Time reversibility and nonequilibrium thermodynamics of second-order stochastic processes.Nonequilibrium thermodynamics of a general second-order stochastic system is investigated.  Comparing the two conditions together, we find 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.  We prove that such an entropy production rate can be decomposed into two non-negative terms, expressed respectively by the conditional mean and variance of the thermodynamic force associated with the irreversible velocity flux at any given spatial coordinate.  Furthermore, regarding the connection between the first law and second law, we find 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.
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