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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.

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... Moreover, such equations have properties that are relevant and/or interesting from both a physical and mathematical point of view. For instance, they admit exact analytical solutions of the q-Gaussian form that can be interpreted as maximum entropy densities obtainable from the optimization (under appropriate constraints) of S q ; they have been studied in the context of entropy production [13,14]; with different drift forces [15,16]; they obey an H-theorem formulated in terms of a free-energy-like quantity [17,18], and so on. In particular, we highlight an experimental work on granular media [5], that verified within great precision (2% error) the Tsallis and Bukman's [10] prediction 19 years after the original proposal. ...

... (k) on the right-hand side of (A4) is aq-Gaussian distribution (up to normalization), given by (16). Hence, F q [S N ](k) isq-Gaussian of the form: ...

... with p = 2 + d(1 − q). Comparing the above equation with (16), the variance of S N reads ...

We propose a new approach to describe the effective microscopic dynamics of (power-law) nonlinear Fokker-Planck equations. Our formalism is based on a nonextensive generalization of the Wiener process. This allow us to obtain, in addition to significant physical insights, several analytical results with great simplicity. Indeed, we obtain analytical solutions for a nonextensive version of Brownian free-particle and Ornstein-Uhlenbeck process, and explain anomalous diffusive behaviours in terms of memory effects in a nonextensive generalization of Gaussian white noise. Finally, we apply the develop formalism to model thermal noise in electric circuits.

... Among uncountable other illustrations, we may mention lateral force microscope experiments in silicon, Si(100) surfaces [33], and similar phenomena in bio-inspired, asymmetrically-structured surfaces [34]. In the case of over-damped systems described by NLFPEs, isotropic, non-linear drag forces were considered in [35], and non-isotropic, linear drag forces in [36]. In the present work, we investigate a system with drag forces that are both non-linear and non-isotropic. ...

We investigate a one-dimensional, many-body system consisting of particles interacting via repulsive, short-range forces, and moving in an overdamped regime under the effect of a drag force that depends on direction. That is, particles moving to the right do not experience the same drag as those moving to the left. The dynamics of the system, effectively described by a non-linear, Fokker–Planck equation, exhibits peculiar features related to the way in which the drag force depends on velocity. The evolution equation satisfies an H-theorem involving the Sq nonadditive entropy, and admits particular, exact, time-dependent solutions closely related, but not identical, to the q-Gaussian densities. The departure from the canonical, q-Gaussian shape is related to the fact that in one spatial dimension, in contrast to what occurs in two or more spatial dimensions, the drag’s dependence on direction entails that its dependence on velocity is necessarily (and severely) non-linear. The results reported here provide further evidence of the deep connections between overdamped, many-body systems, non-linear Fokker–Planck equations, and the Sq-thermostatistics.

... The standard dissipative acceleration α disss , equation (36), is a parabolic function of angular velocity ω [93]. Some authors have used power-laws [30,64,[94][95][96][97][98][99]. In order to test if such models provide better fits we considered the following power-law ...

A detailed analysis of pendular motion is presented. Inertial effects, self-oscillation, and memory, together with non-constant moment of inertia, hysteresis, and negative damping are shown to be required for the comprehensive description of the free pendulum oscillatory regime. The effects of very high initial amplitudes, friction in the roller bearing axle, drag, and pendulum geometry are also analyzed and discussed. A model consisting of a fractional differential equation fits and explains high resolution and long-time experimental data gathered from standard action-camera videos.

... Therein, the first correction in ζ 0 introduces a quadratic dependence on v. [34][35][36] In some situations, nonlinearities in the drag coefficient have quite strong physical implications. [36][37][38][39][40] The main goal of this paper is to study the Mpemba effect in the kinetic theory framework we have just described, i.e., the Enskog-Fokker-Planck equation with nonlinear drag. To meet this end, we work using the first Sonine approximation, in which the time evolution of temperature is coupled to that of excess kurtosis. ...

We look into the Mpemba effect—the initially hotter sample cools sooner—in a molecular gas with nonlinear viscous drag. Specifically, the gas particles interact among them via elastic collisions and with a background fluid at equilibrium. Thus, within the framework of kinetic theory, our gas is described by an Enskog–Fokker–Planck equation. The analysis is carried out using the first Sonine approximation, in which the evolution of temperature is coupled to that of excess kurtosis. This coupling leads to the emergence of the Mpemba effect, which is observed at an early stage of relaxation and when the initial temperatures of the two samples are close enough. This allows for the development of a simple theory, linearizing the temperature evolution around a reference temperature, namely, the initial temperature closer to the asymptotic equilibrium value. The linear theory provides a semiquantitative description of the effect, including expressions for crossover time and maximum temperature difference. We also discuss the limitations of our linearized theory.

