Irreversible thermodynamics

We present a maximum-entropy theory of mesoscopic kinetics. The theory gives fully nonlinear nonequilibrium thermodynamic relationships and has no explicit requirement for either microscopic bath variables, an equilibrium energy, or an equilibrium partition function. The entropy maximization process is instead carried out over transition probability distributions with constraints on particle position and velocity updates. The Lagrange multipliers for these constraints express the instantaneous temperature and pressure of external (or microscopic) thermostatic driving systems, with which the distinguished system may or may not eventually reach equilibrium. We show that the analogues of the Gibbs-Maxwell relations and free energy perturbation techniques carry over to fluctuation-dissipation theorems and nonequilibrium ensemble reweighting techniques as should be expected. The result is a fully time-dependent, non-local description of a nonequilibrium ensemble coupled to reservoirs at possibly time-varying thermostatic or mechanical states. We also show that the thermodynamic entropy production extends the generalized fluctuation theorem through the addition of an instantaneous information entropy term for the end-points, leading to a concise statement of the second law of thermodynamics.