Microscopic theory for the time irreversibility and the entropy production

In stochastic thermodynamics, the entropy production of a thermodynamic system is defined by the irreversibility measured by the logarithm of the ratio of the path probabilities in the forward and reverse processes. We derive the relation between the irreversibility and the entropy production starting from the deterministic equations of motion of the whole system consisting of a physical system and a surrounding thermal environment. The derivation assumes the Markov approximation that the environmental degrees of freedom equilibrate instantaneously. Our approach provides a guideline for the choice of the proper reverse process to a given forward process, especially when there exists a velocity-dependent force. We demonstrate our idea with an example of a charged particle in the presence of a time-varying magnetic field.