Non-Additive Entropic Forms and Evolution Equations for Continuous and Discrete Probabilities.

Increasing interest has been shown in the subject of non-additive entropic forms during recent years, which has essentially been due to their potential applications in the area of complex systems. Based on the fact that a given entropic form should depend only on a set of probabilities, its time evolution is directly related to the evolution of these probabilities. In the present work, we discuss some basic aspects related to non-additive entropies considering their time evolution in the cases of continuous and discrete probabilities, for which nonlinear forms of Fokker-Planck and master equations are considered, respectively. For continuous probabilities, we discuss an H-theorem, which is proven by connecting functionals that appear in a nonlinear Fokker-Planck equation with a general entropic form. This theorem ensures that the stationary-state solution of the Fokker-Planck equation coincides with the equilibrium solution that emerges from the extremization of the entropic form. At equilibrium, we show that a Carnot cycle holds for a general entropic form under standard thermodynamic requirements. In the case of discrete probabilities, we also prove an H-theorem considering the time evolution of probabilities described by a master equation. The stationary-state solution that comes from the master equation is shown to coincide with the equilibrium solution that emerges from the extremization of the entropic form. For this case, we also discuss how the third law of thermodynamics applies to equilibrium non-additive entropic forms in general. The physical consequences related to the fact that the equilibrium-state distributions, which are obtained from the corresponding evolution equations (for both continuous and discrete probabilities), coincide with those obtained from the extremization of the entropic form, the restrictions for the validity of a Carnot cycle, and an appropriate formulation of the third law of thermodynamics for general entropic forms are discussed.

Entropic form emergent from superstatistics.

The Beck-Cohen superstatistics became an important theory in the scenario of complex systems because it generates distributions representing regions of a nonequilibrium system, characterized by different temperatures T≡ß^{-1}, leading to a probability distribution f(ß). In superstatistics, some classes have been most frequently considered for f(ß), like χ^{2}, χ^{2} inverse, and log-normal ones. Herein we investigate the superstatistics resulting from a χ_{η}^{2} distribution through a modification of the usual χ^{2} by introducing a real index η (0<η≤1). In this way, one covers two common and relevant distributions as particular cases, proportional to the q-exponential (e_{q}^{-ßx}=[1-(1-q)ßx]^{1/1-q}) and the stretched exponential (e^{-(ßx)^{η}}). Furthermore, an associated generalized entropic form is found. Since these two particular-case distributions have been frequently found in the literature, we expect that the present results should be applicable to a wide range of classes of complex systems.

Finite-size scaling of quasi-stationary-state temperature.

We numerically study, from first principles, the temperature T_{QSS} and duration t_{QSS} of the longstanding initial quasi-stationary state of the isolated d-dimensional classical inertial α-XY ferromagnet with two-body interactions decaying as 1/r_{ij}^{α} (α≥0). It is shown that this temperature T_{QSS} (defined proportional to the kinetic energy per particle) depends, for the long-range regime 0≤α/d≤1, on (α,d,U,N) with numerically negligible changes for dimensions d=1,2,3, with U the energy per particle and N the number of particles. We verify the finite-size scaling T_{QSS}-T_{∞}∝1/N^{ß} where T_{∞}≡2U-1 for U≲U_{c}, and ß appears to depend sensibly only on α/d. Our numerical results indicate that neither the scaling with N of T_{QSS} nor that of t_{QSS} depend on U.

Criticality in the duration of quasistationary state.

The duration of the quasistationary states (QSSs) emerging in the d-dimensional classical inertial α-XY model, i.e., N planar rotators whose interactions decay with the distance r_{ij} as 1/r_{ij}^{α} (α≥0), is studied through first-principles molecular dynamics. These QSSs appear along the whole long-range interaction regime (0≤α/d≤1), for an average energy per rotator UU_{c}. They are characterized by a kinetic temperature T_{QSS}, before a crossover to a second plateau occurring at the Boltzmann-Gibbs temperature T_{BG}>T_{QSS}. We investigate here the behavior of their duration t_{QSS} when U approaches U_{c} from below, for large values of N. Contrary to the usual belief that the QSS merely disappears as U→U_{c}, we show that its duration goes through a critical phenomenon, namely t_{QSS}∝(U_{c}-U)^{-ξ}. Universality is found for the critical exponent ξ≃5/3 throughout the whole long-range interaction regime.

