Statistical-Mechanical Theory of Kinetic Equations

A new method of finding kinetic equations for the fundamental state variables is explored by generalizing the Boltzmann equation in two directions with the aid of the theory of generalized Brownian motions developed previously. First, a nonlinear stochastic equation of motion is derived in the form of the Boltzmann equation with a fluctuating force. This fluctuating force not only determines the thermal fluctuations in gases but also serves to reconcile the macroscopic irreversibility represented by the H-theorem with the mechanical reversibility as one relation between the dissipation and the fluctuation. Secondly, a new nonlinear kinetic equation is derived for a coarse-grained particle density in µ space which is valid not only for liquids but also for dense gases near critical points. The time evolution of macroscopic variables can be determined from this kinetic equation. A relevant decomposition of dynamical variables into slow and rapid processes is found, and the streaming term is expressed in terms of an effective one-particle energy renormalized by the short-range correlation. This kinetic equation provides us with a new molecular basis for studying transport in dense systems, including nonlinear phenomena.