Fluctuation–dissipation relations for chemical systems far from equilibrium

Abstract

A new fluctuation–dissipation relation is suggested for constant step, one intermediate chemical processes far from equilibrium. It establishes a relationship between the net reaction rate t̃(x), the probability diffusion coefficient D(x) in the composition space, and the species-specific affinity A(x): t̃(x)=2D(x)tanh[−A(x)/2kT], where x is the concentration of the active intermediate, k is Boltzmann’s constant, and T is the absolute temperature. The theory is valid for nonlinear fluctuations of arbitrary size. For macroscopic systems the fluctuation–dissipation relation may be viewed as a force-flux relationship. We distinguish four fluctuation–dissipation regimes which correspond to the decrease of the absolute value of the species-specific affinity. The passage from high ‖A(x)‖ to small ‖A(x)‖ corresponds to a crossover from a linear dependence of the species-specific dissipation rate φ̇(x) on ‖A(x‖)‖, φ̇(x)∼−‖A(x)‖, to a square one: φ̇(x)∼−A2(x). A main feature of the fluctuation–dissipation relation is its symmetry with respect to the contributions of the forward and backward chemical processes to fluctuation and relaxation. Two new physical interpretations of the probability diffusion coefficient are given: one corresponds to a measure of the strength of fluctuations at a steady state, and the other to a measure of the instability of a given fluctuation state. The dispersion of the number q of reaction events in a given time interval is given by a generalized Einstein relation: 〈Δq2〉=2VD(x)t, where V is the volume of the system. The diffusion coefficient D(x) is proportional to the reciprocal value of the mean age 〈τ(x)〉 of a fluctuation state characterized by the concentration x: D(x)=1/[2V〈τ(x)〉]. These interpretations are not related to the use of a Fokker–Planck approximation of the chemical Master Equation.