In this paper we proposed a new fundamental equation of statistical physics in place of the Liouville equation. That is the anomalous Langevin equation in Γ space or its equivalent Liouville diffusion equation of time-reversal asymmetry. This equation reflects that the form of motion of particles in statistical thermodynamic systems has the drift-diffusion duality and the law of motion of statistical thermodynamics is stochastic in essence, but not completely deterministic. Starting from this equation the BBGKY diffusion equation hierarchy, the law of entropy increase, the theorem of minimum entropy production, the balance equations of Gibbs and Boltzmann nonequilibrium entropy are derived and presented here. Furthermore we have derived a nonlinear evolution equation of Gibbs and Boltzmann nonequilibrium entropy density. To our knowledge, this is the first treatise on them. The evolution equation shows that the change of nonequilibrium entropy density originates together from drift, typical diffusion and complicated inherent source production. Contrary to conventional viewpoint, the entropy production density σ≥0 everywhere for any systems cannot be proved universally. Conversely, σ may be negative in some local space of some inhomogeneous systems far from equilibrium. The hydrodynamic equations, such as the generalized Navier–Stokes equation, the mass drift-diffusion equation and the thermal conductivity equation have been derived succinctly from the BBGKY diffusion equation hierarchy. The Liouville diffusion equation has the same equilibrium solution as that of the Liouville equation. All these derivations and results are unified and rigorous from the new fundamental equation without adding any extra assumption.
On the fundamental equation of nonequilibrium statistical physics—Nonequilibrium entropy evolution equation and the formula for entropy production rate