Dynamical Instability and Transport Coefficient in Deterministic Diffusion

TL;DR

This paper investigates the relationship between dynamical instability and transport coefficients in deterministic diffusion, establishing a universal connection between Lyapunov exponents and velocity in biased systems.

Contribution

It introduces a theoretical framework linking Lyapunov exponents and transport properties in biased deterministic diffusions, confirmed by numerical simulations.

Findings

Difference in generalized Lyapunov exponents relates to normalized velocity.

Ratios of time-averaged velocity to Lyapunov exponent converge to a universal constant.

Numerical simulations support the theoretical predictions.

Abstract

We construct both normal and anomalous deterministic biased diffusions to obtain the Einstein relation for their time-averaged transport coefficients. We find that the difference of the generalized Lyapunov exponent between biased and unbiased deterministic diffusions is related to the normalized velocity based on the ensemble average. By Hopf's ergodic theorem, the ratios between the time-averaged velocity and the Lyapunov exponent for single trajectories converge to a universal constant, which is proportional to the strength of the bias. We confirm this theory using numerical simulations.