Entropy, extropy and information potential in stochastic systems far from equilibrium

The relations between information, entropy and energy, which are well known in equilibrium thermodynamics, are not clear far from equilibrium. Moreover, the usual expression of the classical thermodynamic potentials is only valid near equilibrium. In previous publications, we showed for a chemical system maintained far from equilibrium, that a new thermodynamic potential, the information potential, can be defined by using the stochastic formalism of the Master Equation. Here, we extend this theory to a completely general discrete stochastic system. For this purpose, we use the concept of extropy, which is defined in classical thermodynamics as the total entropy produced in a system and in the reservoirs during their equilibration. We show that the statistical equivalent of the thermodynamic extropy is the relative information. If a coarse-grained description by means of the thermodynamic extensive variables is available (which is the case for many macroscopic systems) the coarse-grained statistical extropy allows one to define the information potential in the thermodynamic limit. Using this potential, we study the evolution of such systems towards a non-equilibrium stationary state. We derive a general thermodynamic inequality between energy dissipation and information dissipation, which sharpens the law of the maximum available work for non-equilibrium systems.