... The standard dissipative acceleration α disss , equation (35), is a parabolic function of angular velocity ω [91]. Some authors have used power-laws [30,61,[92][93][94][95][96][97]. In order to test if such models provide better fits we considered the following power-law ...

A detailed analysis of three pendular motion models is presented. Inertial effects, self-oscillation, and memory, together with non-constant moment of inertia, hysteresis and negative damping are shown to be required for the comprehensive description of the free pendulum oscillatory regime. The effects of very high initial amplitudes, friction in the roller bearing axle, drag, and pendulum geometry are also analysed and discussed. The model that consists of a fractional differential equation provides both the best explanation of, and the best fits to, experimental high resolution and long-time data gathered from standard action-camera videos.
Please, download from https://arxiv.org/pdf/2006.15665

... Therein, the first correction in ζ 0 introduces a quadratic dependence on v. [31][32][33] In some situations, nonlinearities in the drag coefficient have quite strong physical implications. [33][34][35][36][37] The main goal of this paper is to study the Mpemba effect in the kinetic theory framework we have just described, i.e., the Enskog-Fokker-Planck equation with nonlinear drag. To meet this end, we work in the first Sonine approximation, in which the time evolution of the temperature is coupled to that of the excess kurtosis. ...

We look into the Mpemba effect---the initially hotter sample cools sooner---in a molecular gas with nonlinear viscous drag. Specifically, the gas particles interact among them via elastic collisions and also with a background fluid at equilibrium. Thus, within the framework of kinetic theory, our gas is described by an Enskog--Fokker--Planck equation. The analysis is carried out in the first Sonine approximation, in which the evolution of the temperature is coupled to that of the excess kurtosis. This coupling leads to the emergence of the Mpemba effect, which is observed in an early stage of the relaxation and when the initial temperatures of the two samples are close enough. This allows for the development of a simple theory, linearizing the temperature evolution around a reference temperature---namely the initial temperature closer to the asymptotic equilibrium value. The linear theory provides a semiquantitative description of the effect, including expressions for the crossover time and the maximum temperature difference. We also discuss the limitations of our linearized theory.

The principle of least action is a variational principle that states an object will always take the path of least action as compared to any other conceivable path. This principle can be used to derive the equations of motion of many systems, and therefore provides a unifying equation that has been applied in many fields of physics and mathematics. Hamilton’s formulation of the principle of least action typically only accounts for conservative forces, but can be reformulated to include non-conservative forces such as friction by using the approach suggested by Wang and co-workers (Lin and Wang, 2014; Wang, 2015). However, it turns out that this modfied action is dependent upon the nature and amount of damping in the system and the optimality argument fails to hold for sufficiently large values of the damping coefficient. In this paper, we specifically investigate the modified action principle for the cases of a linearly and cubically damped, nonlinear pendulum.

We propose an approach to describe the effective microscopic dynamics of (power-law) nonlinear Fokker-Planck equations. Our formalism is based on a nonextensive generalization of the Wiener process. This allows us to obtain, in addition to significant physical insights, several analytical results with great simplicity. Indeed, we obtain analytical solutions for a nonextensive version of the Brownian free-particle and Ornstein-Uhlenbeck processes, and we explain anomalous diffusive behaviors in terms of memory effects in a nonextensive generalization of Gaussian white noise. Finally, we apply the developed formalism to model thermal noise in electric circuits.

In the present paper we make the transition from the Becker–Döring system of equations to the hybrid (discrete and continuum) description. This new type of system of equations consists of the equation of the Fokker–Plank–Einstein–Kolmogorov type added by the Becker–Döring equations. We consider the H-theorem for it. We also consider the H-theorem for the Becker–Döring system of equations with discrete time and showed that it is true for some partially implicit discretization in time. Due to generality of the kinetic approach the present work can be useful for specialists in different spheres engaged in modeling the evolution of structures differing by properties.

This review article aims to stress and reunite some of the analytic formalism of the anomalous diffusive processes that have succeeded in their description. Also, it has the objective to discuss which of the new directions they have taken nowadays. The discussion is started by a brief historical report that starts with the studies of thermal machines and combines in theories such as the statistical mechanics of Boltzmann-Gibbs and the Brownian Movement. In this scenario, in the twentieth century, a series of experiments were reported that were not described by the usual model of diffusion. Such experiments paved the way for deeper investigation into anomalous diffusion. These processes are very abundant in physics, and the mechanisms for them to occur are diverse. For this reason, there are many possible ways of modelling the diffusive processes. This article discusses three analytic approaches to investigate anomalous diffusion: fractional diffusion equation, nonlinear diffusion equation and Langevin equation in the presence of fractional, coloured or multiplicative noises. All these formalisms presented different degrees of complexity and for this reason, they have succeeded in describing anomalous diffusion phenomena.

