Phase Transitions in Small Systems: Microcanonical vs. Canonical Ensembles

TL;DR

This paper compares phase transition phenomena in small classical systems within microcanonical and canonical ensembles, revealing rich oscillatory behavior and singularities in microcanonical thermodynamics, and smooth transitions in the canonical ensemble.

Contribution

It demonstrates how small systems exhibit microcanonical phase-like behavior with singularities, contrasting with smooth canonical phase transitions analyzed via partition function zeros.

Findings

Microcanonical thermodynamics shows oscillations and singularities in small systems.

Canonical ensemble exhibits smooth phase transitions identified through partition function zeros.

Observable effects in evaporation/dissociation experiments are predicted for microcanonical systems.

Abstract

We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (non-analyticities) separating different microscopic phases. These microscopic phases may be identified as different microphysical dissociation states of the small system. The microscopic oscillations of microcanonical thermodynamic quantities (e.g. temperature, heat capacity, or pressure) should in principle be observable in suitably designed evaporation/dissociation experiments (which must realize the physical preconditions of the microcanonical ensemble). By contrast, singular phase transitions cannot occur, if a small system is embedded into an infinite heat bath (thermostat), corresponding to the canonical ensemble. For the simple model systems under consideration, it is nevertheless possible to identify a smooth canonical phase transition by studying the distribution of complex zeros of the canonical partition function.