Boltzmann's H-function and diffusion processes.

There exists a generalization of Boltzmann's H-function that allows for nonuniformly populated stationary states, which may exist far from thermodynamic equilibrium. Here we describe a method for obtaining a generalized or collective diffusion coefficient D directly from this H-function, the only constraints being that the relaxation process is Markov (short memory), continuous in the reaction coordinate, and local in the sense of a flux/force relationship. As an application of this H-function method, we simulate the self-consistent extraction of D via Langevin/Fokker-Planck (L/FP) dynamics on various potential energy landscapes. We observe that the initial epoch of relaxation, which is far removed from the stationary state, provides the most reliable estimates of D. The construction of an H-function that guarantees conformity with the second law of thermodynamics has been generalized to allow for diffusion coefficients that may depend on both the reaction coordinate and time, and the extension to an arbitrary number of reaction coordinates is straightforward. For this multidimensional case, the diffusion tensor must be positive definite in the sense that its eigenvalues must be real and positive. To illustrate the behavior of the proposed collective diffusion coefficient, we simulate the H-function for a variety of Langevin systems. In particular, the impacts on H and D of landscape shape, sample size, selection of an initial distribution, finite dynamic observation range, stochastic correlations, and short/long-term memory effects are examined.