External Stimuli on Neural Networks: Analytical and Numerical Approaches.

Based on the behavior of living beings, which react mostly to external stimuli, we introduce a neural-network model that uses external patterns as a fundamental tool for the process of recognition. In this proposal, external stimuli appear as an additional field, and basins of attraction, representing memories, arise in accordance with this new field. This is in contrast to the more-common attractor neural networks, where memories are attractors inside well-defined basins of attraction. We show that this procedure considerably increases the storage capabilities of the neural network; this property is illustrated by the standard Hopfield model, which reveals that the recognition capacity of our model may be enlarged, typically, by a factor 102. The primary challenge here consists in calibrating the influence of the external stimulus, in order to attenuate the noise generated by memories that are not correlated with the external pattern. The system is analyzed primarily through numerical simulations. However, since there is the possibility of performing analytical calculations for the Hopfield model, the agreement between these two approaches can be tested-matching results are indicated in some cases. We also show that the present proposal exhibits a crucial attribute of living beings, which concerns their ability to react promptly to changes in the external environment. Additionally, we illustrate that this new approach may significantly enlarge the recognition capacity of neural networks in various situations; with correlated and non-correlated memories, as well as diluted, symmetric, or asymmetric interactions (synapses). This demonstrates that it can be implemented easily on a wide diversity of models.

Quasi-stationary-state duration in the classical d-dimensional long-range inertial XY ferromagnet.

A classical α-XY inertial model, consisting of N two-component rotators and characterized by interactions decaying with the distance r_{ij} as 1/r_{ij}^{α} (α≥0) is studied through first-principle molecular-dynamics simulations on d-dimensional lattices of linear size L (N≡L^{d} and d=1,2,3). The limits α=0 and α→∞ correspond to infinite-range and nearest-neighbor interactions, respectively, whereas the ratio α/d>1 (0≤α/d≤1) is associated with short-range (long-range) interactions. By analyzing the time evolution of the kinetic temperature T(t) in the long-range-interaction regime, one finds a quasi-stationary state (QSS) characterized by a temperature T_{QSS}; for fixed N and after a sufficiently long time, a crossover to a second plateau occurs, corresponding to the Boltzmann-Gibbs temperature T_{BG} (as predicted within the BG theory), with T_{BG}>T_{QSS}. It is shown that the QSS duration (t_{QSS}) depends on N, α, and d, although the dependence on α appears only through the ratio α/d; in fact, t_{QSS} decreases with α/d and increases with both N and d. Considering a fixed energy value, a scaling for t_{QSS} is proposed, namely, t_{QSS}∝N^{A(α/d)}e^{-B(N)(α/d)^{2}}, analogous to a recent analysis carried out for the classical α-Heisenberg inertial model. It is shown that the exponent A(α/d) and the coefficient B(N) present universal behavior (within error bars), comparing the XY and Heisenberg cases. The present results should be useful for other long-range systems, very common in nature, like those characterized by gravitational and Coulomb forces.

d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies.

We numerically study the first-principle dynamics and thermostatistics of a d-dimensional classical inertial Heisenberg ferromagnetic model ( d = 1 , 2 , 3 ) with interactions decaying with the distance r i j as 1 / r i j α ( α ≥ 0 ), where the limit α = 0 ( α → ∞ ) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio α / d > 1 ( 0 ≤ α / d ≤ 1 ) characterizes the short-ranged (long-ranged) regime. By means of first-principle molecular dynamics we study: (i) The scaling with the system size N of the maximum Lyapunov exponent λ in the form λ ∼ N - κ , where κ ( α / d ) depends only on the ratio α / d ; (ii) The time-averaged single-particle angular momenta probability distributions for a typical case in the long-range regime 0 ≤ α / d ≤ 1 (which turns out to be well fitted by q-Gaussians), and (iii) The time-averaged single-particle energies probability distributions for a typical case in the long-range regime 0 ≤ α / d ≤ 1 (which turns out to be well fitted by q-exponentials). Through the Lyapunov exponents we observe an intriguing, and possibly size-dependent, persistence of the non-Boltzmannian behavior even in the α / d > 1 regime. The universality that we observe for the probability distributions with regard to the ratio α / d makes this model similar to the α -XY and α -Fermi-Pasta-Ulam Hamiltonian models as well as to asymptotically scale-invariant growing networks.

