Stability under Perturbations of the Large Time Average Motion of Dynamical Systems with Conserved Phase Space Volume

TL;DR

This paper demonstrates that the long-term average behavior of volume-preserving dynamical systems remains stable under small perturbations, supporting the robustness of statistical physics assumptions and numerical methods.

Contribution

It proves a general continuity theorem showing the stability of time and ensemble averages in weakly perturbed ergodic systems, extending the applicability of ergodic theory.

Findings

Double averages are only slightly affected by perturbations.

Supports the validity of the Boltzmann Ergodic Hypothesis.

Justifies numerical approximations in molecular dynamics.

Abstract

The stability against perturbations of a dynamical system conserving a generalized phase-space volume is studied by exploiting the similarity between statistical physics formalism and that of ergodic theory. A general continuity theorem is proved. Resulting from this theorem the double average - time and ensemble - of an observable of a weakly perturbed ergodic dynamical system is only slightly changed, even in the infinite time limit. Consequences of this statistical analogue of the structural stability are: extension of the range of practical applicability of the Boltzmann Ergodic Hypothesis, justification of the perturbation method in statistical physics, justification of the numerical approximations in molecular dynamic calculations and smoothness of the transition from bounded to unbounded motion as observed in numerical simulation of anomalous transport in tokamaks.