Understanding the motion of a tracer particle in a rarefied gas is of fundamental and practical importance. We report the experimental investigation of individual Cs atoms impinging on a dilute cloud of ultracold Rb atoms with variable density. We study the nonequilibrium relaxation of the initial nonthermal state and detect the effect of single collisions which has eluded observation so far. We show that after few collisions, the measured spatial distribution of the light tracer atoms is correctly described by a generalized Langevin equation with a velocity-dependent friction coefficient, over a large range of Knudsen numbers.

In this letter, we address the relationship between the statistical
fluctuations of grain displacements for a full quasistatic plane shear
experiment, and the corresponding anomalous diffusion exponent, $\alpha$. We
experimentally validate a particular case of the so-called Tsallis-Bukman
scaling law, $\alpha = 2 / (3 - q)$, where $q$ is obtained by fitting the
probability density function (PDF) of the measured fluctuations with a
$q$-Gaussian distribution, and the diffusion exponent is measured independently
during the experiment. Applying an original technique, we are able to evince a
transition from an anomalous diffusion regime to a Brownian behavior as a
function of the length of the strain-window used to calculate the displacements
of grains in experiments. The outstanding conformity of fitting curves to a
massive amount of experimental data shows a clear broadening of the fluctuation
PDFs as the length of the strain-window decreases, and an increment in the
value of the diffusion exponent - anomalous diffusion. Regardless of the size
of the strain-window considered in the measurements, we show that the
Tsallis-Bukman scaling law remains valid, which is the first experimental
verification of this relationship for a classical system at different diffusion
regimes. We also note that the spatial correlations show marked similarities to
the turbulence in fluids, a promising indication that this type of analysis can
be used to explore the origins of the macroscopic friction in confined granular
materials.

A method of finding entropic form for a given stationary probability distribution and specified potential field is discussed, using the steady-state Fokker-Planck equation. As examples, starting with the Boltzmann and Tsallis distribution and knowing the force field, we obtain the Boltzmann-Gibbs and Tsallis entropies. Also, the associated entropy for the gamma probability distribution is found, which seems to be in the form of the gamma function. Moreover, the related Fokker-Planck equations are given for the Boltzmann, Tsallis, and gamma probability distributions.

We deduce a nonlinear and inhomogeneous Fokker-Planck equation within a
generalized Stratonovich, or stochastic $\alpha$-, prescription ($\alpha=0$,
$1/2$ and $1$ respectively correspond to the It\^o, Stratonovich and anti-It\^o
prescriptions). We obtain its stationary state $p_{st}(x)$ for a class of
constitutive relations between drift and diffusion and show that it has a
$q$-exponential form, $p_{st}(x) = N_q[1 - (1-q)\beta V(x)]^{1/(1-q)}$, with an
index $q$ which does not depend on $\alpha$ in the presence of any nonvanishing
nonlinearity. This is in contrast with the linear case, for which the index $q$
is $\alpha$-dependent.

With the use of a quantity normally scaled in multifractals, a generalized form is postulated for entropy, namelyS
q
k [1 –
i=1
W
p
i
q
]/(q-1), whereq characterizes the generalization andp
i are the probabilities associated withW (microscopic) configurations (W). The main properties associated with this entropy are established, particularly those corresponding to the microcanonical and canonical ensembles. The Boltzmann-Gibbs statistics is recovered as theq1 limit.

We study scaling properties of energy spreading in disordered strongly
nonlinear Hamiltonian lattices. Such lattices consist of nonlinearly coupled
local linear or nonlinear oscillators, and demonstrate a rather slow,
subdiffusive spreading of initially localized wave p ackets. We use a
fractional nonlinear diffusion equation as a heuristic model of this process,
and confirm that the scaling predictions resulting from a self-similar solution
of this equation are indeed applicable to all studied cases. We s how that the
spreading in nonlinearly coupled linear oscillators slows down compared to a
pure power law, while for nonlinear local oscillators a power law is valid in
the whole studied range of parameters.

