Entropy production in continuous phase space systems

We propose an alternative method to compute the environmental entropy production of a classical underdamped nonequilibrium system, not necessarily in detailed balance, in a continuous phase space. It is based on the idea that the Hamiltonian orbits of the corresponding isolated system can be regarded as microstates and that entropy is generated in the environment whenever the system moves from one microstate to another. This approach has the advantage that it is not necessary to distinguish between even and odd-parity variables. We show that the method leads to a different expression for the differential entropy production along an infinitesimal stochastic path. However, when integrating over all possible paths the local entropy production turns out to be the same as in previous studies. This demonstrates that the differential entropy production in continuous phase space systems is not uniquely defined.