Towards a microscopic theory of nonisothermal stochastic processes

Using a nonlinear projection operator technique, the Markovian stochastic dynamics of a classical Brownian particle is derived in the case that the effective temperature is not constant. After recapitulating the general formalism and its application to isothermal Brownian motion, an isolated ‘particle plus environment’ system is analyzed in the microcanomical ensemble. The ensuing generalized Kramers equation is shown to involve a microcanonical potential of mean force different from the conditional free energy, and the conditional entropy emerges as the relevant equilibrium potential. The Smoluchowski limit is presented, and the effective spatial diffusion coefficient is calculated. Upon coupling the system to a heat bath, the microcanonical theory is seen to correspond to the case of small heat conductance. The final result for the stochastic process in the position, momentum, and energy variables is shown to be consistent with both isothermal Kramers theory and nonlinear equilibrium thermodynamics.