Time reversibility and nonequilibrium thermodynamics of second-order   stochastic processes
  Nonequilibrium thermodynamics of a general second-order stochastic system is investigated. We prove that at steady state, under inversion of velocities, the condition of time-reversibility over the phase space is equivalent to the antisymmetry of spatial flux and the symmetry of velocity flux. Then we show that the condition of time-reversibility alone could not always guarantee the Maxwell-Boltzmann distribution. Comparing the two conditions together, we found that the frictional force naturally emerges as the unique odd term of the total force at thermodynamic equilibrium, and is followed by the Einstein relation. The two conditions respectively correspond to two previously reported different entropy production rates. In case that the external force is only position-dependent, the two entropy production rates become one. We prove that such an entropy production rate can be decomposed into two nonnegative terms, expressed respectively by the conditional mean and variance of the thermodynamic force associated with the irreversible velocity flux at any given spatial coordinate. In the small inertia limit, the former term becomes the entropy production rate of the overdamped dynamics; while the anomalous entropy production rate originated from the latter term. Furthermore, regarding the connection between the First Law and Second Law, we found that in the steady state of such a limit, the anomalous entropy production rate is also the leading order of the Boltzmann-factor weighted difference between the spatial heat dissipation densities of the underdamped and overdamped dynamics, while their unweighted difference always tends to vanish. 
Hao Ge
cond-mat.stat-mech