Stochastic models of cotransport; by R. F. Streater; symport, antiport, Shannon entropy, static head.

Stochastic models of cotransport

by R. F. Streater; Transport Theory and Statistical Physics, 22, 1-37, 1993.

ABSTRACT

We construct discrete Boltzmann equations describing the reactions between any number of chemicals. By including a heat particle to balance the energy, we arrive at a model for the chemistry of liquids. The ratio of forward to backward reaction rates exactly obeys an exponential law, showing the Boltzmann factor; this is what we called the "Arrhenius" law, but is only part of it: Arrhenius wanted the rates themselves to obey an exponential law depending on the temperature and the activation energies of the forward or backward processes. This is only approximately true in experiment, and our model suggests why.

We prove that all isolated systems within this class of models converge to equilibrium.

We use the models to describe cotransport through membranes, and prove that the systems converge to the static head equilibrium, for both symport and antiport. For open systems, we find that either the static head is trivial or is numerically ambiguous. For the continuum limit we obtain heat equations with non-linear boundary conditions across the membrane, in which the ambiguity reappears as being due to the lack of boundary conditions at $\infty$. We find a stationary solution representing flow in which the concentrations do not satisfy the predictions of equilibrium thermodynamics.

This theory of chemical reactions has been extended and explained in my book Statistical Dynamics. This incorporates improvements worked out with Rondoni and Koseki.

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