Density matrix formulation of dynamical systems

Abstract

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.

Section "Case studies": damped Harmonic oscillator discussion expanded, a new plot added in appendix F

References in corpus (5)

• Applied Koopmanism
• Characterizing dynamics with covariant Lyapunov vectors
• Probing rare physical trajectories with Lyapunov weighted dynamics
• Quantum mechanics as classical statistical mechanics with an ontic extension and an epistemic restriction
• What does dynamical systems theory teach us about fluids?

Cited by in corpus (7)

• Thermodynamic speed limits for mechanical work
• Quantum simulation of dissipation for Maxwell equations in dispersive media
• A geometrical description of non-Hermitian dynamics: speed limits in finite rank density operators
• Spectral bounds on the entropy flow rate and Lyapunov exponents in differentiable dynamical systems
• Maximum speed of dissipation
• Phase space contraction rate for classical mixed states
• A time-marching quantum algorithm for simulation of the nonlinear Lorenz dynamics