Second-order fast-slow dynamics of non-ergodic Hamiltonian systems: Thermodynamic interpretation and simulation

TL;DR

This paper derives second-order corrections for non-ergodic fast-slow Hamiltonian systems, interprets their energy thermodynamically, and validates the approximations through numerical simulations, enhancing understanding of their dynamics beyond the homogenised limit.

Contribution

It provides a rigorous derivation of second-order corrections in fast-slow Hamiltonian systems and links these corrections to thermodynamic principles, offering improved approximation methods.

Findings

Second-order corrections decompose into oscillatory and average motion components.

Thermodynamic quantities like temperature and entropy satisfy energy relations to second order.

Numerical simulations confirm the accuracy and efficiency of the second-order approximations.

Abstract

A class of fast-slow Hamiltonian systems with potential $U_\varepsilon$ describing the interaction of non-ergodic fast and slow degrees of freedom is studied. The parameter $\varepsilon$ indicates the typical timescale ratio of the fast and slow degrees of freedom. It is known that the Hamiltonian system converges for $\varepsilon\to0$ to a homogenised Hamiltonian system. We study the situation where $\varepsilon$ is small but positive. First, we rigorously derive the second-order corrections to the homogenised (slow) degrees of freedom. They can be decomposed into explicitly given terms that oscillate rapidly around zero and terms that trace the average motion of the corrections, which are given as the solution to an inhomogeneous linear system of differential equations. Then, we analyse the energy of the fast degrees of freedom expanded to second-order from a thermodynamic point of view. In particular, we define and expand to second-order a temperature, an entropy and external forces and show that they satisfy to leading-order, as well as on average to second-order, thermodynamic energy relations akin to the first and second law of thermodynamics. Finally, we analyse for a specific fast-slow Hamiltonian system the second-order asymptotic expansion of the slow degrees of freedom from a numerical point of view. Their approximation quality for short and long time frames and their total computation time are compared with those of the solution to the original fast-slow Hamiltonian system of similar accuracy.