  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.
