Boltzmann-Gibbs Research Articles

Diffusion with preferential relocation in a confining potential

Abstract We study the relaxation of a diffusive particle confined in an arbitrary external potential and subject to a non-Markovian resetting protocol. With a constant rate r, a previous time τ between the initial time and the present time t is chosen from a given probability distribution K ( τ , t ) , and the particle is reset to the position that it occupied at time τ. Depending on the shape of K ( τ , t ) , the particle either relaxes toward the Gibbs–Boltzmann distribution or toward a non-trivial stationary distribution that breaks ergodicity and depends on the initial position and the resetting protocol. From a general asymptotic theory, we find that if the kernel K ( τ , t ) is sufficiently localized near τ = 0, i.e. mostly the initial part of the trajectory is remembered and revisited, the steady state is non-Gibbs–Boltzmann. Conversely, if K ( τ , t ) decays slowly enough or increases with τ, i.e. recent positions are more likely to be revisited, the probability distribution of the particle tends toward the Gibbs–Boltzmann state at large times. In the latter case, however, the temporal approach to the stationary state is generally anomalously slow, following for instance an inverse power law or a stretched exponential, if K ( τ , t ) is not too strongly peaked at the current time t. These findings are verified by the analysis of several exactly solvable cases and by numerical simulations.

Riemann surfaces and winding numbers of Rényi phase structure of charged-flat black holes

It’s widely recognized that the free energy landscape captures the essentials of thermodynamic phase transitions. In this work, we extend the findings of [1] by incorporating the nonextensive nature of black hole entropy. Specifically, the connection between black hole phase transitions and the winding number of Riemann surfaces derived through complex analysis is extended to the Rényi entropy framework. This new geometrical and nonextensive formalism is employed to predict the phase portraits of charged-flat black holes within both the canonical and grand canonical ensembles. Furthermore, we elucidate novel relations between the number of sheets comprising the Riemann surface of the Hawking–Page and Van der Waals transitions and the dimensionality of black hole spacetimes. Notably, these new numbers are consistent with those found for charged-AdS black holes in Gibbs–Boltzmann statistics, providing another significant example of the potential connection between the cosmological constant and the nonextensive Rényi parameter.

Impact of Barrow’s entropy on topological classification of higher-dimensional black holes

In this paper, we explore the impact of Barrow’s entropy on the topological classification of d-dimensional black holes (BHs). Specifically, we examine the Gauss–Bonnet (GB) and singly rotating Kerr black holes (KBHs) as well as their AdS counterparts. We notice that Barrow’s entropy significantly influences the topological numbers of these BHs. To understand these effects, we use the thermodynamic domain — the space defined by thermodynamic variables like temperature and pressure — to study topologically inspired defects in the BHs. These defects are points in the thermodynamic domain where singular or discontinuous behavior occurs. By computing the winding numbers of these defects, which are integers that count how many times a loop around the defect encircles the origin, we can understand the local and global topology of the BHs. Our analysis suggest that the topological number [Formula: see text] derived from Barrow’s statistics differs from the one obtained using Gibbs–Boltzmann statistics. Furthermore, we observe that BHs can undergo topological phase transitions that characterize them in different thermodynamic topological classes. Overall, these findings demonstrate the importance of Barrow’s entropy in understanding the topological classification of BHs, and highlight the potential for further research in this area.

Topology of black hole thermodynamics via Rényi statistics

In this paper, we investigate the topological numbers of the four-dimensional Schwarzschild black hole, d-dimensional Reissner–Nordström (RN) black hole, d-dimensional singly rotating Kerr black hole and five-dimensional Gauss–Bonnet black hole via the Rényi statistics. We find that the topological number calculated via the Rényi statistics is different from that obtained from the Gibbs–Boltzmann (GB) statistics. However, what is interesting is that the topological classifications of different black holes are consistent in both the Rényi and GB statistics: the four-dimensional RN black hole, four-dimensional and five-dimensional singly rotating Kerr black holes, five-dimensional charged and uncharged Gauss–Bonnet black holes belong to the same kind of topological class, and the four-dimensional Schwarzschild black hole and d(>5)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(>5)$$\\end{document}-dimensional singly rotating Kerr black holes belong to another kind of topological class. In addition, our results suggest that the topological numbers calculated via the Rényi statistics in asymptotically flat spacetime background are equal to those calculated from the standard GB statistics in asymptotically AdS spacetime background, which provides more evidence for the connection between the nonextensivity of the Rényi parameter λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document} and the cosmological constant Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}.

