Typicality from concentration of measure in models with binary state variables

Abstract

We study the emergence of typicality in classical systems with a large number of binary state variables. We show analytically that for sufficiently large subsets of the complete state space, state functions which can be associated with macroscopic observables, such as density or energy, are sharply concentrated around a typical value, i.e., the vast majority of microscopic states show the same macroscopic behavior. From this static typicality result we obtain the dynamical counterpart, which states that if two initial conditions are drawn from a sufficiently large subset of the state space, the dynamical trajectories of the macroscopic observables would be approximately the same, provided that the dynamics does not compress abruptly the size of sets of states under evolution. The ensuing dynamical typicality phenomenon is very similar to what was recently found in the context of the dynamics of isolated quantum systems. We illustrate and apply our analytical results by analyzing one-dimensional cellular automata.