On the Fundamental Equation of Nonequilibrium Statistical Physics --   Nonequilibrium Entropy Evolution Equation and the Formula for Entropy   Production Rate
  In this paper we presented an overview on our works. More than ten years ago, we proposed a new fundamental equation of nonequilibrium statistical physics in place of the present Liouville equation. That is the stochastic velocity type's Langevin equation in 6N dimensional phase space or its equivalent Liouville diffusion equation. This equation is time-reversed asymmetrical. It shows 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 expressed by a superposition of both the law of dynamics and stochastic velocity and possesses both determinism and probability. Hence it is different with the law of motion of particles in dynamical systems. Starting from this fundamental equation the BBGKY diffusion equation hierarchy, the Boltzmann collision diffusion equation, the hydrodynamic equations such as the mass drift-diffusion equation, the Navier-Stokes equation and the thermal conductivity equation have been derived and presented here. What is more important, we first constructed a nonlinear evolution equation of nonequilibrium entropy density in 6N, 6 and 3 dimensional phase space, predicted the existence of entropy diffusion. This evolution equation reveals that the time rate of change of nonequilibrium entropy density originates together from its drift, diffusion and production in space. Furthermore, we presented a formula for entropy production rate (i.e. the law of entropy increase), proved that internal attractive force in nonequilibrium system can result in entropy decrease while internal repulsive force leads to another entropy increase, obtained an unified theoretical expression for thermodynamic degradation and self-organizing evolution, and revealed that the entropy diffusion mechanism caused the system to approach to equilibrium. 
Xiu-San Xing