Starting with the relative entropy based on a previously proposed entropy
function $S_q[p]=\int dx p(x)(-\ln p(x))^q$, we find the corresponding Fisher's
information measure. After function redefinition we then maximize the Fisher
information measure with respect to the new function and obtain a differential
operator that reduces to a space coordinate second derivative in the $q\to 1$
limit. We then propose a simple differential equation for anomalous diffusion
and show that its solutions are a generalization of the functions in the
Barenblatt-Pattle solution. We find that the mean squared displacement, up to a
$q$-dependent constant, has a time dependence according to $<x^2>\sim
K^{1/q}t^{1/q}$, where the parameter $q$ takes values $q=\frac{2n-1}{2n+1}$
(superdiffusion) and $q=\frac{2n+1}{2n-1}$ (subdiffusion), $\forall n\geq 1$.

Spin relaxation close to the glass temperature of CuMn and AuFe spin glasses is shown, by neutron spin echo, to follow a generalized exponential function which explicitly introduces hierarchically constrained dynamics and macroscopic interactions. The interaction parameter is directly related to the normalized Tsallis nonextensive entropy parameter q and exhibits universal scaling with reduced temperature. At the glass temperature q=5/3 corresponding, within Tsallis' q statistics, to a mathematically defined critical value for the onset of strong disorder and nonlinear dynamics.

We investigate the escape behavior of systems governed by the one-dimensional nonlinear diffusion equation theta(t)rho=theta(x)[theta(x)Urho]+Dtheta(x)2rho(nu), where the potential of the drift, U(x), presents a double well and D,nu are real parameters. For systems close to the steady state, we obtain an analytical expression of the mean first-passage time, yielding a generalization of Arrhenius law. Analytical predictions are in very good agreement with numerical experiments performed through integration of the associated Ito-Langevin equation. For nu not equal to 1, important anomalies are detected in comparison to the standard Brownian case. These results are compared to those obtained numerically for initial conditions far from the steady state.

An N-dimensional nonlinear Fokker-Planck equation is investigated here by considering the time dependence of the coefficients, where drift-controlled and source terms are present. We exhibit the exact solution based on the generalized Gaussian function related to the Tsallis statistics. Furthermore, we show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained by changing the time dependence of the coefficients.

Phase space can be constructed for N equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated. If they are locally correlated, we expect the Boltzmann-Gibbs entropy S(BG) identical with -k Sigma(i) p(i) ln p(i) to be extensive, i.e., S(BG)(N) proportional, variant N for N --> infinity. In particular, if they are independent, S(BG) is strictly additive, i.e., S(BG)(N) = NS(BG)(1), for allN. However, if the subsystems are globally correlated, we expect, for a vast class of systems, the entropy S(q) identical with k[1 - Sigma(i) p(q)(i)]/(q - 1) (with S(1) = S(BG)) for some special value of q not equal 1 to be the one which is extensive [i.e., S(q)(N) proportional, variant N for N --> infinity]. Another concept which is relevant is strict or asymptotic scale-freedom (or scale-invariance), defined as the situation for which all marginal probabilities of the N-system coincide or asymptotically approach (for N --> infinity) the joint probabilities of the (N - 1)-system. If each subsystem is a binary one, scale-freedom is guaranteed by what we hereafter refer to as the Leibnitz rule, i.e., the sum of two successive joint probabilities of the N-system coincides or asymptotically approaches the corresponding joint probability of the (N - 1)-system. The kinds of interplay of these various concepts are illustrated in several examples. One of them justifies the title of this paper. We conjecture that these mechanisms are deeply related to the very frequent emergence, in natural and artificial complex systems, of scale-free structures and to their connections with nonextensive statistical mechanics. Summarizing, we have shown that, for asymptotically scale-invariant systems, it is S(q) with q not equal 1, and not S(BG), the entropy which matches standard, clausius-like, prescriptions of classical thermodynamics.

We demonstrated experimentally that the momentum distribution of cold atoms in dissipative optical lattices is a Tsallis distribution. The parameters of the distribution can be continuously varied by changing the parameters of the optical potential. In particular, by changing the depth of the optical lattice, it is possible to change the momentum distribution from Gaussian, at deep potentials, to a power-law tail distribution at shallow optical potentials.

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.

We use the H theorem to establish the entropy and the entropic additivity law for a system composed of subsystems, with the dynamics governed by the Klein-Kramers equations, by considering relations among the dynamics of these subsystems and their entropies. We start considering the subsystems governed by linear Klein-Kramers equations and verify that the Boltzmann-Gibbs entropy is appropriated to this dynamics, leading us to the standard entropic additivity, SBG(1∪2)=SBG1+SBG2, consistent with the fact that the distributions of the subsystem are independent. We then extend the dynamics of these subsystems to independent nonlinear Klein-Kramers equations. For this case, the results show that the H theorem is verified for a generalized entropy, which does not preserve the standard entropic additivity for independent distributions. In this scenario, consistent results are obtained when a suitable coupling among the nonlinear Klein-Kramers equations is considered, in which each subsystem modifies the other until an equilibrium state is reached. This dynamics, for the subsystems, results in the Tsallis entropy for the system and, consequently, verifies the relation Sq(1∪2)=Sq1+Sq2+(1−q)Sq1Sq2/k, which is a nonadditive entropic relation.

