Macroscopic Determinism in Interacting Systems Using Large Deviation Theory

TL;DR

This paper extends large deviation theory to analyze the macroscopic deterministic behavior of large interacting systems with vector-valued observables, establishing a map that predicts system evolution with high probability as system size grows.

Contribution

It introduces a novel approach using a level-1 large deviation principle for joint observables, generalizing previous results and providing conditions for the existence of a macrostate evolution map.

Findings

Existence of a macrostate evolution map derived from a generalized dynamic free energy.

Conditions under which the macrostate map is well-defined, finite, and differentiable.

Application to a simple lattice model with an exact macroscopic solution.

Abstract

We consider the quasi-deterministic behavior of systems with a large number, $n$, of deterministically interacting constituents. This work extends the results of a previous paper [J. Stat. Phys. 99:1225-1249 (2000)] to include vector-valued observables on interacting systems. The approach used here, however, differs markedly in that a level-1 large deviation principle (LDP) on joint observables, rather than a level-2 LDP on empirical distributions, is employed. As before, we seek a mapping $\psi_{t}$ on the set of (possibly vector-valued) macrostates such that, when the macrostate is given to be $a_0$ at time zero, the macrostate at time $t$ is $\psi_{t}(a_0)$ with a probability approaching one as $n$ tends to infinity. We show that such a map exists and derives from a generalized dynamic free energy function, provided the latter is everywhere well defined, finite, and differentiable. We discuss some general properties of $\psi_{t}$ relevant to issues of irreversibility and end with an example of a simple interacting lattice, for which an exact macroscopic solution is obtained.