Two famous results of Einstein derived from the Jarzynski equality

Abstract

The Jarzynski equality (JE) is a remarkable statement relating transient irreversible processes to infinite-time free energy differences. Although twenty years old, the JE remains unfamiliar to many; nevertheless it is a robust and powerful law. We examine two of Einstein's most simple and well-known discoveries, one classical and one quantum, and show how each of these follows from the JE. Our first example is Einstein's relation between the drag and diffusion coefficients of a particle in Brownian motion. In this context we encounter a paradox in the macroscopic limit of the JE which is fascinating, but also warns us against using the JE too freely outside of the microscopic domain. Our second example is the equality of Einstein's $B$ coefficients for absorption and stimulated emission of quanta. Here resonant light does irreversible work on a sample, and the argument differs from Einstein's equilibrium reasoning using the Planck black-body spectrum. We round out our examples with a brief derivation and discussion of Jarzynski's remarkable equality.