Spectral bounds on the entropy flow rate and Lyapunov exponents in differentiable dynamical systems

TL;DR

This paper derives spectral bounds on entropy flow rate and Lyapunov exponents in dynamical systems, linking microscopic stability properties to macroscopic irreversibility and transport phenomena, with applications to electrical conductivity.

Contribution

It introduces a method to compute bounds on entropy production and transport coefficients using spectral properties of the local stability matrix, extending fundamental dynamical bounds.

Findings

Bounds on electrical conductivity for charged particles under electric field

Spectral properties of the local stability matrix can be used to estimate entropy flow rate

Bounds are numerically computable from molecular dynamics simulations

Abstract

Some microscopic dynamics are also macroscopically irreversible, dissipating energy and producing entropy. For many-particle systems interacting with deterministic thermostats, the rate of thermodynamic entropy dissipated to the environment is the average rate at which phase space contracts. Here, we use this identity and the properties of a classical density matrix to derive upper and lower bounds on the entropy flow rate with the spectral properties of the local stability matrix. These bounds are an extension of more fundamental bounds on the Lyapunov exponents and phase space contraction rate of continuous-time dynamical systems. They are maximal and minimal rates of entropy production, heat transfer, and transport coefficients set by the underlying dynamics of the system and deterministic thermostat. Because these limits on the macroscopic dissipation derive from the density matrix and the local stability matrix, they are numerically computable from the molecular dynamics. As an illustration, we show that these bounds are on the electrical conductivity for a system of charged particles subject to an electric field.