Dynamical typicality in classical lattice systems

Considering deterministic classical lattice systems with continuous dynamical variables, we show that, if the initial conditions are sampled according to a probability distribution in which the dynamical variables are statistically independent, the temporal evolution of any macroscopic observable is approximately the same for the vast majority of the randomly sampled initial states. 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 requirement on the dynamics is that the influence of a local perturbation in the initial state decays, at any given instance of time, sufficiently fast with distance. Our results are relevant, in particular, to classical Hamiltonian systems on a lattice, but also apply to a large class of dissipative dynamics. We illustrate our general results with two analytical examples, a coupled map lattice and a Hamiltonian system of coupled rotors with long-range interactions.