Equilibrium States in Two-Temperature Systems.

Systems characterized by more than one temperature usually appear in nonequilibrium statistical mechanics. In some cases, e.g., glasses, there is a temperature at which fast variables become thermalized, and another case associated with modes that evolve towards an equilibrium state in a very slow way. Recently, it was shown that a system of vortices interacting repulsively, considered as an appropriate model for type-II superconductors, presents an equilibrium state characterized by two temperatures. The main novelty concerns the fact that apart from the usual temperature T, related to fluctuations in particle velocities, an additional temperature θ was introduced, associated with fluctuations in particle positions. Since they present physically distinct characteristics, the system may reach an equilibrium state, characterized by finite and different values of these temperatures. In the application of type-II superconductors, it was shown that θ ≫ T , so that thermal effects could be neglected, leading to a consistent thermodynamic framework based solely on the temperature θ . In the present work, a more general situation, concerning a system characterized by two distinct temperatures θ 1 and θ 2 , which may be of the same order of magnitude, is discussed. These temperatures appear as coefficients of different diffusion contributions of a nonlinear Fokker-Planck equation. An H-theorem is proven, relating such a Fokker-Planck equation to a sum of two entropic forms, each of them associated with a given diffusion term; as a consequence, the corresponding stationary state may be considered as an equilibrium state, characterized by two temperatures. One of the conditions for such a state to occur is that the different temperature parameters, θ 1 and θ 2 , should be thermodynamically conjugated to distinct entropic forms, S 1 and S 2 , respectively. A functional Λ [ P ] ≡ Λ ( S 1 [ P ] , S 2 [ P ] ) is introduced, which presents properties characteristic of an entropic form; moreover, a thermodynamically conjugated temperature parameter γ ( θ 1 , θ 2 ) can be consistently defined, so that an alternative physical description is proposed in terms of these pairs of variables. The physical consequences, and particularly, the fact that the equilibrium-state distribution, obtained from the Fokker-Planck equation, should coincide with the one from entropy extremization, are discussed.

Associating an Entropy with Power-Law Frequency of Events.

Events occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are intimately related to Tsallis entropy S q . The relevant parameters become: (i) The entropic index q, which is directly related to the power of the corresponding distribution; (ii) The ground-state energy ε 0 , in terms of which all energies are rescaled. One verifies that the corresponding processes take place at a temperature T q with k T q ∝ ε 0 (i.e., isothermal processes, for a given q), in analogy with those in the class of self-organized criticality, which are known to occur at fixed temperatures. Typical examples are analyzed, like earthquakes, avalanches, and forest fires, and in some of them, the entropic index q and value of T q are estimated. The knowledge of the associated entropic form opens the possibility for a deeper understanding of such phenomena, particularly by using information theory and optimization procedures.

Thermodynamic framework for the ground state of a simple quantum system.

