On practical applicability of the Jarzynski relation in statistical mechanics: a pedagogical example

TL;DR

This paper examines the practical use of the Jarzynski relation in statistical mechanics through a simple ideal gas model, highlighting its validity and limitations in non-equilibrium conditions and large systems.

Contribution

It provides a pedagogical example demonstrating the conditions under which the Jarzynski identity holds and discusses the exponential growth of experimental runs needed for large systems.

Findings

Jarzynski identity is valid due to rapid molecules in the Maxwell tail.

For large systems with fast piston movement, the number of runs needed grows exponentially.

The model illustrates the practical challenges in applying the Jarzynski relation.

Abstract

We suggest and discuss a simple model of an ideal gas under the piston to gain an insight into the workings of the Jarzynski identity connecting the average exponential of the work over the non-equilibrium trajectories with the equilibrium free energy. We show that the Jarzynski identity is valid for our system due to the very rapid molecules belonging to the tail of the Maxwell distribution. For the most interesting extreme, when the system volume is large, while the piston is moving with large speed (compared to thermal velocity) for a very short time, the necessary number of independent experimental runs to obtain a reasonable approximation for the free energy from averaging the non-equilibrium work grows exponentially with the system size.