Statistical mechanics and thermodynamics of large and small systems

Sheldon Goldstein, David A. Huse, Joel L. Lebowitz, Pablo Sartori

Published 2017-12-24Version 1

Thermodynamics makes definite predictions about the thermal behavior of macroscopic systems in and out of equilibrium. For example, it provides relations between various measurable quantities such as the specific heat or compressibility of equilibrium systems, and the direction of energy flow in systems with non-uniform temperatures. Statistical mechanics aims to derive this behavior from the dynamics and statistics of the atoms and molecules making up these systems. A key element in this derivation is the large number of microscopic degrees of freedom of macroscopic systems. For such systems, the law of large numbers gives "almost sure" (definite) predictions for the behavior of an individual "typical" macroscopic system. The extension of these concepts and considerations to small systems, isolated or in contact with reservoirs, raises many questions. Here we will reexamine the various definitions of entropy for nonequilibrium systems. These include thermodynamic (hydrodynamic), Boltzmann, and Gibbs-Shannon entropies. We will then consider how they are applicable to systems with only a few relevant degrees of freedom in contact with thermal reservoirs, such as a nano-particle in a fluid, where they may be subject to measurement.