Speed limits on deterministic chaos and dissipation

Abstract

Uncertainty in the initial conditions of dynamical systems can cause exponentially fast divergence of trajectories, a signature of deterministic chaos, or be suppressed by the dissipation of energy. Here, we derive a classical uncertainty relation that sets a speed limit on the rates of local observables underlying these behaviors. For systems with a time-invariant stability matrix, the speed limit we derive simplifies to a classical analog of the Mandelstam-Tamm versions of the time-energy uncertainty relation. These classical bounds are set by fluctuations in the local stability of state space. To measure these fluctuations, we introduce a definition of the Fisher information in terms of Lyapunov vectors in tangent space, analogous to the quantum Fisher information defined in terms of wave vectors in Hilbert space. This information sets an upper bound on the speed at which classical dynamical systems and their observables, instantaneous Lyapunov exponents and dissipation, evolve. This speed limit applies to systems that are open or closed, conservative or dissipative, actively driven or passively evolving, and directly connects the geometries of phase space and information.