Evaldo M. F. Curado

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.

Consequences of the H theorem from nonlinear Fokker-Planck equations.

A general type of nonlinear Fokker-Planck equation is derived directly from a master equation, by introducing generalized transition rates. The H theorem is demonstrated for systems that follow those classes of nonlinear Fokker-Planck equations, in the presence of an external potential. For that, a relation involving terms of Fokker-Planck equations and general entropic forms is proposed. It is shown that, at equilibrium, this relation is equivalent to the maximum-entropy principle. Families of Fokker-Planck equations may be related to a single type of entropy, and so, the correspondence between well-known entropic forms and their associated Fokker-Planck equations is explored. It is shown that the Boltzmann-Gibbs entropy, apart from its connection with the standard--linear Fokker-Planck equation--may be also related to a family of nonlinear Fokker-Planck equations.

A general nonlinear Fokker-Planck equation and its associated entropy

Abstract.A recently introduced nonlinear Fokker-Planck equation, derived directly from a master equation, comes out as a very general tool to describe phenomenologically systems presenting complex behavior, like anomalous diffusion, in the presence of external forces. Such an equation is characterized by a nonlinear diffusion term that may present, in general, two distinct powers of the probability distribution. Herein, we calculate the stationary-state distributions of this equation in some special cases, and introduce associated classes of generalized entropies in order to satisfy the H-theorem. Within this approach, the parameters associated with the transition rates of the original master-equation are related to such generalized entropies, and are shown to obey some restrictions. Some particular cases are discussed.

ROUNDOFF-INDUCED COALESCENCE OF CHAOTIC TRAJECTORIES

Numerical experiments recently discussed in the literature show that identical nonlinear chaotic systems linked by a common noise term (or signal) may synchronize after a finite time. We study the process of synchronization as function of precision of calculations. Two generic behaviors of the average coalescence time are identified: exponential or linear. In both cases no synchronization occurs if iterations are done with {\em infinite} precision.

General aspects of the thermodynamical formalism

We present some recent developments concerning general aspects of the thermodynamical formalism. Through simple arguments, we exhibit some basic entropies that have most of the thermodynamic properties of the Shannon entropy. Their stabilities are also analysed, and several points concerning nonextensive thermodynamics are discussed.

Classes of N-Dimensional Nonlinear Fokker-Planck Equations Associated to Tsallis Entropy

Several previous results valid for one-dimensional nonlinear Fokker-Planck equations are generalized to N-dimensions. A general nonlinear N-dimensional Fokker-Planck equation is derived directly from a master equation, by considering nonlinearitiesin the transition rates. Using nonlinear Fokker-Planck equations, the H-theorem is proved;for that, an important relation involving these equations and general entropic forms is introduced. It is shown that due to this relation, classes of nonlinear N-dimensional Fokker-Planck equations are connected to a single entropic form. A particular emphasis is given to the class of equations associated to Tsallis entropy, in both cases of the standard, and generalized definitions for the internal energy.

q-moments remove the degeneracy associated with the inversion of the q-Fourier transform

It was recently proven [Hilhorst, JSTAT, P10023 (2010)] that the q-generalization of the Fourier transform is not invertible in the full space of probability density functions for q > 1. It has also been recently shown that this complication disappears if we dispose of the q-Fourier transform not only of the function itself, but also of all of its shifts [Jauregui and Tsallis, Phys. Lett. A 375, 2085 (2011)]. Here we show that another road exists for completely removing the degeneracy associated with the inversion of the q-Fourier transform of a given probability density function. Indeed, it is possible to determine this density if we dispose of some extra information related to its q-moments.

Thermodynamic framework for compact q-Gaussian distributions

Abstract Recent works have associated systems of particles, characterized by short-range repulsive interactions and evolving under overdamped motion, to a nonlinear Fokker–Planck equation within the class of nonextensive statistical mechanics, with a nonlinear diffusion contribution whose exponent is given by ν = 2 − q . The particular case ν = 2 applies to interacting vortices in type-II superconductors, whereas ν > 2 covers systems of particles characterized by short-range power-law interactions, where correlations among particles are taken into account. In the former case, several studies presented a consistent thermodynamic framework based on the definition of an effective temperature θ (presenting experimental values much higher than typical room temperatures T , so that thermal noise could be neglected), conjugated to a generalized entropy s ν (with ν = 2 ). Herein, the whole thermodynamic scheme is revisited and extended to systems of particles interacting repulsively, through short-ranged potentials, described by an entropy s ν , with ν > 1 , covering the ν = 2 (vortices in type-II superconductors) and ν > 2 (short-range power-law interactions) physical examples. One basic requirement concerns a cutoff in the equilibrium distribution P eq ( x ) , approached due to a confining external harmonic potential, ϕ ( x ) = α x 2 ∕ 2 ( α > 0 ). The main results achieved are: (a) The definition of an effective temperature θ conjugated to the entropy s ν ; (b) The construction of a Carnot cycle, whose efficiency is shown to be η = 1 − ( θ 2 ∕ θ 1 ) , where θ 1 and θ 2 are the effective temperatures associated with two isothermal transformations, with θ 1 > θ 2 ; (c) Thermodynamic potentials, Maxwell relations, and response functions. The present thermodynamic framework, for a system of interacting particles under the above-mentioned conditions, and associated to an entropy s ν , with ν > 1 , certainly enlarges the possibility of experimental verifications.

A thermodynamical formalism describing mechanical interactions

The dynamical behavior of an overdamped mechanical model devoid of any usual thermal effects is analyzed by a formalism that is similar to usual thermodynamics, and completely independent of any ad hoc assumption of a probability distribution of states in phase space of the mechanical model. It leads to the definition of a new entropy function, which does not coincide with the usual thermodynamical entropy. The new step making the difference to previous studies of this system is the identification of two non-equivalent mechanical interaction mechanisms, which are defined and identified as work and pseudo-heat. Together with the introduced effective temperature θ, they make it possible to characterize the equivalent to isothermal, adiabatic, isobaric, and isochoric processes. Three statements, formally analogous to the zeroth, first, and second law of thermodynamics, are issued. The statement of the second law results from the asymmetry in the way energy can be exchanged along the two processes. A Carnot cycle is defined, for which the efficiency is expressed in terms of θ in the operating pseudo-heat reservoirs. The analogous Clausius theorem for the system operating an arbitrary reversible cycle is proved, leading to the new entropy function. Consequences of the extension of thermodynamic formalism to mechanical models with different processes of transferring energy are discussed.

Stock exchange: A statistical model

Abstract In the spirit of microfoundations of macroeconomic theory, we introduce a coupled map lattice model to describe time evolution of relevant quantities associated with stock exchange. In particular, computer simulations exhibit the stabilizing effect, on stock exchange, of dispersion of the external parameters which control the coupled map.