An extended liouville equation for variable particle number systems

TL;DR

This paper extends the classical Liouville equation to systems with variable particle numbers, enabling the study of open and particle-creating/annihilating systems within a unified statistical mechanics framework.

Contribution

It introduces a generalized Liouville equation for variable particle number systems and derives a corresponding conservation equation, including particle creation and annihilation effects.

Findings

Grand canonical distribution is a stationary solution

Extended equation applies to nonequilibrium systems with particle number changes

Provides a theoretical foundation for open system statistical mechanics

Abstract

It is well-known that the Liouville equation of statistical mechanics is restricted to systems where the total number of particles (N) is fixed. In this paper, we show how the Liouville equation can be extended to systems where the number of particles can vary, such as in open systems or in systems where particles can be annihilated or created. A general conservation equation for an arbitrary dynamical variable is derived from the extended Liouville equation following Irving and Kirkwood's2 technique. From the general conservation equation, the particle number conservation equation is obtained that includes general terms for the annihilation or creation of particles. It is also shown that the grand canonical ensemble distribution function is a particular stationary solution of the extended Liouville equation, as required. In general, the extended Liouville equation can be used to study nonequilibrium systems where the total number of particles can vary.