Density matrix formulation of dynamical systems*

Some physical systems respond to perturbations with cascading failures and others respond by transducing flows of energy and entropy to form structures or do work. The statistical evolution of perturbations is critical to mitigating disasters and to the ability to function dynamically. Laws governing the statistical evolution of ensembles are central to classical mechanics, quantum mechanics, and dynamical systems, but do not explicitly describe the spread of perturbations. Here, we establish a density matrix formalism for this purpose that applies to classical dynamical systems and is analogous to the density matrix formulation of quantum mechanics. In this statistical-dynamical framework, the classical density matrix describes the statistical state of a system, with the time evolution of unit Lyapunov vectors in the tangent space giving unitary dynamics, and the classical Liouville equation corresponding to the preservation of a trace, a feature typically associated with quantization. We illustrate the theory with conservative and dissipative model dynamics.