Random Walkers Cellular Automata for Diffusion Processes

7 years ago

Abstract

Smooth concentration fields and balance equations for systems of random walkers are obtained by using the coarsegrained space‐time averaging method from ]5[ and averages over the statistical ensemble. This yields a new cellular automaton numerical model for diffusion processes. The number of particles and the averaging space‐time scale needed to approximate the concentration, with a given precision, are obtained. Applications were made for systems with small concentrations and diffusion in random fields.

Authors

Calin Vamoş
“T. Popoviciu” Institute of Numerical Analysis, Romanian Academy

Nicolae Suciu
“T. Popoviciu” Institute of Numerical Analysis, Romanian Academy

Harry Vereecken
ZE, Forschungszentrum Julich ICG-4, D52425-Julich, Deutschland

Uwe Jaekel
ZE, Forschungszentrum Julich ICG-4, D52425-Julich, Deutschland

Holger Schwarze
ZE, Forschungszentrum Julich ICG-4, D52425-Julich, Deutschland

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Vamoş, C., Suciu, N., Vereecken, H., Jaekel, U. and Schwarze, H. (2000), Random Walkers Cellular Automata for Diffusion Processes. Z. angew. Math. Mech., 80: 367-368.

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2000 Vamos Random Walkers Cellular Automata for Diffusion Processes

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ZAMM. Z. Angew. Math. Mech.

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https://doi.org/10.1002/zamm.2000080145510.1002/zamm.20000801455

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References

[1] Gao, J. B., Hwanq, S. K., Lm, J. M. : When Can Chaos Induce Noise?; Phys. Rev. Lett., 82(6) (1999), 1132-1135.

[2] Gaylord, R., Nishidate, K.: Modeling Nature: Cellular Automata and Computer Simulations with Mathematica; Springer, New York, 1996.

[3] Kirkwood, J. G.: Selected Topics in Statistical Mechanics; Gordon and Breach, New York, 1967.

[4] Matheron, G., De Marsly, G.: Is “hamport in Porous Media Always Diffusive?; Water Resour. k., 16 (1980), 901-917.

[5] Schwarze, L, Jaekel; U., Vereecken, H., Estimation of Macrodispersion by Different Approximation Methods for Flow and ‘Ikansport in Randomly Heterogeneous Media; (1999), (to be published).

[6] VAMOS, C., GEORGESCU, A., Suciu, N., Turcu, I.: Balance Equations for Physical Systems with Corpuscular Structure; Physica A 227 (1996), 81-92.

[7] 7 VAMOS, C., SUCIU, N., M.PECULEA: Numerical Modelling of the One-Dimensional Diffueion by Random Walkers; Rev. Springer, New York, 1996. Flow and ‘Ikansport in Randomly Heterogeneous Media; (1999), (to be published). Physica A 227 (1996), 81-92. Anal. NumC. ThCorie Approximation 26, 1-2 (1997), 237-247.

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Random Walkers Cellular Automata for Diffusion Processes

Section 1-7 S367 C~LIN VAMOS, NICOLAE SUCIU, HARRY VEREECKEN, UWE JAEKEL, HOLQER SCHWARZE Random Walkers Cellular Automata for Diffusion Processes Smooth concentmtion fields and balance equations for systems of mndom walkers are obtained by using the warse- gmined space-time avemging method f m m [ti] and averages over the statistical ensemble. This yields a new cellular automaton numerical model for diffusion processes. The number of particles and the averaging space-time scale needed to appmximate the concentmtion, with a given precision, are obtained. Applications were made for systems with small concentrations and diffusion in mndom fields. 1. Continuous modeling by space-time averages Let us consider a system of N particles and a kinematic description by piece-wise analytic trajectories, pi : I e R, I = [0, TJ C IR, (1 5 i 5 N). It was proved [6], that the coorse-gmined space-time avemge (9) : IR3 x (7,T-r) c--$ IR, where V = 43ra3/3 is the volume of the sphere S(r, a) and H+ the Heaviside left continuous function, has a.e. continuous derivatives in IR3 x (7, T - T) and satisfies the identity &(lop> + 4Y(cp5a) = (dloP/dt). (2) If Pi(t) = vi(q(t,U)), where q = (rl,...,r~,~l,~~.,~~) are the trajectories of the stochastic process into the positions-velocities space we find that the ensemble averages of (cp), for a 4 O,T + 0, give the usual ’fine-grained’continuous fields of statistical mechanics [3]. Defining the concentmtion field c(r,t) = (l)(r,t) and the Eulerian velocity field u,(r,t) = (&)(r,t)/c(r,t), (1) gives the continuity equation 8tc +8,(cu,) = 0. With cu, = (dx,/dt), defining the Lagmngian velocity field v,(rlt) = &(z,)(r,t) + u&(za)(r,t), and the diffusion tensor a&t) = I(2cr)(r,t)up(r,t) - (za€p)(r,t)/c(rrt)l, (3) from the continuity equation and (2) we obtain 8tc + B,(cv,) = S,Sp(cD,p). Thus, the positivity of coefficients (3) is a test for diffusive behavior as described by an advection-diffusion equation. 2. Random walkers cellular automata Cellular automata describe complex systems by ”fictitious particles moving in a grid, according to simple and local rules, undergoing simultaneous changes of states” [2]. We consider N ndependent mndom walkers inside the 1-dimensionalgrid {zk = k6t I -m 5 k m). Then the number of particles at the site k is (4) 1 2 nk(t +at) = -[nk-l(t) + nk+l(t)] - 6nk+l(t) + ank-l(t), where 6nk+l and 6nk-l are small for large N. For nk(z,t) = c(z,t)62, (4) becomes the finite-differences diffusion equation, with D = 6x2/26t, the corresponding diffusion coefficient. The global estimation, (C;!, Ink - c(zk,t)6zI)/N, is of order of for N = lo6 and m 2 30. Coarse-grained averages (1) over the trajectories of the cellular automaton give the concentration not only in knots but also in any (z,t), reduce the computing time and improve the approximation. By averages over an ensemble of 200 identical cellular automata, it waa found that the precision in all the grid is at least if a is between 62/8 and 862 and correspondingly, T is between lo6& and1066t. If the particles reaching the boundaries are instantaneously removed, one finds the mean time needed to eliminate the N particles from the grid to be At - l n N (i.e. c - e-tl at small concentrations for systems placed in the previous boundary conditions ) [7]. Also, by using a variable step grid, we have verified that there are space-time scales 80 that (3) gives precise estimations for diffusion coefficients (defined aa functions on grid steps by D = 6z2/26t).

