Stochastic dynamics beyond the weak coupling limit: thermalization

Quasirelativistic Langevin equation

We address the problem of a microscopic derivation of the Langevin equation for a weakly relativistic Brownian particle. A noncovariant Hamiltonian model is adopted, in which the free motion of particles is described relativistically while their interaction is treated classically, i.e., by means of action-to-a-distance interaction potentials. Relativistic corrections to the classical Langevin equation emerge as nonlinear dissipation terms and originate from the nonlinear dependence of the relativistic velocity on momentum. On the other hand, similar nonlinear dissipation forces also appear as classical (nonrelativistic) corrections to the weak-coupling approximation. It is shown that these classical corrections, which are usually ignored in phenomenological models, may be of the same order of magnitude, if not larger than, relativistic ones. The interplay of relativistic corrections and classical beyond-the-weak-coupling contributions determines the sign of the leading nonlinear dissipation term in the Langevin equation and thus is qualitatively important.