Kinetic equations and brownian motion for a solid in a stationary temperature gradient

A kinetic equation for the distribution function for a subsystem s interacting with a bath b maintained in a stationary non-isothermal state by reservoirs is derived by using a projection operator formalism and a perturbation expansion parameter λ appropriate to some brownian motion problems. Thus, when s is a heavy particle of mass m ₀ and b a lattice of light particles of mass m then λ = (m m ₀)^1/2. By means of an assumption about the decay of correlations of b variables in the field of s, the terms are classified as destruction terms which vanish to arbitrary order in λ for long enough times and collision terms which give well-defined integrals for long times. For the heavy particle in a lattice the leading collision terms, after linearization in the temperature gradient, lead to an equation equivalent to the generalized nonisothermal Fokker-Planck equation of Nicolis.