Domingo Prato

Nonextensive Statistical Mechanics: Some Links with Astronomical Phenomena

A variety of astronomical phenomena appear to not satisfy the ergodic hypothesis in the relevant stationary state, if any. As such, there is no reason for expecting the applicability of Boltzmann–Gibbs (BG) statistical mechanics. Some of these phenomena appear to follow, instead, nonextensive statistical mechanics. In the same manner that the BG formalism is based on the entropy SBG=−k∑ipi ln pi, the nonextensive one is based on the form Sq=k(1 −∑ipiq)/(q− 1) (with S1=SBG). The stationary states of the former are characterized by an exponential dependence on the energy, whereas those of the latter are characterized by an (asymptotic) power law. A brief review of this theory is given here, as well as of some of its applications, such as the solar neutrino problem, polytropic self-gravitating systems, galactic peculiar velocities, cosmic rays and some cosmological aspects. In addition to these, an analogy with the Keplerian elliptic orbits versus the Ptolemaic epicycles is developed, where we show that optimizing Sq with a few constraints is equivalent to optimizing SBG with an infinite number of constraints.

Superdiffusion in decoupled continuous time random walks

Abstract Continuous time random walk models with decoupled waiting time density are studied. When the spatial one-jump probability density belongs to the Levy distribution type and the total time transition is exponential a generalized superdiffusive regime is established. This is verified by showing that the square width of the probability distribution (appropriately defined) grows as t2/γ with 0

Bulk-mediated surface diffusion: non-Markovian desorption dynamics

Here we analyse the dynamics of adsorbed molecules within the bulk-mediated surface diffusion framework, when the particles desorption mechanism is characterized by a non-Markovian process, while the particles adsorption as well as its motion in the bulk is governed by Markovian dynamics. We study the diffusion of particles in both semi-infinite and finite cubic lattices, analysing the conditional probability to find the system on the reference absorptive plane as well as the surface dispersion as functions of time. The results are compared with known Markovian cases showing the differences that can be exploited to distinguish between Markovian and non-Markovian desorption mechanisms in experimental situations.

Polychromatic majority model: Criticality and real space renormalization group

Abstract A generalization of a simple-majority rule model is presented. The system, say a d -dimensional hypercubic checkerboard, whose elements are coloured with one out of q colours with probabilities p 1 , p 2 ,..., p q presents a continuous phase transition. Using a real space renormalization group (RG) approach, we establish the phase diagram as well as the correlation length critical exponent v . The various types of convergence of the RG numerical values for v towards the (presumably) exact answer are analyzed in connection with finite size scalings.

Bulk mediated surface diffusion: the infinite system case

Abstract.An analytical soluble model based on a Continuous Time Random Walk (CTRW) scheme for the adsorption-desorption processes at interfaces, called bulk-mediated surface diffusion, is presented. The time evolution of the effective probability distribution width on the surface is calculated and analyzed within an anomalous diffusion framework. The asymptotic behavior for large times shows a sub-diffusive regime for the effective surface diffusion but, depending on the observed range of time, other regimes may be obtained. Monte Carlo simulations show excellent agreement with analytical results. As an important byproduct of the indicated approach, we present the evaluation of the time for the first visit to the surface.

A simple numerical method for solving problems in electrostatics

A very simple numerical method for solving boundary problems in electrostatics is presented. The approximation scheme is based on the substitution of a series expansion of the electrostatic potential by a finite sum and a discretization of the boundary where the conditions are given. These boundary conditions may be of the mixed type. In this way the original problem is reduced to the solution of a system of linear equations. As an application of the method proposed, the problem of a hollow cap of a sphere set to a constant potential is treated.

Bulk mediated surface diffusion: finite bulk case

Within the framework of a Master Equation scheme, we address the dynamics of adsorbed molecules (a fundamental issue in surface physics) and study the diffusion of particles in a finite cubic lattice whose boundaries are at the z = 1 and the z = L planes where L = 2; 3; 4; ..., while the x and y directions are unbounded. As we are interested in the effective diffusion process at the interface z = 1, we calculate analytically the conditional probability for finding the particle on the z = 1 plane as well as the surface dispersion as a function of time and compare these results with Monte Carlo simulations finding an excellent agreement. These results show that: there exists an optimal number of layers that maximizes 〈r2(t)〉 on the interface; for a small number the layers the long-time effective diffusivity on the interface is normal, crossing over abruptly towards a subdiffusive behavior as the number of layers increases.

Specific heat revisited

The correlation between potential shape and specific heat is generally absent from textbook discussions. We present a detailed analysis of the specific heat contribution due to independent particles subject to one‐dimensional classical and quantum model potentials. For the classical models, we use phase space concepts to develop a clear physical interpretation of the temperature dependence of the specific heat. For the quantum models, we make the interpretation in terms of the differences in quantum levels.

Uniform function constants of motion and their first-order perturbation

The main purpose of this work is to present some uniform function constants of motion rather than the well-known quantities arising from spacetime symmetries. These constants are usually associated with the intrinsic characteristics of the trajectories of a particle in a central potential field. We treat two cases. The first is the Lenz vector which sometimes is found in the literature [1, 2]; the other is associated with the isotropic harmonic oscillator, of relative importance in some simple models of the classical molecular interaction. The first example is applied to describe the perturbation of the trajectories in the Rutherford scattering and the precession of the Keplerian orbit of a planet. In the other case the conserved quantity is a symmetric tensor. We find the eigenvectors and eigenvalues of that tensor while at the same time we obtain the solution to the problem of calculating the rotation rate of the orbits in first order of a perturbation parameter in the potential energy, by performing a simple coordinate transformation in the Cartesian plane. We think that the present work addresses many aspects of mechanics with a didactical interest in other physics or mathematics courses.

An alternative approach to the Terwiel cumulants expansion in disordered media

In the context of the study of diffusion in disordered media we present an alternative way to obtain the Terwiel cumulants expansion. Our approach starts from a formal solution of the master equation (ME) associated with the model of the nearest-neighbour random walk in a one-dimensional disordered chain. We apply our formalism to the analysis of a finite-effective-medium-like approximation.