S368 ZAMM . Z. Angew. Math. Mech. 80 (2000) S2 3. Diffusion in random fields In a similar manner, we built a 2-dimensional random walkers cellular automaton. On it, we superposed an advection given by the samples of a random field and we computed ensemble averages. For horieontal advection and constant velocities in each layer, we get the numerical simulation of the model proposed by Matheron and de Marsily [4], dz('+) (t) = V(y(t),wV)dt + Ddw (t) , dy (t) = Ddw (t) , where w is the Wiener process and w, labels the realizations of the random field. Diffusivebehavior corresponds to - t, where uz is the longitudind dispersion, defined by mean square displacements. Using the method from [5] we produced several random advection fields with different correlations. The picture contains dispersions curves given by cellular automaton, for different random advection fields. Unbiased random walk (nubrw") obviously has a diffusive behavior. For Gaussian (- e-g') correlated longitudinal field we get the nondiffusive behavior reported by Matheron and de Marsily [3], uz - t8i2 ("ubrw+vx(y)") and a diffusive long-time behavior, when a constant transversal velocity is added (nubrwv"). For random fields with identical values in a given number of neighbor layers (the last curve correspond to 2 layesrs correlation), ui goes to CY ta dependence for increasing correlation length. TIME 4. Remarks The agreement with 1-dimensional diffusion equation (section 2) and the model of Matheron and de Marsily (section 3) are first proofs of the new numerical model for diffusion processes. These encourage us to use it to study more complex processes, as the challenging problem of transport in heterogeneous porous media. At the same time, we also stress here the possibility to obtain information on the behavior of disordered systems using (3) as a test for diffusive behavior. So, our approach belongs to the attempts to give quantitative definitions of noise and chaos in terms of "divergence between nearby trajectories" [l]. Acknowledgements One of us (N. S.) thanks for the kind hospitality and fhitfil discusstons during his work stage at Forschungszentrum Julich. 5. References 1 GAO, J. B., HWANQ, S. K., Lm, J. M. : When Can Chaos Induce Noise?; Phys. Rev. Lett., 82(6) (1999), 1132-1135. 2 GAYLORD, R., NISHIDATE, K.: Modeling Nature: Cellular Automata and Computer Simulations with Mathematica; 3 KIRKWOOD, J. G.: Selected Topics in Statistical Mechanics; Gordon and Breach, New York, 1967. 4 MATHERON~G., DE MARSLY, G.: Is "hamport in Porous Media Always Diffusive?; Water Resour. k., 16 (1980), 901-917. 5 SCHWARZE, L, JAEKEL; U., VEREECKEN, H.: Estimation of Macrodispersion by Different Approximation Methods for 6 VAMOS, C., GEORGESCU, A., S U C ~ , N., ~ c u , I.: Balance Equations for Physical Systems with Corpuscular Structure; 7 VAMOS, C., SUCIU, N., M.PECULEA: Numerical Modelling of the One-Dimensional Diffueion by Random Walkers; Rev. Springer, New York, 1996. Flow and 'Ikansport in Randomly Heterogeneous Media; (1999), (to be published). Physica A 227 (1996), 81-92. Anal. NumC. ThCorie Approximation 26, 1-2 (1997), 237-247. Addresses: CXLINVAMOS, NICOLAE SUCIU, "T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68, 3400 Cluj-Napoca 1, Romhia. HARRY VEREECKEN , UWE JAEKEL, HOLGER SCHWARZE, Forschungszentrum Julich ICG-4, D52425- Julich, Deutschland.