Speed limits on deterministic chaos and dissipation

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.

Figure 1

Speed limit on the dissipation rate of the van der Pol oscillator. (a) The Fisher information ${\tau }^{-1}=\sqrt{{\mathcal{I}}_F}=2\mathrm{\Delta }{\mathit{A}}^{\top }$ upper bounds (dashed black curve) the intrinsic speed ${\tau }_{\mathit{V}}^{-1}$ of dissipation $\mathrm{\Lambda }=\langle \mathit{V}\rangle$ with respect to a normalized perturbation $\mathit{\varrho }$ (solid blue): ${\tau }^{-1}\ge {\tau }_{\mathit{V}}^{-1}$. (b) The inequality also sets a lower bound on the magnitude of dissipation $\left|\mathrm{\Lambda }\right|\ge \dot{\mathcal{V}}\tau$. Panel (b) shows $\left|\mathrm{\Lambda }\right|$ (solid blue curve) is bounded from below by $\dot{\mathcal{V}}\tau$ (dashed black curve). The trajectory is on the stable limit cycle with damping parameter $\mu =1.5$. Shaded regions mark regimes inaccessible to ${\tau }_{\mathit{V}}^{-1}$ and $\left|\mathrm{\Lambda }\right|$.

Figure 2

Speed limit on chaos. The instantaneous Lyapunov exponent, $r\left(t\right)=\langle {\mathit{A}}_{+}\rangle$, for phase-space orbits of (a) the Lorenz-Fetter model and (b) the Hénon-Heiles model. Square root of the tangent-space Fisher information (dashed black) $\sqrt{{\mathcal{I}}_F}=2\mathrm{\Delta }{\mathit{A}}^{\top }$ upper bounds the speed ${\tau }_{{\mathit{A}}_{+}}^{-1}$ (solid red and green). Shaded regions mark speeds not accessible to the observable. The parameters of the Lorenz-Fetter model are: $\sigma =10, \beta =8/3$, and $\eta =22$ (transient chaos). For the Hénon-Heiles system, the chaotic trajectory corresponds to the energy $E=1/6$.