Generalized relation between the relative entropy and dissipation for nonequilibrium systems

Recently, Kawai, Parrondo, and Van den Broeck have related dissipation to time-reversal asymmetry. We generalized the result by considering a protocol where the physical system is driven away from an initial thermal equilibrium state with temperature $\beta_0$ to a final thermal equilibrium state at a different temperature. We illustrate the result using a model with an exact solution, i.e., a particle in a moving one-dimensional harmonic well.