Escape-Rate Formalism, Decay to Steady States, and Divergences in the Entropy-Production Rate

According to thermodynamics the irreversible entropy production of diffusive relaxation processes diverges at the boundary to the vacuum, i.e., to a state of vanishing particle density. By means of a multibaker map we point out that this divergence is not present in the spatially discrete dynamics, which brings forth the evolution equations of irreversible thermodynamics in the continuum limit. In addition, we show that the irreversible entropy production of relaxation towards a nonempty steady state is proportional to the decay rate of the thermodynamic system subjected to absorbing boundary conditions. This generalizes results of the escape rate formalism.