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Boltzmann's H-Function and Diffusion Processes



Steven P. Lund, Joseph B. Hubbard, Michael W. Halter


There exists a generalization of Boltzmann's $H$-function that allows for non-uniformly 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 multi-dimensional case, the diffusion tensor must be positive definite in the sense that its eigenvalues must be real and positive. In order to mimic a biological signaling process, we simulate the $H$-function for a system of Langevin "particles" interacting via a stochastic potential, and observe the influence of these correlations on the collective diffusion coefficient. We examine the impacts on $H$ and $D$ of landscape shape, sample size, selection of an initial distribution, finite dynamic observation range and short/long term memory effects.
Journal of Physical Chemistry


Boltzmann’s H-function, diffusion coefficient, Langevin, Fokker-Planck, Onsager symmetry.


Lund, S. , Hubbard, J. and Halter, M. (2013), Boltzmann's H-Function and Diffusion Processes, Journal of Physical Chemistry, [online], (Accessed June 17, 2024)


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Created May 15, 2013, Updated January 27, 2020