Boltzmann's H theorem for systems with frictional dissipation

By use of Boltzmann's equation to describe an ensemble of particles under the influence of a friction force, Boltzmann's H theorem is refined to explicitly include frictional dissipation, the accompanying fluctuations being modeled via an added diffusive, Fokker-Planck term. If the friction force per particle mass is proportional to velocity, as is the case with viscous drag with a friction coefficient γ, Boltzmann's H theorem for the time rate of change of the quantity H reads dH/dt ≤ γ. The classical formulation stating that H can never increase is thus replaced by the statement that H cannot increase at a rate higher than γ, a general result but of particular relevance when fluctuations are negligible and the system is far from equilibrium. When the particles are not far from thermal equilibrium, an alternative, more suitable expression emerges which can be written in the form of a Clausius inequality.