Small-System Group: Thermodynamics as a Complete Self-Similarity Limit

Abstract

We revisit the Rayleigh--Riabouchinsky paradox in dimensional analysis by making explicit the bridge between thermodynamics and the mechanical interpretation of temperature. Boltzmann's constant $k_B$ acts as a dimensional unifier, leading to an augmented $\Pi$-theorem with an additional dimensionless group that encodes system size. In the macroscopic thermodynamic limit this small-system group, $\Pi_B = k_B/(c\,\ell^3)$ -- the inverse heat capacity of a control volume of size $\ell^3$ in units of $k_B$ -- becomes irrelevant as the response becomes self-similar with respect to it, recovering Rayleigh's formulation. Under suitable conditions, macroscopic limits make the fluctuations of the observables of interest negligible compared to their expected values, hence the state of a system is characterized by a reduced set of parameters. We thus recast thermodynamics as the complete-similarity limit of statistical mechanics with respect to $\Pi_B$, which also controls thermodynamic fluctuations. We also discuss second-order phase transitions from the viewpoint of incomplete similarity.