  Physical systems that dissipate, mix and develop turbulence also irreversibly
transport statistical density. In statistical physics, laws for these processes
have a mathematical form and tractability that depends on whether the
description is classical or quantum mechanical. Here, we establish a theory for
density transport in any classical dynamical system that is analogous to the
density matrix formulation of quantum mechanics. Defining states in terms of a
classical density matrix leads to generalizations of Liouville's theorem and
Liouville's equation, establishing an alternative computationally-tractable
basis for nonequilibrium statistical mechanics. The formalism is complete with
classical commutators and anti-commutators that embed measures of local
instability and chaos and are directly related to Poisson brackets when the
dynamics are Hamiltonian. It also recovers the traditional Liouville equation
and the Liouville theorem by imposing trace preservation or Hamiltonian
dynamics. Applying to systems that are driven, transient, dissipative, regular,
and chaotic, this formalism has the potential for broad applications.