The ground state of a two-level system (associated with probabilities p and 1-p, respectively) defined by a general Hamiltonian H[over ̂]=H[over ̂]_{0}+λV[over ̂] is studied. The simple case characterized by λ=0, whose Hamiltonian H[over ̂]_{0} is represented by a diagonal matrix, is well established and solvable within Boltzmann-Gibbs statistical mechanics; in particular, it follows the third law of thermodynamics, presenting zero entropy (S_{BG}=0) at zero temperature (T=0). Herein it is shown that the introduction of a perturbation λV[over ̂] (λ>0) in the Hamiltonian may lead to a nontrivial ground state, characterized by an entropy S[p] (with S[p]≠S_{BG}[p]), if the Hermitian operator V[over ̂] is represented by a 2×2 matrix, defined by nonzero off-diagonal elements V_{12}=V_{21}=-z, where z is a real positive number. Hence, this new term in the Hamiltonian, presenting V_{12}≠0, may produce physically significant changes in the ground state, and especially, it allows for the introduction of an effective temperature θ (θ∝λz), which is shown to be a parameter conjugated to the entropy S. Based on this, one introduces an infinitesimal heatlike quantity, δQ=θdS, leading to a consistent thermodynamic framework, and by proposing an infinitesimal form for the first law, a Carnot cycle and thermodynamic potentials are obtained. All results found are very similar to those of usual thermodynamics, through the identification T↔θ, and particularly the form for the efficiency of the proposed Carnot Cycle. Moreover, S also follows a behavior typical of a third law, i.e., S→0, when θ→0.

Repulsive particles under a general external potential: Thermodynamics by neglecting thermal noise.

A recent proposal of an effective temperature θ, conjugated to a generalized entropy s_{q}, 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)=αx^{2}/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 s_{2}. (ii) 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}. 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.

H theorem for generalized entropic forms within a master-equation framework.

The H theorem is proven for generalized entropic forms, in the case of a discrete set of states. The associated probability distributions evolve in time according to a master equation, for which the corresponding transition rates depend on these entropic forms. An important equation describing the time evolution of the transition rates and probabilities in such a way as to drive the system towards an equilibrium state is found. In the particular case of Boltzmann-Gibbs entropy, it is shown that this equation is satisfied in the microcanonical ensemble only for symmetric probability transition rates, characterizing a single path to the equilibrium state. This equation fulfils the proof of the H theorem for generalized entropic forms, associated with systems characterized by complex dynamics, e.g., presenting nonsymmetric probability transition rates and more than one path towards the same equilibrium state. Some examples considering generalized entropies of the literature are discussed, showing that they should be applicable to a wide range of natural phenomena, mainly those within the realm of complex systems.

Anisotropic four-state clock model in the presence of random fields.

A four-state clock ferromagnetic model is studied in the presence of different configurations of anisotropies and random fields. The model is considered in the limit of infinite-range interactions, for which the mean-field approach becomes exact. Both representations of Cartesian spin components and two Ising variables are used, in terms of which the physical properties and phase diagrams are discussed. The random fields follow bimodal probability distributions and the richest criticality is found when the fields, applied in the two Ising systems, are not correlated. The phase diagrams present new interesting topologies, with a wide variety of critical points, which are expected to be useful in describing different complex phenomena.

Critical properties of short-range Ising spin glasses on a Wheatstone-bridge hierarchical lattice.

An Ising spin-glass model with nearest-neighbor interactions, following a symmetric probability distribution, is investigated on a hierarchical lattice of the Wheatstone-bridge family characterized by a fractal dimension D≈3.58. The interaction distribution considered is a stretched exponential, which has been shown recently to be very close to the fixed-point coupling distribution, and such a model has been considered lately as a good approach for Ising spin glasses on a cubic lattice. An exact recursion procedure is implemented for calculating site magnetizations, mi=〈Si〉T, as well as correlations between pairs of nearest-neighbor spins, 〈SiSj〉T (〈〉T denote thermal averages), for a given set of interaction couplings on this lattice. From these local magnetizations and correlations, one can compute important physical quantities, such as the Edwards-Anderson order parameter, the internal energy, and the specific heat. Considering extrapolations to the thermodynamic limit for the order parameter, such as a finite-size scaling approach, it is possible to obtain directly the critical temperature and critical exponents. The transition between the spin-glass and paramagnetic phases is analyzed, and the associated critical exponents ß and ν are estimated as ß=0.82(5) and ν=2.50(4), which are in good agreement with the most recent results from extensive numerical simulations on a cubic lattice. Since these critical exponents were obtained from a fixed-point distribution, they are universal, i.e., valid for any coupling distribution considered.

Fixed-point distributions of short-range Ising spin glasses on hierarchical lattices.

