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Chapman-Enskog development of the multivariate master equation

Author

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  • Van den Broeck, C.
  • Houard, J.
  • Malek Mansour, M.

Abstract

A systematic development for a Multivariate Master Equation (MME), describing reaction diffusion systems, is presented along the same lines as the Chapman-Enskog development of kinetic gas theory. Diffusion, which occurs at a very fast rate, brings the system near a state of diffusional equilibrium, the analogue of local equilibrium for gases. In diffusional equilibrium, the global Master Equation (ME) is shown to be an exact consequence of the MME. For finite systems, corrections to the global ME, resulting from the finiteness of the diffusion times, are calculated. The development is verified on an exactly solvable model and illustrated on the Schlögl model. The difficulties encountered in the thermodynamic limit are discussed, and possible outcomes suggested.

Suggested Citation

  • Van den Broeck, C. & Houard, J. & Malek Mansour, M., 1980. "Chapman-Enskog development of the multivariate master equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 101(1), pages 167-184.
    • Handle: RePEc:eee:phsmap:v:101:y:1980:i:1:p:167-184
    DOI: 10.1016/0378-4371(80)90107-7
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    References listed on IDEAS

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    1. Borckmans, P. & Dewel, G. & Walgraef, D., 1977. "Long time behaviour of correlation functions in Heisenberg paramagnets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 88(2), pages 261-282.
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    Cited by:

    1. Borgis, D. & Moreau, M., 1984. "On the stochastic theory of a bistable chemical reaction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 123(1), pages 109-130.
    2. Frankowicz, M. & Gudowska-Nowak, E., 1982. "Stochastic simulation of a bistable chemical system: The two-box model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 116(1), pages 331-344.
    3. Boissonade, J., 1982. "Molecular dynamics studies of metastability and nucleation in a non-equilibrium chemical system with multiple steady states," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 113(3), pages 607-618.
    4. Moreau, M. & Vicente, L., 1982. "On the stochastic theory of polymerization reactions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 111(1), pages 139-160.
    5. Garcia, Alejandro L. & van den Broeck, Christian & Aertsens, Marc & Serneels, Roger, 1987. "A Monte Carlo simulation of coagulation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 143(3), pages 535-546.

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    1. Borckmans, P. & Walgraef, D. & Dewel, G., 1978. "Critical dynamics of paramagnets with dipolar coupling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 91(3), pages 411-423.

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