Entropy and density of states from isoenergetic nonequilibrium processes

TL;DR

This paper derives two identities in statistical mechanics relating entropy differences and density of states at constant energy, extending the Jarzynski equality to the microcanonical ensemble and providing practical tools for numerical simulations.

Contribution

It introduces a novel extension of the Jarzynski equality and a thermodynamic integration formula for microcanonical entropy calculations.

Findings

Derived a microcanonical Jarzynski-like identity.

Established a thermodynamic integration formula for entropy.

Discussed practical numerical simulation applications.

Abstract

Two identities in statistical mechanics involving entropy differences (or ratios of density of states) at constant energy are derived. The first provides a nontrivial extension of the Jarzynski equality to the microcanonical ensemble [C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)], which can be seen as a ``fast-switching'' version of the adiabatic switching method for computing entropies [M. Watanabe, W. P. Reinhardt, Phys. Rev. Lett. 65, 3301 (1990)]. The second is a thermodynamic integration formula analogous to a well-known expression for free energies, and follows after taking the quasistatic limit of the first. Both identities can be conveniently used in conjunction with a scaling relation (herein derived) that allows one to extrapolate measurements taken at a single energy to a wide range of energy values. Practical aspects of these identities in the context of numerical simulations are discussed.