Local entropy production and Gibbs relation from the nonlinear revised Enskog equation

A local formulation of the Boltzmann $H$ theorem associated with the revised Enskog equation is presented. For weak spatial gradients, one can prove that the entropy production is positive. If we further restrict ourselves to near local equilibrium states, the entropy production takes the form of products of thermodynamic forces by fluxes, i.e., the Gibbs relation, and the entropy flux reduces to the heat flux divided by the temperature.