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Lattice Free Stochastic Dynamics

Published online by Cambridge University Press:  20 August 2015

Alexandros Sopasakis
Alexandros Sopasakis*
Affiliation:
Center for Mathematical Sciences, Lund University, Box 118, 22100 Lund, Sweden
*
*Corresponding author.Email:[email protected]
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Abstract

We introduce a lattice-free hard sphere exclusion stochastic process. The resulting stochastic rates are distance based instead of cell based. The corresponding Markov chain build for this many particle system is updated using an adaptation of the kinetic Monte Carlo method. It becomes quickly apparent that due to the lattice-free environment, and because of that alone, the dynamics behave differently than those in the lattice-based environment. This difference becomes increasingly larger with respect to particle densities/temperatures. The well-known packing problem and its solution (Palasti conjecture) seem to validate the resulting lattice-free dynamics.

Keywords

82C2082C2282C8082B2082D9990B20Lattice-freemicroscopic stochastic dynamicskinetic Monte Carlo
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 3 , September 2012 , pp. 691 - 702
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Biham, O., Middleton, A. A., and Levine, D.. Self-organization and a dynamical transition in traffic-flow models. Phys. Rev. A, 10:6124, 1992.CrossRefGoogle Scholar
[2]Bortz, A. B., Kalos, M. H., and Lebowitz, J. L.. A new algorithm for Monte Carlo simulations of Ising spin systems. J. Comput. Phys., 17:10, 1975.Google Scholar
[3]Gidas, B.. Metropolis-type Monte Carlo simulation algorithms and simulated annealing. In J. Laurie Snell, editor, Topics in Contemporary Probability and its Applications. CRC Press, 1995.Google Scholar
[4]Gillespie, D. T.. A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. J. Comput. Phys., 22:403, 1976.Google Scholar
[5]Henson, S. M., Costantino, R. F., Cushing, J. M., Desharnais, R. A., Dennis, B., and King, A. A.. Lattice effects observed in chaotic dynamics of experimental populations. Science, 294:602–605, 2001.CrossRefGoogle ScholarPubMed
[6]Hubbard, D., The Failure of Risk Management: Why It’s Broken and How to Fix It, John Wiley &Sons, 2009.Google Scholar
[7]Jackson, E. A.. Perspectives of nonlinear dynamics. Cambridge Univ. Press, 1:216–219, 1989.Google Scholar
[8]Katsoulakis, M. A., Majda, A. J., and Sopasakis, A.. Multiscale couplings in prototype hybrid deterministic/stochastic systems: Part I, deterministic closures. Comm. Math. Sci., 2:255–294, 2004.CrossRefGoogle Scholar
[9]Katsoulakis, M. A., Majda, A. J., and Vlachos, D. G.. Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems. J. Comput. Phys., 186(1):250–278, 2003.CrossRefGoogle Scholar
[10]Katsoulakis, M. A., Plecháč, P., and Sopasakis, A.. Error analysis of coarse-graining for stochastic lattice dynamics. SIAM J. Num. Anal., 44(6), 2006.Google Scholar
[11]Katz, S. and Lebowitz, J. and Spohn, H.. Stationary nonequilibrium states for stochastic lattice gas models of ionic superconductors. J. Statist. Phys., 34:497, 1984.Google Scholar
[12]Kipnis, C. and Landim, C.. Scaling Limits of Interacting Particle Systems. Springer-Verlag, 1999.Google Scholar
[13]Krug, J. and Spohn, H.. Universality classes for deterministic surface growth. Phys. Rev. A, 38:4271, 1988.CrossRefGoogle ScholarPubMed
[14]Kurtze, D. A. and Hong, D. S.. Traffic jams, granular flow, and soliton selection. Phys. Rev. E, 52:218–221, 1995.CrossRefGoogle ScholarPubMed
[15]Liggett, T. M.. Interacting Particle Systems. Springer, 1985.Google Scholar
[16]Nagel, K. and Schreckenberg, M.. A cellular automaton model for freeway traffic. J. Phys. I France, 2:2221, 1992.Google Scholar
[17]Palasti, I.. On some random space filling problems. Publ. Math. Inst. Hungar. Acad. Sci., 5:353–360, 1960.Google Scholar
[18]Renyi, A.. On a one-dimensional problem concerning random space-filling. Publ. Math. Inst. Hungar. Acad. Sci., 3:109–127, 1958.Google Scholar
[19]Schreckenberg, M. and Wolf, D. E.. Traffic and Granular Flow. Springer Singapore, 1998.Google Scholar
[20]Schulze, T. P.. Efficient kinetic Monte Carlo simulation. Journal of Computational Physics, 227(4):2455–2462, February 2008.CrossRefGoogle Scholar
[21]Slepoy, A., Thompson, A. P., and Plimpton, S. J.. A constant-time kinetic Monte Carlo algorithm for simulation of large biochemical reaction networks. Journal of Chemical Physics, 128(20):205101, 2008.Google Scholar
[22]Sopasakis, A. and Katsoulakis, M. A.. Stochastic modeling and simulation of traffic flow: ASEP with Arrhenius look-ahead dynamics. SIAM J. on Applied Mathematics, 2005.Google Scholar
[23]Spohn, H.. Large scale dynamics of interacting particles. Springer-Verlag, 1991.CrossRefGoogle Scholar
[24]Vlachos, D. G. and Katsoulakis, M. A.. Derivation and validation of mesoscopic theories for diffusion of interacting molecules. Phys. Rev. Let., 85(18):3898, 2000.CrossRefGoogle ScholarPubMed
[25]Wolfram, S., Cellular Automata as Simple Self-Organizing Systems, Caltech Preprint CALT-68-938 (submitted to Nature), 1982.Google Scholar
[26]Wolfram, S., A New Kind of Science, Champaign, IL: Wolfram Media, Inc., 2002.Google Scholar
[27]Guo, Z., Zheng, C., and Shi, B.. Discrete lattice effects on the forcing term in the lattice Boltz-mann method. Phys. Rev. E, 65(4):046308, 2002.CrossRefGoogle ScholarPubMed