The diffusive pair annihilation model with embedded topological domains and archaeological data are applied in an analysis of the hypothetical Trojan–Greek war during the late Bronze Age. Estimations of parameter are explicitly made for critical dynamics of the model. In particular, the 8-meter walls of Troy could be viewed as the effective shield that provided the technological difference between the two armies. Suggestively, the numbers in The Iliad are quite sound, being in accord with Lanchester’s laws of warfare.

Nonlinear Fokker-Planck equations endowed with curl drift forces are investigated. The conditions under which these evolution equations admit stationary solutions, which are $q$-exponentials of an appropriate potential function, are determined. It is proved that when these stationary solutions exist, the nonlinear Fokker-Planck equations satisfy an $H$-theorem in terms of a free-energy like quantity involving the $S_q$ entropy. A particular two dimensional model admitting analytical, time-dependent, $q$-Gaussian solutions is discussed in detail. This model describes a system of particles with short-range interactions, performing overdamped motion under drag effects, due to a rotating resisting medium. It is related to models that have been recently applied to the study of type-II superconductors. The relevance of the present developments to the study of complex systems in physics, astronomy, and biology, is discussed.

A recent proposal of an effective temperature θ, conjugated to a generalized entropy sq, typical of nonextensive statistical mechanics, has led to a consistent thermodynamic framework in the case q=2. The proposal was explored for repulsively interacting vortices, currently used for modeling type-II superconductors. In these systems, the variable θ presents values much higher than those of typical room temperatures T, so that the thermal noise can be neglected (T/θ≃0). The whole procedure was developed for an equilibrium state obtained after a sufficiently long-time evolution, associated with a nonlinear Fokker-Planck equation and approached due to a confining external harmonic potential, ϕ(x)=αx2/2 (α>0). Herein, the thermodynamic framework is extended to a quite general confining potential, namely ϕ(x)=α|x|z/z (z>1). It is shown that the main results of the previous analyses hold for any z>1: (i) The definition of the effective temperature θ conjugated to the entropy s2. (ii) The construction of a Carnot cycle, whose efficiency is shown to be η=1−(θ2/θ1), where θ1 and θ2 are the effective temperatures associated with two isothermal transformations, with θ1>θ2. The special character of the Carnot cycle is indicated by analyzing another cycle that presents an efficiency depending on z. (iii) Applying Legendre transformations for a distinct pair of variables, different thermodynamic potentials are obtained, and furthermore, Maxwell relations and response functions are derived. The present approach shows a consistent thermodynamic framework, suggesting that these results should hold for a general confining potential ϕ(x), increasing the possibility of experimental verifications.

We propose a general coarse-graining method to derive a continuity equation that describes any dissipative system of repulsive particles interacting through short-ranged potentials. In our approach, the effect of particle-particle correlations is incorporated to the overall balance of energy, and a non-linear diffusion equation is obtained to represent the overdamped dynamics. In particular, when the repulsive interaction potential is a short-ranged power-law, our approach reveals a distinctive correspondence between particle-particle energy and the generalized thermostatistics of Tsallis for any non-positive value of the entropic index q. Our methodology can also be applied to microscopic models of superconducting vortices and complex plasma, where particle-particle correlations are pronounced at low concentrations. The resulting continuum descriptions provide elucidating and useful insights on the microdynamical behavior of these physical systems. The consistency of our approach is demonstrated by comparison with molecular dynamics simulations.

The nonextensive entropic measure proposed by Tsallis [C. Tsallis, J. Stat. Phys. 52, 479 (1988)JSTPBS0022-471510.1007/BF01016429] introduces a parameter, q, which is not defined but rather must be determined. The value of q is typically determined from a piece of data and then fixed over the range of interest. On the other hand, from a phenomenological viewpoint, there are instances in which q cannot be treated as a constant. We present two distinct approaches for determining q depending on the form of the equations of constraint for the particular system. In the first case the equations of constraint for the operator O[over ̂] can be written as Tr(F^{q}O[over ̂])=C, where C may be an explicit function of the distribution function F. We show that in this case one can solve an equivalent maxent problem which yields q as a function of the corresponding Lagrange multiplier. As an illustration the exact solution of the static generalized Fokker-Planck equation (GFPE) is obtained from maxent with the Tsallis enropy. As in the case where C is a constant, if q is treated as a variable within the maxent framework the entropic measure is maximized trivially for all values of q. Therefore q must be determined from existing data. In the second case an additional equation of constraint exists which cannot be brought into the above form. In this case the additional equation of constraint may be used to determine the fixed value of q.

