Nonlinear heat equations

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.