Phase transitions in simplified models with long-range interactions.

We study the origin of phase transitions in several simplified models with long-range interactions. For the self-gravitating ring model, we are unable to observe a possible phase transition predicted by Nardini and Casetti [Phys. Rev. E 80, 060103R (2009).] from an energy landscape analysis. Instead we observe a sharp, although without any nonanalyticity, change from a core-halo to a core-only configuration in the spatial distribution functions for low energies. By introducing a different class of solvable simplified models without any critical points in the potential energy we show that a behavior similar to the thermodynamics of the ring model is obtained, with a first-order phase transition from an almost homogeneous high-energy phase to a clustered phase and the same core-halo to core configuration transition at lower energies. We discuss the origin of these features for the simplified models and show that the first-order phase transition comes from the maximization of the entropy of the system as a function of energy and an order parameter, as previously discussed by Hahn and Kastner [Phys. Rev. E 72, 056134 (2005); Eur. Phys. J. B 50, 311 (2006)], which seems to be the main mechanism causing phase transitions in long-range interacting systems.

Discrete atomistic model to simulate etching of a crystalline solid.

A discrete atomistic solid-on-solid model is proposed to describe dissolution of a crystalline solid by a liquid. The model is based on the simple assumption that the probability per unit time of a unit cell being removed is proportional to its exposed area. Numerical simulations in one dimension demonstrate that the model has very good scaling properties. After removal of only about 10(2) monolayers, independently of the substrate size, the etched surface shows almost time-independent short-range correlations and the receding surface presents the Family-Vicsek scaling behavior. The scaling parameters alpha=0.491+/-0.002 and beta=0.330+/-0.001 indicate that the system belongs to the Kardar-Parisi-Zhang universality class. The imposition of periodic boundary conditions on the simulations reduces the effective system size by a factor of 0.68 without changing the exponents alpha and beta. Surprisingly, the periodic condition changes drastically the statistics of the surface height fluctuations and the short-range correlations. Without periodic conditions, that statistics is, up to 3 standard deviations, an asymmetric Lévy distribution with mu=1.82+/-0.01, and outside this region the statistics is Gaussian. With periodic conditions, that statistics is Gaussian, except for large negative fluctuations.

Scaling transformation of random walk distributions in a lattice.

We use a decimation procedure in order to obtain the dynamical renormalization group transformation (RGT) properties of random walk distribution in a 1+1 lattice. We obtain an equation similar to the Chapman-Kolmogorov equation. First we show the existence of invariants through the RGT. We also show the existence of functions which are semi-invariants through the RGT. Second, we show as well that the distribution R(q)(x)=[1+b(q-1)x(2)](1/(1-q)) (q>1), which is an exact solution of a nonlinear Fokker-Planck equation, is a semi-invariant for RGT. We obtain the map q(')=f(q) from the RGT and we show that this map has two fixed points: q=1, attractor, and q=2, repellor, which are the Gaussian and the Lorentzian, respectively. We show the connections between these result and the Levy flights.