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.

A model of superconducting vortices under overdamped motion is currently used for describing type-II superconductors. Recently, this model has been identified to a nonlinear Fokker-Planck equation and associated to an entropic form characteristic of nonextensive statistical mechanics, S2(t)≡Sq=2(t). In the present work, we consider a system of superconducting vortices under overdamped motion, following an irreversible process, so that by using the corresponding nonlinear Fokker-Planck equation, the entropy time rate [dS2(t)/dt] is investigated. Both entropy production and entropy flux from the system to its surroundings are analyzed. Molecular dynamics simulations are carried for this process, showing a good agreement between the numerical and analytical results. It is shown that the second law holds within the present framework, and we exhibit the increase of S2(t) with time, up to its stationary-state value.

The domain of non-extensive thermostatistics has been subject to intensive research over the past twenty years and has matured significantly. Generalised Thermostatistics cuts through the traditionalism of many statistical physics texts by offering a fresh perspective and seeking to remove elements of doubt and confusion surrounding the area. The book is divided into two parts-the first covering topics from conventional statistical physics, whilst adopting the perspective that statistical physics is statistics applied to physics. The second developing the formalism of non-extensive thermostatistics, of which the central role is played by the notion of a deformed exponential family of probability distributions. Presented in a clear, consistent, and deductive manner, the book focuses on theory, part of which is developed by the author himself, but also provides a number of references towards application-based texts. Written by a leading contributor in the field, this book will provide a useful tool for learning about recent developments in generalized versions of statistical mechanics and thermodynamics, especially with respect to self-study. Written for researchers in theoretical physics, mathematics and statistical mechanics, as well as graduates of physics, mathematics or engineering. A prerequisite knowledge of elementary notions of statistical physics and a substantial mathematical background are required.

A thermodynamic formalism is developed for a system of interacting particles under overdamped motion, which has been recently analyzed within the framework of nonextensive statistical mechanics. It amounts to expressing the interaction energy of the system in terms of a temperature θ, conjugated to a generalized entropy sq, with q =2. Since θ assumes much higher values than those of typical room temperatures T ⪡θ, the thermal noise can be neglected for this system (T /θ≃0). This framework is now extended by the introduction of a work term δW which, together with the formerly defined heat contribution (δQ =θdsq), allows for the statement of a proper energy conservation law that is analogous to the first law of thermodynamics. These definitions lead to the derivation of an equation of state and to the characterization of sq adiabatic and θ isothermic transformations. On this basis, a Carnot cycle is constructed, whose efficiency is shown to be η =1-(θ2/θ1), where θ1 and θ2 are the effective temperatures of the two isothermic transformations, with θ1>θ2. The results for a generalized thermodynamic description of this system open the possibility for further physical consequences, like the realization of a thermal engine based on energy exchanges gauged by the temperature θ.

Cold atoms in dissipative optical lattices exhibit an unusual transport behaviour that cannot be described within Boltzmann–Gibbs statistical mechanics. New theoretical tools and concepts need thus be developed to account for their observable macroscopic properties. Here we review recent progress achieved in the study of these processes. We emphasize the generality of the findings for a broad class of physical, chemical and biological systems, and discuss open questions and perspectives for future work.

In the attempt to solve the age-old problem of unifying Langevin, Fokker-Planck and Boltzmann theories for test particles in a dilute gas, the Uhlenbeck and Ornstein’s theory relating Langevin and Fokker-Planck equations is critically analyzed. Agreement and discrepancies between such theory and the results following from the Boltzmann one are also examined. It is concluded that the currently assumed form of the fluctuating-force autocorrelation function, which is extremely successful for Brownian particles in dense fluids, cannot generally guarantee an accurate relaxation law for the mean square velocity components of generic test particles in dilute gases. This difficulty can be overcome in the framework of a more general kinetic approach which is shown to consistently include Langevin, Fokker-Planck, and Boltzmann theories. The advantages of such approach in interpreting experimental results are particularly evident when the test particles move in a (homogeneous) gas in non-equilibrium conditions and when correlations exist between test- and gas-particle velocities.

A non-linear, generalized Fokker-Planck (GFP) equation is studied whose
exact stationary solutions are the maximum entropy distributions
introduced by Tsallis in his generalization of Statistical Mechanics. In
the case of a constant or linearly varying drift, the time dependent
solutions of the GFP equation are seen to obey a suitable form of the
celebrated H-theorem. The temporal changes of Tsallis' entropies are
seen to be given in terms of the Fisher's information measure.
Particular time-dependent solutions of the GFP equation of the maximum
(Tsallis') entropy form are obtained.