Fixed-point distributions for the couplings of Ising spin glasses with nearest-neighbor interactions on hierarchical lattices are investigated numerically. Hierarchical lattices within the Migdal-Kadanoff family with fractal dimensions in the range 2.58≤D≤7, as well as a lattice of the Wheatstone-Bridge family with fractal dimension D≈3.58 are considered. Three initial distributions for the couplings are analyzed, namely, the Gaussian, bimodal, and uniform ones. In all cases, after a few iterations of the renormalization-group procedure, the associated probability distributions approached universal fixed shapes. For hierarchical lattices of the Migdal-Kadanoff family, the fixed-point distributions were well fitted either by stretched exponentials, or by q-Gaussian distributions; both fittings recover the expected Gaussian limit as D→∞. In the case of the Wheatstone-Bridge lattice, the best fit was found by means of a stretched-exponential distribution.

Consistent thermodynamic framework for interacting particles by neglecting thermal noise.

An effective temperature θ, conjugated to a generalized entropy s(q), was introduced recently for a system of interacting particles. Since θ presents values much higher than those of typical room temperatures T≪θ, the thermal noise can be neglected (T/θ≃0) in these systems. Moreover, the consistency of this definition, as well as of a form analogous to the first law of thermodynamics, du=θds(q)+δW, were verified lately by means of a Carnot cycle, whose efficiency was shown to present the usual form, η=1-(θ(2)/θ(1)). Herein we explore further the heat contribution δQ=θds(q) by proposing a way for a heat exchange between two such systems, as well as its associated thermal equilibrium. As a consequence, the zeroth principle is also established. Moreover, we consolidate the first-law proposal by following the usual procedure for obtaining different potentials, i.e., applying Legendre transformations for distinct pairs of independent variables. From these potentials we derive the equation of state, Maxwell relations, and define response functions. All results presented are shown to be consistent with those of standard thermodynamics for T>0.

Second law and entropy production in a nonextensive system.

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, S(2)(t)≡S((q)=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 [dS(2)(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 S(2)(t) with time, up to its stationary-state value.

Carnot cycle for interacting particles in the absence of thermal noise.

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 s(q), 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 (δ Q = θ ds(q)), 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 s(q) adiabatic and θ isothermic transformations. On this basis, a Carnot cycle is constructed, whose efficiency is shown to be η = 1-(θ(2)/θ(1)), where θ(1) and θ(2) are the effective temperatures of the two isothermic transformations, with θ(1)>θ(2). 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.

Probability distributions and associated nonlinear Fokker-Planck equation for the two-index entropic form S(q,δ).

The probability distributions and associated Fokker-Planck equation of the recently postulated entropic form, S(q,δ), are investigated. This entropy was proposed as an unification of the well-known S(q) of nonextensive-statistical mechanics and S(δ), which appeared lately as a possibly appropriate candidate for the black-hole entropy. The connection between S(q,δ) and a nonlinear Fokker-Planck equation, such as to satisfy an H-theorem, is explored. The stationary-state probability distribution follows a transcendental equation, which is solved numerically for typical values of q and δ. The same transcendental equation is obtained through the maximum-entropy principle, showing that the two procedures are equivalent.

Comment on "Vortex distribution in a confining potential".

A system of interacting vortices is considered as an appropriate model for describing properties of type-II superconductors, and it has been shown lately to be deeply associated with nonextensive statistical mechanics. Herein we comment on a recent investigation of this model [M. Girotto, A. P. dos Santos, and Y. Levin, Phys. Rev. E 88, 032118 (2013)], which tried to contradict this assertion, based on a mean-field type of solution, compared with numerical-simulation data that correspond typically to a regime characterized by low concentrations of particles, as well as very high temperatures. It is shown that the physical situations analyzed differ significantly from those of a real superconducting phase. The analytical solution obtained from such a mean-field approximation shows a discrepancy with respect to the results of molecular-dynamics numerical simulations, which increases as the temperature is lowered towards the superconducting phase, as expected. We demonstrate that these results, when interpreted properly by means of an analytical solution within the framework of nonextensive statistical mechanics, present a remarkable agreement between molecular-dynamics simulations and theoretical results, for all temperatures, specially for those temperatures associated with the existence of type-II superconductivity.