Non-Uniform Convergence in Moment Expansions of Integral Work Relations

TL;DR

This paper investigates how moment series in integral fluctuation theorems can fail to converge uniformly in certain limits, emphasizing the importance of limit order in non-equilibrium thermodynamics calculations.

Contribution

It demonstrates non-uniform convergence in moment expansions using models of measurement feedback and gas expansion, revealing the significance of limit ordering.

Findings

Low order moments approximate the limit well

High order moments deviate significantly from the limit

Transition to convergence shifts to higher moments as the limit is approached

Abstract

Exponential averages that appear in integral fluctuation theorems can be recast as a sum over moments of thermodynamic observables. We use two examples to show that such moment series can exhibit non-uniform convergence in certain singular limits. The first example is a simple model of a process with measurement and feedback. In this example, the limit of interest is that of error-free measurements. The second system we study is an ideal gas particle inside an (infinitely) fast expanding piston. Both examples show qualitative similarities; the low order moments are close to their limiting value, while high order moments strongly deviate from their limit. As the limit is approached the transition between the two groups of moments is pushed toward higher and higher moments. Our findings highlight the importance of the ordering of limits in certain non-equilibrium related calculations.