Nonlinear Heat Equations

by R. F. Streater, Reports on Mathematical Physics, 40, 557-564, 1997.

ABSTRACT

We show that the Smoluchowski equation for a Brownian particle in a potential can be supplemented by an equation for the dynamics of the temperature, such that the first and second laws of thermodynamics are obeyed. We find a stationary solution far from the Gibbsian state, representing a cloud of particles under gravity, in a horizontal slab, whose two faces are maintained at different temperatures. We find the configuration for which the rate of entropy production is smallest, and show that it is not stationary, contrary to the claim of the Prigogine-Glansdorff theorem. We consider a model studied by David Smith, in which the Brownian particle is a two-level atom. We show that under isothermal conditions, the free energy can be given a natural definition out of equilibrium, and is a decreasing function of time.

D. A. Smith has applied his model, called the dumbbell model, to a describe a myosin molecule, in the paper "Direct tests of muscle cross-bridge theories: predictions of a Brownian dumbbell model for position-dependent cross-bridge lifetimes and step sizes with an optically trapped actin filament", Biophys. Journal, 75, 2996-3007, 1998.

Various generalisations of this work are described by Wojnar, Dolbeault and Esteban.

During my recent (2003) visit to Helsinki, I discovered that Markku Lampinen had already shown the invalidity of the Glansdorff-Prigogine theorem for the heat equation; this work appeared as "A problem of the principle of minimum entropy production", TKK Lampotekniikka ja Koneoppi, No. 47, 1990. He does exactly the same functional optimisation as in the present paper.

Subscribers to Elsevier can read the full paper Nonlinear Heat Equations. It has been reviewed by Lorenza Viola.