Landau free energy, Landau entropy, phase transitions and limits of metastability in an analytical model with a variable number of degrees of freedom

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Abstract
In this paper the statistical mechanical properties are investigated of a simple analytical model which is based loosely on a cluster of atoms. The model has a phase transition from a solid-like to a liquid-like phase. Particular interest is paid to the dependence of the nature of this phase transition on the number of degrees of freedom. As expected the transition becomes sharper as the number of degrees of freedom increases. The phase properties of the model are discussed conveniently in terms of Landau functions. In the canonical ensemble the relevant function is the Landau free energy F(Q, T), a function of the order parameter Q and the temperature T. The corresponding function in a microcanonical ensemble is minus the Landau entropy, a function of the order parameter and total energy. These functions can, in general, be calculated from probability distributions which are available from simulations. In both cases stable or metastable phases correspond to valleys in the function; continuous transitions to merging valleys and first-order phase changes to valleys separated by barriers. The N dependence of the Landau functions per degree of freedom is very weak, so that the best criterion for determining the order of a phase transition, which is valid for large and small systems, is the presence or absence of a barrier in the relevant Landau function. The limits of metastability of phases occur at the heads of the valleys in the Landau functions. It is shown that these can be related to the properties of the system in the potential ensemble. The model exhibits a region of negative specific heat (i.e., an S-band in the caloric curve) in the microcanonical ensemble which becomes vanishingly small in the limit of large N. Finally, the relation of these results to other proposed criteria for the limits of metastability of solid phases is discussed.

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