Hamilton's equations for relaxation function

TL;DR

This paper introduces a Hamilton's equations-based method to calculate the relaxation function in thermally isolated systems, simplifying the process by avoiding initial ensemble averages, demonstrated on a quartic oscillator.

Contribution

A novel Hamilton's equations approach for computing relaxation functions in isolated systems, reducing computational complexity and initial condition averaging.

Findings

Method successfully applied to quartic oscillator

Potential extension to quantum and stochastic thermodynamics

Simplifies calculation of relaxation functions

Abstract

The relaxation function is the cornerstone to perform calculations in weakly driven processes. Properties that such a function should obey are already established, but the difficulty in its calculation is still an issue to be overcome. In this work, I proposed a new method to determine such a function for thermally isolated systems, based on a Hamilton's equations approach. Observing that the microscopic relaxation function can be turned into a canonical variable, one can choose the initial conditions of the solutions of Hamilton's equations to avoid the calculation of the average in the initial canonical ensemble. The unbearable example of the quartic oscillator is solved to corroborate the method. Extensions to the quantum realm and stochastic thermodynamics are mandatory.