The relationship between H-theorems and free energies is studied on the basis of generalized entropies. Two kinds of nonlinear Fokker-Planck equations with different nonlinear diffusion terms that exhibit the power-law-type equilibrium distributions of Tsallis thermostatistics are investigated from the viewpoint of nonequilibrium free energies and stability analysis of their solutions. Using the generalized entropies Liapunov functions are constructed to show H-theorems, which ensure uniqueness of and convergence to the equilibrium distributions of the nonlinear Fokker-Planck equations.

The problem of the limits of validity of the Langevin equation is considered in detail in the case of (microscopic) test-particles in very dilute gases. It is shown that, in this case, the current Langevin equation follows from the Newton’s law in an exact way only in the Maxwell test-particle–gas-particle interaction model, and in an approximate way only in the Rayleigh-gas limit and in the low-velocity limit, while in any other interaction model, or limit, only a Langevin-like equation with speed-dependent friction coefficient and speed-dependent fluctuating force can be written. Such a circumstance, although probably limited to the particular physical situation considered in this paper, suggests that, in general, some preliminary, specific check of the validity of the Langevin equation should be performed before using the said equation to interpret laboratory experiments.

We consider nonextensive systems that are related to the nonextensive entropy proposed by Tsallis and can be described by means of the nonlinear porous medium equation and the nonlinear Fokker–Planck equation proposed by Plastino and Plastino. We show how to determine the degree of nonextensivity of these systems from experimental data. Both transient and stationary cases are addressed.

A model is developed of the interstellar diffusion of galactic civilizations which takes into account the population dynamics of such civilizations. The problem is formulated in terms of potential theory, with a family of nonlinear partial differential and difference equations specifying population growth and diffusion for an organism with advantageous genes that undergoes random dispersal while increasing in population locally, and a population at zero population growth. In the case of nonlinear diffusion with growth and saturation, it is found that the colonization wavefront from the nearest independently arisen galactic civilization can have reached the earth only if its lifetime exceeds 2.6 million years, or 20 million years if discretization can be neglected. For zero population growth, the corresponding lifetime is 13 billion years. It is concluded that the earth is uncolonized not because interstellar spacefaring civilizations are rare, but because there are too many worlds to be colonized in the plausible colonization lifetime of nearby civilizations, and that there exist no very old galactic civilizations with a consistent policy of the conquest of inhabited worlds.

The formalism of statistical mechanics can be generalised by starting from more general measures of information than the Shannon entropy and maximising those subject to suitable constraints. We discuss some of the most important examples of information measures that are useful for the description of complex systems. Examples treated are the Rényi entropy, Tsallis entropy, Abe entropy, Kaniadakis entropy, Sharma?Mittal entropies, and a few more. Important concepts such as the axiomatic foundations, composability and Lesche stability of information measures are briefly discussed. Potential applications in physics include complex systems with long-range interactions and metastable states, scattering processes in particle physics, hydrodynamic turbulence, defect turbulence, optical lattices, and quite generally driven nonequilibrium systems with fluctuations of temperature.

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2008

A system of interacting vortices under overdamped motion, which has been commonly used in the literature to model flux-front penetration in disordered type-II superconductors, was recently related to a nonlinear Fokker-Planck equation, characteristic of nonextensive statistical mechanics, through an analysis of its stationary state. Herein, this connection is extended by means of a thorough analysis of the time evolution of this system. Numerical data from molecular-dynamics simulations are presented for both position and velocity probability distributions P(x,t) and P(v(x),t), respectively; both distributions are well fitted by similar q-Gaussian distributions, with the same index q=0, for all times considered. Particularly, the evolution of the system occurs in such a way that P(x,t) presents a time behavior for its width, normalization, and second moment, in full agreement with the analytic solution of the nonlinear Fokker-Planck equation. The present results provide further evidence that this system is deeply associated with nonextensive statistical mechanics.

We briefly discuss the state of the art on the anomalous dynamics of the Hamiltonian mean field (HMF) model. We stress the important role of the initial conditions for understanding the microscopic nature of the intriguing metastable quasi-stationary states (QSS) observed in the model and the connections to Tsallis statistics and glassy dynamics. We also present new results on the existence of metastable states in the Kuramoto model and discuss the similarities with those found in the HMF model. The existence of metastability seems to be quite a common phenomenon in fully coupled systems, whose origin could be also interpreted as a dynamical mechanism preventing or hindering synchronization.

