Thermodynamic and stochastic theory of nonequilibrium systems: Fluctuation probabilities and excess work

Abstract

For a nonequilibrium system described at the mesoscopic level by the master equation, we prove that the probability of fluctuations about a steady state is governed by a thermodynamic function, the ‘‘excess work.’’ The theory applies to systems with one or more nonequilibrium steady states, for reactions in a compartment that contains intermediates Xj of variable concentration, along with a reactant A and product B whose concentrations are held constant by connection of the reaction chamber to external reservoirs. We use a known relation between the stationary solution Ps(X) of the master equation and an underlying stochastic Hamiltonian H: to logarithmic accuracy, the potential that gives Ps(X) is the stochastic action S evaluated along fluctuational trajectories, obtained by solving Hamilton’s equations of motion starting at a steady state. We prove that the differential action dS equals a differential excess work dφ0, and show that dφ0 can be measured experimentally in terms of total free energy changes for the reaction compartment and the reservoirs. Thus we connect the probability of concentration fluctuations in an open reaction compartment to thermodynamic functions for the entire closed system containing the compartment. The excess work dφ0 is the difference between the total free energy change for a specified change in the quantities of A, X, Y, and B in the state of interest, and the free energy change for the same changes in species numbers, imposed on the same system in a reference state (A,X0,Y0,B). The reference-state concentration for species Xj is derived from the momentum pj canonically conjugate to Xj along the fluctuational trajectory. For systems with linear rate laws, the reference state (A,X0,Y0,B) is the steady state, and φ0 is equivalent to the deterministic excess work φdet* introduced in our previous work. For nonlinear systems, (A,X0,Y0,B) differs from the deterministic reference state (A,X*,Y*,B) in general, and φ0≠φdet*. If the species numbers change by ±1 or 0 in each elementary step and if the overall reaction is a conversion A→X→Y→B, the reference state (A,X0,Y0,B) is the steady state of a corresponding linear system, identified in this work. In each case, dφ0 is an exact differential. Along the fluctuational trajectory away from the steady state, dφ0≳0. Along the deterministic kinetic trajectory, dφ0≤0, and φ0 is a Liapunov function. For two-variable systems linearized about a steady state, we establish a separate analytic relation between Ps(X), φdet*, and a scaled temperature T*.