Dynamical typicality in classical lattice systems

TL;DR

This work extends dynamical typicality, previously observed in quantum systems, to deterministic classical lattice systems with continuous variables by proving concentration of measure for time-evolved macroscopic observables under independent initial data. Using bounded-differences arguments and a locality propagation hypothesis, it derives that Var[$f_t$] $\le\tfrac14\sum_i c_i(t)^2$, with $c_i(t)$ capturing the bounded influence of individual coordinates. The theory is concretely applied to a one-dimensional lattice of coupled rotors with long-range interactions ($\alpha>1$), where explicit bounds show energy-density fluctuations scale as $O(1/N)$ and numerical simulations corroborate trajectory collapse as $N$ grows. These results imply that macroscopic non-equilibrium dynamics in such classical systems are largely insensitive to microscopic initial conditions, with potential extensions to weakly correlated initial states.

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.

Figures (2)

• Figure 1: Numerically obtained results for the potential energy density $u(\vec{x}(t))$ from (\ref{['22']}) (see also Eq. (\ref{['6']})) versus time $t$ for the model of coupled rotors from (\ref{['15']}) and (\ref{['16']}) with $J=1$, $\alpha=2$ and different system sizes, namely $N=10$ in (a) and $N=4000$ in (b). The initial conditions $\vec{x}(0)=(\vec{q}(0), \vec{p}(0))$ (see also below Eq. (\ref{['16']})) have been randomly sampled such that each component $p_i(0)$ and $q_i(0)$ was uniformly and independently distributed inside a finite interval, namely $-1.5\leq p_i(0)<1.5$ and $-1\leq q_i(0)<1$. The different colors correspond to six different realizations of those random initial conditions.
• Figure 2: Variance of $u(\vec{x}(t))$ versus system size $N$ at times $t=0$ and $t=20$ for the same model as in Fig. \ref{['fig:fig1']} but now with $0\leq p_i(0)<0.1$, $0\leq q_i(0)<0.1$, and sampling over $5\times10^{3}$ random initial conditions. Note the logarithmic scales. The red line serves as a guide to the eye.