Multispecies effects in the equilibrium and out-of-equilibrium thermostatistics of overdamped motion.

Progress has been recently made, both theoretical and experimental, regarding the thermostatistics of complex systems of interacting particles or agents (species) obeying a nonlinear Fokker-Planck dynamics. However, major advances along these lines have been restricted to systems consisting of only one type of species. The aim of the present contribution is to overcome that limitation, going beyond single-species scenarios. We investigate the dynamics of overdamped motion in interacting and confined many-body systems having two or more species that experience different intra- and interspecific forces in a regime where forces arising from standard thermal noise can be neglected. Even though these forces are neglected, the behavior of the system can be analyzed in terms of an appropriate thermostatistical formalism. By recourse to a mean-field treatment, we derive a set of coupled nonlinear Fokker-Planck equations governing the behavior of these systems. We obtain an H theorem for this Fokker-Planck dynamics and discuss in detail an example admitting an exact, analytical stationary solution.

Nonlinear drag forces and the thermostatistics of overdamped motion.

Diverse processes in statistical physics are usually analyzed on the assumption that the drag force acting on a test particle moving in a resisting medium is linear on the velocity of the particle. However, nonlinear drag forces do appear in relevant situations that are currently the focus of experimental and theoretical work. Motivated by these developments, we explore the consequences of nonlinear drag forces for the thermostatistics of systems of interacting particles performing overdamped motion. We derive a family of nonlinear Fokker-Planck equations for these systems, taking into account the effects of nonlinear drag forces. We investigate the main properties of these evolution equations, including an H-theorem, and obtain exact solutions of the stretched q-exponential form.

From the nonlinear Fokker-Planck equation to the Vlasov description and back: Confined interacting particles with drag.

Nonlinear Fokker-Planck equations endowed with power-law diffusion terms have proven to be valuable tools for the study of diverse complex systems in physics, biology, and other fields. The nonlinearity appearing in these evolution equations can be interpreted as providing an effective description of a system of particles interacting via short-range forces while performing overdamped motion under the effect of an external confining potential. This point of view has been recently applied to the study of thermodynamical features of interacting vortices in type II superconductors. In the present work we explore an embedding of the nonlinear Fokker-Planck equation within a Vlasov equation, thus incorporating inertial effects to the concomitant particle dynamics. Exact time-dependent solutions of the q-Gaussian form (with compact support) are obtained for the Vlasov equation in the case of quadratic confining potentials.

Nonlinear Kramers equation associated with nonextensive statistical mechanics.

Stationary and time-dependent solutions of a nonlinear Kramers equation, as well as its associated nonlinear Fokker-Planck equations, are investigated within the context of Tsallis nonextensive statistical mechanics. Since no general analytical time-dependent solutions are found for such a nonlinear Kramers equation, an ansatz is considered and the corresponding asymptotic behavior is studied and compared with those known for the standard linear Kramers equation. The H-theorem is analyzed for this equation and its connection with Tsallis entropy is investigated. An application is discussed, namely the motion of Hydra cells in two-dimensional cellular aggregates, for which previous measurements have verified q-Gaussian distributions for velocity components and superdiffusion. The present analysis is in quantitative agreement with these experimental results.

Nonlinear Ehrenfest's urn model.

Ehrenfest's urn model is modified by introducing nonlinear terms in the associated transition probabilities. It is shown that these modifications lead, in the continuous limit, to a Fokker-Planck equation characterized by two competing diffusion terms, namely, the usual linear one and a nonlinear diffusion term typical of anomalous diffusion. By considering a generalized H theorem, the associated entropy is calculated, resulting in a sum of Boltzmann-Gibbs and Tsallis entropic forms. It is shown that the stationary state of the associated Fokker-Planck equation satisfies precisely the same equation obtained by extremization of the entropy. Moreover, the effects of the nonlinear contributions on the entropy production phenomenon are also analyzed.

Generalized entropy production phenomena: a master-equation approach.