Tsallis’ q-triplet [C. Tsallis, Dynamical scenario for nonextensive statistical mechanics, Physica A 340 (2004) 1–10] is the best empirical quantifier of nonextensivity. Here we study it with reference to an experimental time-series related to the daily depth-values of the stratospheric ozone layer. Pertinent data are expressed in Dobson units and range from 1978 to 2005. After the evaluation of the three associated Tsallis’ indices one concludes that nonextensivity is clearly a characteristic of the system under scrutiny.

Tsallis [Physica A 340 (2004) 1] identified a set of numbers, the “q-triplet” ≡{qstat, qsen, qrel}, for a system described by nonextensive statistical mechanics. The deviation of the q's from unity is a measure of the departure from thermodynamic equilibrium. We present observations of the q-triplets derived from two sets of daily averages of the magnetic field strength B observed by Voyager 1 in the solar wind near 40 AU during 1989 and near 85 AU during 2002, respectively. The results for 1989 do not differ significantly from those for 2002. We find qstat=1.75±0.06, qsen=-0.6±0.2, and qrel=3.8±0.3.

Recently, Compte and Jou derived nonlinear diffusion equations by applying the principles of linear nonequilibrium thermodynamics to the generalized nonextensive entropy proposed by Tsallis. In line with this study, stochastic processes in isolated and closed systems characterized by arbitrary generalized entropies are considered and evolution equations for the process probability densities are derived. It is shown that linear nonequilibrium thermodynamics based on generalized entropies naturally leads to generalized Fokker–Planck equations.

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.

Driven anomalous diffusions (such as those occurring in some surface growths) are currently described through the nonlinear Fokker-Planck-like equation (∂/∂t)pmu=-(∂/∂x)[F(x)pmu]+D(∂2/∂x2 )pnu [(mu,nu)∈R2 F(x)=k1-k2x is the external force; k2>=0]. We exhibit here the (physically relevant) exact solution for all (x,t). This solution was found through an ansatz based on the generalized entropic form Sq[p]=\{1-∫du[p(u)]q\}/(q-1) (with q∈R), in a completely analogous manner through which the usual entropy S1[p]=-∫dup(u)lnp(u) is known to provide the correct ansatz for exactly solving the standard Fokker-Planck equation (mu=nu=1). This remarkably simple unification of normal diffusion (q=1), superdiffusion (q>1) and subdiffusion (q

We investigate probability density functions of velocity differences at different distances r measured in a Couette-Taylor flow for a range of Reynolds numbers Re. There is good agreement with the predictions of a theoretical model based on nonextensive statistical mechanics (where the entropies are nonadditive for independent subsystems). We extract the scale-dependent nonextensitivity parameter q(r,Re) from the laboratory data.

Multidimensional nonlinear Fokker-Planck equations of mean-field type are proposed within the framework of generalized thermostatistics to develop a general formulation of stability analysis of their solutions. Two types of eigenvalue equations are studied. The nonlinear Fokker-Planck equations are shown to exhibit an H theorem with a Liapunov functional that takes the form of a free energy involving generalized entropies of Tsallis. The second-order variation of the Liapunov functional is computed to conduct local stability analysis and the associated eigenvalue equation is derived for an arbitrary form of mean-field coupling potential. Assuming quasiequilibrium for the velocity distribution, the reduced eigenvalue equation with space coordinates alone is also obtained. The alternative type of eigenvalue equation based on the linearization of the nonlinear Fokker-Planck equations is presented. Taking the mean-field coupling potential to be the gravitational one, the nonlinear Fokker-Planck equation in terms of three-dimensional velocity and space coordinates together with the framework of stability analysis is shown to be applicable to a mean-field model of self-gravitating system. By solving the eigenvalue equation for the eigenfunction with 0 eigenvalue, the occurrence of stability change of the equilibrium probability density with spherical symmetry is discussed.

We introduce a class of generalized Fokker-Planck equations that conserve energy and mass and increase a generalized entropy functional until a maximum entropy state is reached. Nonlinear Fokker-Planck equations associated with Tsallis entropies are a special case of these equations. Applications of these results to stellar dynamics and vortex dynamics are proposed. Our prime result is a relaxation equation that should offer an easily implementable parametrization of two-dimensional turbulence. Usual parametrizations (including a single turbulent viscosity) correspond to the infinite temperature limit of our model. They forget a fundamental systematic drift that acts against diffusion as in Brownian theory. Our generalized Fokker-Planck equations can have applications in other fields of physics such as chemotaxis for bacterial populations. We propose the idea of a classification of generalized entropies in "classes of equivalence" and provide an aesthetic connection between topics (vortices, stars, bacteria, em leader ) which were previously disconnected.

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.