Nonlinear heat equations. (English) Zbl 0907.60094
Summary: The modified Smoluchowski equation, coupled to a temperature field, leads to a pair of nonlinear heat equations obeying the first and second laws of thermodynamics. We obtain a solution representing a particle under gravity, moving in a slab and maintained in stasis away from the Gibbs state by a temperature gradient. A two-state atom in a potential in isothermal conditions is described by coupled equations satisfying detailed balance. It is shown that the free energy is a monotonic decreasing function of time.
MSC:
| 60K40 | Other physical applications of random processes |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 82C35 | Irreversible thermodynamics, including Onsager-Machlup theory |
Keywords:
nonequilibrium thermodynamics; entropy; free energy; stochastic differential equation; Brownian motion; diffusionReferences:
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