The time rate of generalized entropic forms, defined in terms of discrete probabilities following a master equation, is investigated. Both contributions, namely entropy production and flux, are obtained, extending works carried previously for the Boltzmann-Gibbs entropy to a wide class of entropic forms. Particularly, it is shown that the entropy-production contribution is always non-negative for such entropies. Some illustrative examples for known generalized entropic forms in the literature are also worked out. Since generalized entropies have been lately associated with several complex systems in nature, the present analysis should be applicable to irreversible processes in these systems.

Entropy production and nonlinear Fokker-Planck equations.

The entropy time rate of systems described by nonlinear Fokker-Planck equations--which are directly related to generalized entropic forms--is analyzed. Both entropy production, associated with irreversible processes, and entropy flux from the system to its surroundings are studied. Some examples of known generalized entropic forms are considered, and particularly, the flux and production of the Boltzmann-Gibbs entropy, obtained from the linear Fokker-Planck equation, are recovered as particular cases. Since nonlinear Fokker-Planck equations are appropriate for the dynamical behavior of several physical phenomena in nature, like many within the realm of complex systems, the present analysis should be applicable to irreversible processes in a large class of nonlinear systems, such as those described by Tsallis and Kaniadakis entropies.

Nonlinear relativistic and quantum equations with a common type of solution.

Generalizations of the three main equations of quantum physics, namely, the Schrödinger, Klein-Gordon, and Dirac equations, are proposed. Nonlinear terms, characterized by exponents depending on an index q, are considered in such a way that the standard linear equations are recovered in the limit q→1. Interestingly, these equations present a common, solitonlike, traveling solution, which is written in terms of the q-exponential function that naturally emerges within nonextensive statistical mechanics. In all cases, the well-known Einstein energy-momentum relation is preserved for arbitrary values of q.

Thermostatistics of overdamped motion of interacting particles.

We show through a nonlinear Fokker-Planck formalism, and confirm by molecular dynamics simulations, that the overdamped motion of interacting particles at T=0, where T is the temperature of a thermal bath connected to the system, can be directly associated with Tsallis thermostatistics. For sufficiently high values of T, the distribution of particles becomes Gaussian, so that the classical Boltzmann-Gibbs behavior is recovered. For intermediate temperatures of the thermal bath, the system displays a mixed behavior that follows a novel type of thermostatistics, where the entropy is given by a linear combination of Tsallis and Boltzmann-Gibbs entropies.

The two-dimensional site-diluted Ising model: a short-time-dynamics approach.

The site-diluted Ising ferromagnet is investigated on a square lattice, within short-time-dynamics numerical simulations, for different site concentrations. The dynamical exponents θ and z are obtained and it is shown that these exponents do depend strongly on the disorder, exhibiting a clear breakdown of universality, characterized by relative variations of nearly 100% in the range of site concentrations investigated. In what concerns the static exponents ß and ν, universality is preserved within the error bars.

Spin-1 Ising model: exact damage-spreading relations and numerical simulations.

The nearest-neighbor-interaction spin-1 Ising model is investigated within the damage-spreading approach. Exact relations involving quantities computable through damage-spreading simulations and thermodynamic properties are derived for such a model, defined in terms of a very general Hamiltonian that covers several spin-1 models of interest in the literature. Such relations presuppose translational invariance and hold for any ergodic dynamical procedure, leading to an efficient tool for obtaining thermodynamic properties. The implementation of the method is illustrated through damage-spreading simulations for the ferromagnetic spin-1 Ising model on a square lattice. The two-spin correlation function and the magnetization are obtained, with precise estimates of their associated critical exponents and of the critical temperature of the model, in spite of the small lattice sizes considered. These results are in good agreement with the universality hypothesis, with critical exponents in the same universality class of the spin- 12 Ising model. The advantage of the present method is shown through a significant reduction of finite-size effects by comparing its results with those obtained from standard Monte Carlo simulations.

