Abstract
We develop a concise method to compute the corrections to the master equation for chemically reacting systems in particle number space that arise if the system is not a well-stirred tank reactor, but the transport occurs by diffusion. Starting from the master equation in theR N space of all reactant particle positions, we expand in inverse powers of the diffusion constant and eliminate all transport modes adiabatically. It is found that the overall effect of spatially nonuniform fluctuations cannot be treated as a mere renormalization of the reaction rate constants. From second order on there appear correction terms with a new structure that corresponds formally to additional virtual reaction paths. An intuitive interpretation along this line is impeded, however, by the formal occurrence of negative reaction rate constants in these terms, i.e., the reaction rate may depend on the concentrations of the final products of the virtual reaction rather than on the ingoing products. We also identify Avogadro's constant as the suitableΩ parameter and extend van Kampen'sΩ-expansion systematically, to spatially continuous systems. This secondary expansion then serves to interpret the corrections to the rate equation, and the average and autocorrelation of the density in the stationary state. It is seen that the limitsD→∞ andΩ→∞ do not commute. The relevant length and time scales are discussed.
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References
N. G. van Kampen, Phys.Rep. 124(2):69 1985.
N. G. van Kampen,Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
A. Suna,Phys. Rev. B 1:1716 (1970).
N. G. van Kampen,Int. J. Quant. Chem: Quant. Chem. Symp. 16:101 (1982).
Chapter 12 of Ref. 2.
C. van den Broeck, J. Houard, and M. Malek-Mansour,Physica 101A:167 (1980).
G. Nicolis and M. Malek-Mansour,Phys. Rev. 29A:2845 (1984).
C. W. Gardiner,Handbook of Stochastic Methods (Springer, Berlin, 1983).
U. M. Titulaer,Physica 91A:321 (1978).
W. Hess and R. Klein,Adv. Phys. 32(2):173–283 (1983).
U. Kneller and U. M. Titulaer,Physica 129A:514 (1985).
J. A. M. Janssen,Physica 133A:497 (1985).
C. Bloch,Nucl. Phys. 6:329 (1958).
P. Mazur and D. Bedeaux,Physica 67:23 (1973).
M. A. Burschka,Fluctuations in diffusion reaction systems, Ph.D. Thesis, Utrecht (1985).
U. M. Titulaer,Physica 100A:234 (1980).
Y. Kuramoto,Prog. Theor. Phys. 49:1782 (1973),52:711 (1974).
D. Toussaint and F. Wilczek,J. Chem. Phys. 78(5):2642 (1983).
M. A. Burschka and E. G. D. Cohen, in preparation.
J. M. J. van Leeuwen and F. van Dieren, in:Fundamental Problems in Statistical Mechanics, VI, E. G. D. Cohen, ed. (North-Holland, Amsterdam, 1985).
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Burschka, M.A. Fluctuations in diffusion reaction systems. I: Adiabatic elimination of transport modes from a mesoscopicN-body system and the Ω-expansion. J Stat Phys 45, 715–744 (1986). https://doi.org/10.1007/BF01021092
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DOI: https://doi.org/10.1007/BF01021092