Probability distribution of classical observables

TL;DR

This paper derives a general framework for the time-dependent probability distribution of classical observables using Hamiltonian mechanics, extending fluctuation theorems and deriving specific cases like Gaussian distributions.

Contribution

It introduces a Hamiltonian mechanics-based method to obtain the probability density functions of classical observables, including thermodynamic work, and proves key fluctuation theorems.

Findings

Derived the time-dependent probability density function for classical observables.

Proved Jarzynski's equality and Crook's fluctuation theorem within this framework.

Obtained Gaussian distribution as a special case for slow processes.

Abstract

In this work, I derive the time-dependent probability density function of classical observables using the Hamiltonian mechanics approach, extending the notion of fluctuation theorems for any observables. In particular, the time-dependent probability density function of the thermodynamic work evolving in the switching time is solved as a special case. From such a relation, I prove Jarzynski's equality and Crook's fluctuation theorem. The particular case of a Gaussian distribution for slowly-varying processes is derived as well at the end.