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The theory of small-system thermodynamics was originally developed to extendthe laws of thermodynamics to length scales of nanometers. Here we review this"nanothermodynamics," and stress how it also applies to large systems thatsubdivide into a heterogeneous distribution of internal subsystems that we call"regions." We emphasize that the true thermal equilibrium of most systems oftenrequires that these regions are in the fully-open generalized ensemble, with adistribution of region sizes that is not externally constrained, which we callthe "nanocanonical" ensemble. We focus on how nanothermodynamics impacts thestatistical mechanics of specific models. One example is an ideal gas ofindistinguishable atoms in a large volume that subdivides into an ensemble ofsmall regions of variable volume, with separate regions containing atoms thatare distinguishable from those in other regions. Combining such subdividedregions yields the correct entropy of mixing, avoiding Gibbs paradox withoutresorting to macroscopic quantum symmetry for semi-classical particles. Othermodels are based on Ising-like spins (binary degrees of freedom), which aresolved analytically in one-dimension, making them suitable examples forintroductory courses in statistical physics. A key result is to quantify thenet increase in entropy when large systems subdivide into small regions ofvariable size. Another result is to show similarity in the equilibriumproperties of a two-state model in the nanocanonical ensemble and a three-statemodel in the canonical ensemble. Thus, emergent phenomena may alter the thermalbehavior of microscopic models, and the correct ensemble is necessary foraccurate predictions.

An introduction to nanothermodynamics: Thermal equilibrium for heterogeneous and finite-sized systems