Using exact relations in damage-spreading simulations: the Baxter line of the two-dimensional Ashkin-Teller model.

The nearest-neighbor-interaction ferromagnetic Ashkin-Teller model is investigated on a square lattice through a powerful computational method for dealing with correlation functions in magnetic systems. This technique, which is based on damage-spreading numerical simulations, makes use of exact relations involving special kinds of damage and correlation functions, as well as the corresponding order parameters of the model. The computation of correlation functions, which represents usually a hard task in standard Monte Carlo simulations, due to large fluctuations, turns out to be much simpler within the present approach. We concentrate our analysis along the Baxter line, well known for its continuously varying critical exponents; seven different points along this line are investigated. The critical exponents associated with correlation functions along the Baxter line are successfully evaluated, by means of numerical methods, within damage-spreading simulations. The efficiency of this method is confirmed through precise estimates of the critical exponents associated with the order parameters (magnetization and polarization), as well as with their corresponding correlation functions, in spite of the small lattice sizes considered.

Damage-spreading simulations through exact relations for the two-dimensional Potts ferromagnet.

A powerful computational method for dealing with correlation functions in magnetic systems, based on damage-spreading simulations, is reviewed and tested, by investigating the q-state Potts ferromagnet, on a square lattice, at criticality. Exact relations involving special kinds of damage and the spin-spin correlation function, as well as the magnetization, are used. The efficiency of the method arises with a significant reduction of the finite-size effects, with respect to conventional Monte Carlo simulations. Correlation functions, which represent usually a hard task within this latter procedure, appear to be much more easily estimated through the present damage-spreading simulations. The effectiveness of the technique is illustrated by an accurate estimate of the exponent eta, of the spin-spin correlation function, for q=2, 3, and 4, with rather small lattice sizes. In the cases q > or = 5, an analysis of the magnetization is consistent with the well-known first-order phase transition.

Phase diagram of the two-dimensional +/-J Ising spin glass.

The +/-J Ising spin glass [probabilities p and (1-p) associated with ferromagnetic and antiferromagnetic couplings, respectively] is studied by applying a real-space renormalization-group technique on a hierarchical lattice that approaches the square lattice. Within such a procedure, there is no spin-glass phase and only two finite-temperature phases are found, namely, the paramagnetic and ferromagnetic ones. In spite of a reasonably small computational effort, an accurate paramagnetic-ferromagnetic boundary is presented: the estimate for the slope at p=1 is in very good agreement with the well-known exact result, whereas the coordinates of the Nishimori point are determined within a high precision. Below the Nishimori point, such a boundary is not strictly vertical-contrary to the usual belief-in such a way that a small reentrance is found at low temperatures.

Ground-state degeneracies of Ising spin glasses on diamond hierarchical lattices.

The total number of ground states for short-range Ising spin glasses, defined on diamond hierarchical lattices of fractal dimensions d=2, 3, 4, 5, and 2.58, is estimated by means of analytic calculations (three last hierarchy levels of the d=2 lattice) and numerical simulations (lower hierarchies for d=2 and all remaining cases). It is shown that in the case of continuous probability distributions for the couplings, the number of ground states is finite in the thermodynamic limit. However, for a bimodal probability distribution (+/-J with probabilities p and 1-p, respectively), the average number of ground states is maximum for a wide range of values of p around p=1 / 2 and depends on the total number of sites at hierarchy level n, Nn. In this case, for all lattices investigated, it is shown that the ground-state degeneracy behaves like exp[h(d)Nn], in the limit Nn large, where h(d) is a positive number which depends on the lattice fractal dimension. The probability of finding frustrated cells at a given hierarchy level n, Fn(p), is calculated analytically (three last hierarchy levels for d=2 and the last hierarchy of the d=3 lattice, with 0>1, only the last hierarchies contribute significantly to the ground-state degeneracy; such a dominant behavior becomes stronger for high fractal dimensions. The exponential increase of the number of ground states with the total number of sites is in agreement with the mean-field picture of spin glasses.