摘要
Some derivations based on the anomalous Langevin equation in Liouville space (i.e. Γ space) or its equivalent Liouville diffusion equation of time reversal asymmetry are presented. The time rate of change, the balance equation, the entropy flow, the entropy production and the law of entropy increase of Gibbs nonequilibrium entropy and Boltzmann nonequilibrium entropy are rigorously derived and presented here. Furthermore, a nonlinear evolution equation of Gibbs nonequilibrium entropy density and Boltzmann nonequilibrium entropy density is first derived. The evolution equation shows that the change of nonequilibrium entropy density originates from not only drift, but also typical diffusion and inherent source production. Contrary to conventional knowledge, the entropy production density σ ≥0 everywhere for all the inhomogeneous systems far from equilibrium cannot be proved. Conversely, σ may be negative in some local space of such systems.
Some derivations based on the anomalous Langevin equation in Liouville space (i.e. Γ space) or its equivalent Liouville diffusion equation of time reversal asymmetry are presented. The time rate of change, the balance equation, the entropy flow, the entropy production and the law of entropy increase of Gibbs nonequilibrium entropy and Boltzmann nonequilibrium entropy are rigorously derived and presented here. Furthermore, a nonlinear evolution equation of Gibbs nonequilibrium entropy density and Boltzmann nonequilibrium entropy density is first derived. The evolution equation shows that the change of nonequilibrium entropy density originates from not only drift, but also typical diffusion and inherent source production. Contrary to conventional knowledge, the entropy production density σ ≥0 everywhere for all the inhomogeneous systems far from equilibrium cannot be proved. Conversely, σ may be negative in some local space of such systems.
