms vanish. This happens when the diffusing particles forget their past itinerary. For diffusion in statistically homogeneous velocity fields with finite corre- lation lengths, we found that for finite correlation times of the Lagrangian velocity or convergent Kubo formula, the particles lose the memory in the long-time limit and normal diffusion occurs. We found a similar behavior for a discrete diffusion process with finite memory generated by autore- gressive noise. Memory terms also quantify the persistent influence of the deterministic initial conditions on the behavior of the trans- port process. Large deterministic initial distributions of the cloud of particles cause large memory terms which prevent the use of the one-particle dispersion as a model for the preasymptotic transport regime. Numerical simulations of diffusion in space random velocity fields with finite correla- tion range indicated the extinction of the memory terms after considerably large times, corresponding to hundreds of cor- relation scales, as well as the self-averaging behavior of the variance and its tendency toward normal diffusion. Thus, the issue of memory effects investigated in this pa- per can be partially answered: diffusing particles forget the memory of the deterministic initial position, as well as the memory of their past itinerary, when they evolve in time- independent random environments with finite spatial correla- tion scales. Indefinitely persistent memory can be found, for instance, in case of diffusion in velocity fields with infinite spatial correlation range. The behavior of the diffusion in time-variable environ- ments requires further investigations. For instance, it is known that a sufficiently fast decay of the time correlation functions ensures the convergence toward a normal diffusion even in absence of spatial decorrelation 16,19. However, for velocity fields with oscillating time correlations the con- dition of convergent Kubo formula, although a necessary condition for normal diffusion limit, may be far from being sufficient 19. The challenge is to find the meaning of the memory terms for anomalous diffusion in conditions of non- trivial interplay of temporal and spatial correlations. For continuous processes, analyses of memory effects via Lagrangian velocity correlations assume unique solutions of the Itô and Fokker-Planck equations. Nevertheless, the lack of smoothness of the velocity samples often precludes the existence of unique solutions. Moreover, as shown in Ref. 24, uniqueness is also an essential ingredient in proving the translation invariance of the ensemble average of the funda- mental solution of the Fokker-Planck equation and the equivalent property of statistical homogeneity of the La- grangian velocity field. When these properties are not en- sured, as in case of our numerical setup, deterministic initial conditions induce memory effects on the effective diffusion coefficients. Even though in such cases first-order approxi- mations still capture the asymptotic behavior 24, numerical models have to be developed for the transitory regime. In this paper, memory effects were quantified in a straightforward way by correlations between the displace- ments of the particles and starting positions. In Ref. 37it was shown that memory effects on diffusion coefficients are described in more detail by a hierarchy of Lagrangian corre- -20 0 20 40 60 80 100 120 1 10 100 1000 SD(m 11 ) / (2Dt) Ut / λ 100λ x 100λ, Pe=100 100λ x λ, Pe = 100 100λ x λ, Pe = ∞ -5 0 5 10 15 20 25 1 10 100 1000 SD(m 22 ) / (2Dt) Ut / λ 100λ x 100λ, Pe=100 λ x 100λ, Pe = 100 λ x 100λ, Pe = ∞ FIG. 5. Standard deviations of longitudinal leftand transverse rightmemory terms and the corresponding mean values thin lines. PERSISTENT MEMORY OF DIFFUSING PARTICLES PHYSICAL REVIEW E 80, 061134 2009 061134-9
lations, sampled on increasing paths on the set of trajectories starting at a given initial position. A particular case is the expansion of the diffusion coefficients in sums of correla- tions of increments of the process sampled on a single tra- jectory 28,38. Such representations of the diffusion coeffi- cients, using double summations of correlations, suggest possible connections with the method of memory kernels 2,4,5and motivate further work. ACKNOWLEDGMENTS The research reported in this paper was supported by Deutsche Forschungsgemeinschaft Grant No. SU 415/1-2, Jülich Supercomputing Centre Project No. JICG41, and Ro- manian Ministry of Education and Research Grant No. 2-CEx06-11-96. APPENDIX A: MEAN VALUE AND COVARIANCE COMPONENTS The coefficients of the Fokker-Planck equation defined in Eqs. 5and 6can be related under the conditions formu- lated in Eqs. 4, 7, and 8to the time derivatives of the mean and covariance components. Therefore, to derive the general expressions of the mean and covariances, we first compute their derivatives. The derivative of the first moment can be computed as follows: d dt i t= lim t→0 1 t i t + t- i t = lim t→0 1 t x i cx, t + tdx- x i cx, tdx = lim t→0 1 t x i - x i gx, t + tx, tdx cx, tdx , where we used Eq. 3and the normalization property of g. For t → 0, using Eqs. 4, 5, and 7one obtains d dt i t= V i x, tcx, tdx = V ¯ i t. A1 By integrating Eq. A1we get Eq. 9in the main text. To compute the derivative of the variance we proceed like for the mean, s ij t + t- s ij t= x i x j cx, t + tdx- x i x j cx, tdx - i t + t j t + t- i t j t = cx, tdx x i - x i x j - x j gx, t + tx, tdx+ x i cx, tdx x j - x j gx, t + tx, tdx + x j cx, tdx x i - x i gx, t + tx, tdx- i t + t j t + t - i t j t , and using Eqs. 4–8for t → 0, we obtain the derivative d dt s ij t=2 D ij x, tcx, tdx + x i V j x, t + x j V i x, tcx, tdx - d dt i t j t , which after expressing the last term by Eq. A1takes the form d dt s ij t=2 D ij x, tcx, tdx + x i V j x, t- V ¯ j t + x j V i x, t- V ¯ i tcx, tdx . A2 Next, we highlight the dependence on the initial positions of the second term in Eq. A2and obtain an equivalent expres- sion of the time derivative of s ij t, d dt s ij t=2 D ij x, tcx, tdx + cx 0 , t 0 dx 0 x i - x 0i V j x, t- V ¯ j t + x j - x 0 j V i x, t - V ¯ i tgx, tx 0 , t 0 dx + cx 0 , t 0 dx 0 x 0i - i t 0 V j x, t- V ¯ j t + x 0 j - j t 0  V i x, t- V ¯ i tgx, tx 0 , t 0 dx . A3 To analyze the second term in Eq. A3, we consider a time sequence t 0 t 1 ¯ t k t k+1 ¯ t n = t, t k+1 - t k = t, and the joint probabilities p of the sequence of events x 0 , t 0 ,..., x n-1 , t n-1 , x , t. The contribution of x i - x 0i V j x , tcan be expressed using the Chapman- Kolmogorov equation and the consistency property of p that integrating over intermediate states one obtains reduced or- der joint probabilities as follows: cx 0 , t 0 dx 0 x i - x 0i V j x, tgx, tx 0 , t 0 dx = ... k=0 n-1 x k+1,i - x k,i V j x, t px, t ; x n-1 , t n-1 ;...; x 0 , t 0 dxdx n-1 ... dx 0 = k=0 n-1  x k+1,i - x k,i V j x, t gx, tx k+1 , t k+1 gx k+1 , t k+1 x k , t k cx k , t k dxdx k+1 dx k = k=0 n-1 cx k , t k dx k x k+1,i - x k,i SUCIU et al. PHYSICAL REVIEW E 80, 061134 2009 061134-10
Vt, x k+1 gx k+1 , t k+1 x k , t k dx k+1 , A4 where Vt ; x k+1 , t k+1 = V j x , tgx , t x k+1 , t k+1 dx. Because the velocity defined by Eq. 5is always finite, the function Vt ; x k+1 , t k+1 is bounded, i.e., there exists an M 0 so that Vt ; x k+1 , t k+1 M for all x k+1 R 3 and t k+1 R. By using Eq. 8for i = j and the Cauchy-Schwarz in- equality in the form x-x x i gx, t + tx, tdx 2 x-x x i 2 gx, t + tx, tdx x-x gx, t + tx, tdx one obtains the condition lim t→0 1 t x-x x i gx, t + tx, tdx= 0. A5 Computing the last integral in Eq. A4as sum between the integral over the sphere of radius and the integral outside the sphere, we have x k+1 -x k x k+1,i - x k,i Vt ; x k+1 , t k+1 gx k+1 , t k+1 x k , t k dx k+1 M x k+1 -x k x k+1,i - x k,i  gx k+1 , t k + tx k , t k dx k+1 → t→0 0 A6 due to condition A5. Considering the negative and positive parts and applying the theorem of mean, the integral over the sphere of radius can be computed as x k+1 -x k x k+1,i - x k,i Vt ; x k+1 , t k+1 gx k+1 , t k+1 x k , t k dx k+1 = Vt ; x k , t k+1 x k+1 -x k x k+1,i - x k,i gx k+1 , t k + tx k , t k dx k+1 + O 2 . A7 For t → 0, from Eq. 5and Eqs. A4–A7one obtains cx 0 , t 0 dx 0 x i - x 0i V j x, tgx, tx 0 , t 0 dx = cx 0 , t 0 dx 0 0 t dt V j x, tV i x, t gx, tx, tgx, tx 0 , t 0 dxdx+ O 2 . Finally, passing to the limit → 0, we have cx 0 , t 0 dx 0 x i - x 0i V j x, tgx, tx 0 , t 0 dx = cx 0 , t 0 dx 0 0 t dt V j x, tV i x, t gx, tx, tgx, tx 0 , t 0 dxdx. A8 Replacing in the second term of Eq. A3the contribution Eq. A8 and the similar one obtained by permutation of the indices i and j one obtains Eq. 11in the main text. APPENDIX B: LAGRANGIAN DESCRIPTION The expectation f (Xt)= f xcx , tdx of some func- tion with compact support f xcan be written by using the Dirac distribution as f xx - Xtdx = f xx - Xtdx . Hence, the one-point probability density i.e., the normalized concentrationcan formally be written as cx , t= x - Xt, and the n-point joint densities as expectations of products of delta functions e.g., 39. Similarly, the transition probability density is the condi- tional expectation gx , t x 0 , t 0 = x - Xt Xt 0  = x - Xt D , where by angular brackets with subscript D we denoted the conditional expectation for fixed initial positions X0of the particles. For the purpose of a transparent La- grangian description it is also convenient to use the subscript X 0 for averages with respect to initial positions. With these, the concentration Eq. 3 can be expressed as cx, t= gx, tx 0 , t 0 cx 0 , t 0 dx 0 = x - Xt DX 0 . B1 In the same way, for higher order probability densities one obtains px n , t n ; x n-1 , t n-1 ;...; x 1 , t 1 = px n , t n ; x n-1 , t n-1 ;...; x 1 , t 1 x 0 ,0cx 0 , t 0 dx 0 = x n - Xt n x n-1 - Xt n-1  ... x n - Xt n  DX 0 . B2 PERSISTENT MEMORY OF DIFFUSING PARTICLES PHYSICAL REVIEW E 80, 061134 2009 061134-11
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