Speed limits on the local stability of classical dynamical systems

Uncertainty relations are a prominent feature of quantum mechanics. However, classical systems are also characterized by a type of uncertainty – deterministic chaos – in which the uncertainty in their initial conditions leads to unpredictable behavior. In this presentation, I will discuss our theory of dynamical systems that mirrors the density matrix formulation of quantum mechanics [1]. Central to this formalism is a classical density matrix, with dynamics governed by a von Neumann-like equation of motion. and dynamical observables, such as Lyapunov exponents, that evolve in time under an Ehrenfest-like theorem. Leveraging this formalism, we derive a family of speed limits on observables in the tangent space that are set by the local dynamical (in)stability [2]. These classical speed limits are mechanical in nature and obtained from a Fisher information constructed in terms of Lyapunov vectors and the local stability matrix. For a dynamical system with a time-independent local stability matrix, these speed limits reduce to a classical analog of the Mandelstam-Tamm time-energy uncertainty relation in quantum mechanics. Our analytical and numerical results for model systems show this theory applies to arbitrary deterministic systems including those that are conservative, dissipative and driven.