Temperature Fluctuations for a System in Contact with a Heat Bath

The problem of how to define and compute temperature fluctuations for a small system in contact with a heat bath is an old one and originates from Einstein’s theory of Brownian motion. Only for a small enough system does their relative size allow a straightforward experimental verification. Here we focus on a mesoscopic system in contact with a heat bath at temperature T _° and provide a self-consistent argument showing as to why, and in what sense, the observable standard deviation of temperature from T _° equals $\sqrt{k_{\rm B}/C}T_{\circ}$ where C is the mesoscopic system’s heat capacity. Our argument is based on ergodic decomposition, a simple fact that holds for a system in thermodynamic equilibrium and away from phase-transition points. In this way we close a line of argument opened by de Haas-Lorentz a century ago.