Fluctuations, stability, and generalized state functions at nonequilibrium steady states

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Abstract
Fluctuation–dissipation postulates, which describe the kinetic effects of molecular processes, are used to characterize nonequilibrium steady states. Attention is restricted to stable, noncritical states which develop in systems with inputs that are time independent. For these systems it is shown that the steady state distribution is Gaussian, which provides a generalization of the well‐known Einstein formula for equilibrium states. For certain systems it is shown that the time dependence of the covariance matrix of the extensive variables gives a necessary and sufficient condition for the stability of a noncritical state. These considerations are illustrated for the steady states accompanying diffusion, heat transport, chemical reactions with linear coupling, and certain nonlinear chemical reactions. These examples show that the covariance matrix is not necessarily related to the local equilibrium entropy. When the covariance matrix is invertible, it can be used to construct generalized state functions which reduce to familiar thermodynamic functions at equilibrium. The generalization of the entropy, called the σ function, is related to stability, the probability density, and generalized ’’thermodynamic forces’’ in precisely the same way as the entropy is at equilibrium.