209 research outputs found
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 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 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
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