  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.
