Thermodynamic approach to nonequilibrium chemical fluctuations

Abstract

A generalized thermodynamic description of one-variable complex chemical systems is suggested on the basis of the Ross, Hunt, and Hunt (RHH) theory of nonequilibrium processes. Starting from the stationary solution of a chemical Master Equation, two complimentary, related sets of generalized state functions are introduced. The first set of functions is derived from a generalized free energy F̌X, and is used to compute the moments of stationary and non-Gaussian concentration fluctuations. Exact expressions for the cumulants of concentration are derived; a connection is made between the cumulants and the fluctuation–dissipation relations of the RHH theory. The second set of functions is derived from an excess free energy φ(x); it is used to express the conditions of existence and stability of nonequilibrium steady states. Although mathematically distinct, the formalisms based on the F̌X and φ(x) functions are physically equivalent: both lead to the same type of differential expressions and to similar global equations. A comparison is made between the RHH and Keizer’s theory of nonequilibrium processes. An appropriate choice of the integration constants occurring in Keizer’s theory is made for one-variable systems. The main differences between the two theories are: the constraints for the two theories are different; the stochastic and thermodynamic descriptions are global in RHH, whereas Keizer’s theory is local. However, both theories share some common features. Keizer’s fluctuation–dissipation relation can be recovered by using the RHH approach; it is valid even if the fluctuations are nonlinear. If the thermodynamic constraints are the same, then Keizer’s theory is a first-order approximation of RHH; this approximation corresponds to a Gaussian description of the probability of concentration fluctuations. Keizer’s theory is a good approximation of RHH in the vicinity of a stable steady state: near a steady state the thermodynamic functions of the two theories are almost identical; the chemical potential in the stationary state is of the equilibrium form in both theories. Keizer’s theory gives a very good estimate of the absolute values of the peaks of the stationary probability density of RHH theory. Away from steady states the predictions of the two theories are different; the differences do not vanish in the thermodynamic limit. The shapes of the tails of the stationary probability distributions are different; and hence the predictions concerning the relative stability are different for the two theories.