Fluctuation theorem for constrained equilibrium systems

TL;DR

This paper demonstrates that constrained equilibrium systems, despite non-volume-preserving dynamics, exhibit phase-space contraction fluctuations that obey a fluctuation theorem and are Gaussian distributed over time.

Contribution

It introduces a fluctuation theorem applicable to constrained equilibrium systems with non-preserving dynamics, expanding the understanding of fluctuation properties in such states.

Findings

Finite-time phase-space contraction fluctuations satisfy a Gallavotti-Cohen type theorem.

These fluctuations are Gaussian for sufficiently long times.

The results are demonstrated on Lennard-Jones fluid, Nosé-Hoover thermostatted oscillator, and a hyperbolic map.

Abstract

We discuss the fluctuation properties of equilibrium chaotic systems with constraints such as iso-kinetic and Nos\'e-Hoover thermostats. Although the dynamics of these systems does not typically preserve phase-space volumes, the average phase-space contraction rate vanishes, so that the stationary states are smooth. Nevertheless finite-time averages of the phase-space contraction rate have non-trivial fluctuations which we show satisfy a simple version of the Gallavotti-Cohen fluctuation theorem, complementary to the usual fluctuation theorem for non-equilibrium stationary states, and appropriate to constrained equilibrium states. Moreover we show these fluctuations are distributed according to a Gaussian curve for long-enough times. Three different systems are considered here, namely (i) a fluid composed of particles interacting with Lennard-Jones potentials; (ii) a harmonic oscillator with Nos\'e-Hoover thermostatting; (iii) a simple hyperbolic two-dimensional map.