The ideal gas as an urn model: derivation of the entropy formula

The approach of an ideal gas to equilibrium is simulated through a generalization of the Ehrenfest ball-and-box model. In the present model, the interior of each box is discretized, {\it i.e.}, balls/particles live in cells whose occupation can be either multiple or single. Moreover, particles occasionally undergo random, but elastic, collisions between each other and against the container walls. I show, both analitically and numerically, that the number and energy of particles in a given box eventually evolve to an equilibrium distribution $W$ which, depending on cell occupations, is binomial or hypergeometric in the particle number and beta-like in the energy. Furthermore, the long-run probability density of particle velocities is Maxwellian, whereas the Boltzmann entropy $\ln W$ exactly reproduces the ideal-gas entropy. Besides its own interest, this exercise is also relevant for pedagogical purposes since it provides, although in a simple case, an explicit probabilistic foundation for the ergodic hypothesis and for the maximum-entropy principle of thermodynamics. For this reason, its discussion can profitably be included in a graduate course on statistical mechanics.Comment: 17 pages, 3 figure

On the accuracy of the melting curves drawn from modelling a solid as an elastic medium

An ongoing problem in the study of a classical many-body system is the characterization of its equilibrium behaviour by theory or numerical simulation. For purely repulsive particles, locating the melting line in the pressure-temperature plane can be especially hard if the interparticle potential has a softened core or contains some adjustable parameters. A method is hereby presented that yields reliable melting-curve topologies with negligible computational effort. It is obtained by combining the Lindemann melting criterion with a description of the solid phase as an elastic continuum. A number of examples are given in order to illustrate the scope of the method and possible shortcomings. For a two-body repulsion of Gaussian shape, the outcome of the present approach compares favourably with the more accurate but also more computationally demanding self-consistent harmonic approximation.Comment: 25 pages, 7 figure

A fingerprint of surface-tension anisotropy in the free-energy cost of nucleation

We focus on the Gibbs free energy \u394G for nucleating a droplet of the stable phase (e.g. solid) inside the metastable parent phase (e.g. liquid), close to the first-order transition temperature. This quantity is central to the theory of homogeneous nucleation, since it superintends the nucleation rate. We recently introduced a field theory describing the dependence of \u394G on the droplet volume V, taking into account besides the microscopic fuzziness of the droplet-parent interface, also small fluctuations around the spherical shape whose effect, assuming isotropy, was found to be a characteristic logarithmic term. Here we extend this theory, introducing the effect of anisotropy in the surface tension, and show that in the limit of strong anisotropy \u394G(V) once more develops a term logarithmic on V, now with a prefactor of opposite sign with respect to the isotropic case. Based on this result, we argue that the geometrical shape that large solid nuclei mostly prefer could be inferred from the prefactor of the logarithmic term in the droplet free energy, as determined from the optimization of its near-coexistence profile

Preroughening, Diffusion, and Growth of An FCC(111) Surface

Preroughening of close-packed fcc(111) surfaces, found in rare gas solids, is an interesting, but poorly characterized phase transition. We introduce a restricted solid-on-solid model, named FCSOS, which describes it. Using mostly Monte Carlo, we study both statics, including critical behavior and scattering properties, and dynamics, including surface diffusion and growth. In antiphase scattering, it is shown that preroughening will generally show up at most as a dip. Surface growth is predicted to be continuous at preroughening, where surface self-diffusion should also drop. The physical mechanism leading to preroughening on rare gas surfaces is analysed, and identified in the step-step elastic repulsion.Comment: Revtex + uuencoded figures, to appear in Physical Review Letter

Anomalous melting behavior under extreme conditions: hard matter turning "soft"

We show that a system of particles interacting through the exp-6 pair potential, commonly used to describe effective interatomic forces under high compression, exhibits anomalous melting features such as reentrant melting and a rich solid polymorphism, including a stable BC8 crystal. We relate this behavior to the crossover, with increasing pressure, between two different regimes of local order that are associated with the two repulsive length scales of the potential. Our results provide a unifying picture for the high-pressure melting anomalies observed in many elements and point out that, under extreme conditions, atomic systems may reveal surprising similarities with soft matter.Comment: 10 pages, 4 figure