Dynamical typicality in classical lattice systems

TL;DR

This paper demonstrates that in large classical lattice systems with independent initial conditions, most trajectories of macroscopic observables are nearly identical, based on measure concentration and decay of local perturbations.

Contribution

It establishes a general framework for dynamical typicality in classical lattice systems using measure concentration and decay assumptions, supported by simulations.

Findings

Most initial conditions lead to similar macroscopic dynamics.

The influence of local perturbations diminishes rapidly with distance.

Simulations of coupled rotors confirm theoretical predictions.

Abstract

Considering deterministic classical lattice systems with continuous variables, we show that, if the initial conditions are sampled according to a probability distribution in which the dynamical variables are statistically independent, the dynamical trajectory of any macroscopic observable is approximately the same for the vast majority of the states in the sample. Our proof relies on general concentration of measure results which provide tight bounds for the deviation from typical behavior in the case of large system sizes. The only condition that we assume for the dynamics is that the influence of a local perturbation in the initial state decays sufficiently fast with distance at any finite time. Our results are relevant, in particular, to classical Hamiltonian systems on a lattice. We apply our general results to a system of coupled rotors with long-range interactions, and report dynamical simulations which verify our findings.