Phase transitions in small systems: Microcanonical vs. canonical ensembles

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 (nonanalyticities) 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.

On the Thermodynamics of Classical Micro-Canonical Systems

“…Each of them serves as the heat bath of the other. Both of them are in thermal equilibrium at the same temperature T , which is the temperature obtained in the standard way by taking the derivative of the micro-canonical entropy (3)-see (5). The temperature of the kinetic subsystem follows from the equipartition law.…”

“…The recent interest in the micro-canonical ensemble [1][2][3][4][5][6][7][8][9][10][11][12][13] is driven by the awareness that this ensemble is the cornerstone of statistical mechanics. Phase transitions described in the canonical ensemble could be linked to topological singularities of the micro-canonical energy landscape [14][15][16][17][18][19][20].…”

Thermodynamics, Statistical Mechanics and Entropy

“…However, because small effects of the order of 1/N, where N is the number of particles, have played a significant role in References [1][2][3][4][5][6][7][8][9][10][11], my discussion will also take small effects seriously.…”

“…These include exact adiabatic invariance and the prediction of the mean of an experimental observable rather than the mode, even though the difference is usually of order 1/N. Indeed, some theorists have insisted that these criteria must be satisfied for thermodynamic consistency [3,4]. Although I do not think that they are necessary for consistency, I will include these additional criteria in the discussion to give a complete picture.…”

“…For the Gibbs entropy of classical systems with a monotonically increasing density of states, the predicted energy is exactly correct [4,5,[7][8][9][10][11], although this is not true of quantum systems [22,23].…”