An appropriate statistical approach for non-equilibrium particle production

The incapability of thermal models to accurately reproduce the horn-like structure of the Kaon-to-pion ratio measured at AGS, SPS, and low RHIC energies, as well as confirmed in the beam energy scan program, has long been a persistent problem. This issue is believed to have arisen due to the inappropriate application of statistics, particularly the extensive additive Boltzmann–Gibbs (BG) statistics. The assumption that the analysis of particle production, a dynamic non-equilibrium process, should be primarily conducted using extensive BG or non-extensive Tsallis statistics, has proven to be an unsuccessful approach that has been followed for several decades. By employing generic (non)extensive statistics, two equivalence classes [Formula: see text] emerge, thereby undermining the validity of any ad hoc assumption. Consequently, the degree of (non)extensivity exhibited by the statistical ensemble is determined by its own characteristics. This encompasses both extensive BG statistics, characterized by [Formula: see text], and non-extensive Tsallis statistics, characterized by [Formula: see text]. The energy dependence of light-, [Formula: see text], and strange-quark occupation factor, [Formula: see text], suggests that the produced particles are most appropriately described as a non-equilibrium ensemble. This is evidenced by a remarkable non-monotonic behavior observed in the [Formula: see text] horn, for instance. On the other hand, the resulting equivalence classes [Formula: see text] are associated with a generic non-extensivity related to extended exponential and Lambert-[Formula: see text] exponentially generating distribution function, which evidently arise from free, short- and long-range correlations. The incorporation of generic non-extensive statistics into the hadron resonance gas model yields an impressive ability to rightfully reproduce the non-monotonic [Formula: see text] ratio.

Thermodynamic topology of Kerr-Sen black holes via Rényi statistics

In the present study, we investigate the topological properties of black holes in terms of Rényi statistics as an extension of the Gibbs-Boltzmann (GB) statistics, aiming to characterize the non-Boltzmannian thermodynamic topology of Kerr-Sen and dyonic Kerr-Sen black holes. Through this research, we discover that the topological number derived via Rényi statistics differs from that obtained through GB statistics. Interestingly, although the nonextensive parameter λ changes the topological number, the topological classification of the Kerr-Sen and dyonic Kerr-Sen black holes remains consistent under both GB and Rényi statistics. In addition, the topological numbers associated with these two types of black holes without cosmological constant using Rényi entropy processes are the same as the AdS cases of them by considering the GB entropy, as further evidenced by such a study found here. This indicates the cosmological constant has some potential connections with the nonextensive parameter from the perspective of thermodynamic topology.

Power-law relaxation of a confined diffusing particle subject to resetting with memory

We study the relaxation of a Brownian particle with long range memory under confinement in one dimension. The particle diffuses in an arbitrary confining potential and resets at random times to previously visited positions, chosen with a probability proportional to the local time spent there by the particle since the initial time. This model mimics an animal which moves erratically in its home range and returns preferentially to familiar places from time to time, as observed in nature. The steady state density of the position is given by the equilibrium Gibbs–Boltzmann distribution, as in standard diffusion, while the transient part of the density can be obtained through a mapping of the Fokker–Planck equation of the process to a Schrödinger eigenvalue problem. Due to memory, the approach at late times toward the steady state is critically self-organised, in the sense that it always follows a sluggish power-law form, in contrast to the exponential decay that characterises Markov processes. The exponent of this power-law depends in a simple way on the resetting rate and on the leading relaxation rate of the Brownian particle in the absence of resetting. We apply these findings to several exactly solvable examples, such as the harmonic, V-shaped and box potentials.

Extensive entropy: the case of zero entropy defect

This paper shows that the Rényi and Boltzmann-Gibbs (BG) extensive entropies share the same functional relationship with the nonextensive entropy associated with kappa distributions, which coincides with the well-known Havrda/Charvát/Daróczy/Tsallis (HCDT) entropy. We find that while the relationship between kappa/HCDT and Rényi entropies is merely a mathematical identity between their entropic statistical definitions, the relationship between kappa/HCDT and BG entropies is based on their thermodynamic connection. The latter connects the entropy between a system characterized by correlations among their all constituents (kappa/HCDT entropy) and the entropy of the same system but with no correlations among their constituents (BG entropy). The origin of this relationship, and its connection with thermodynamics, is examined using the concept of entropy defect, that is, the decrease in a system’s entropy caused by the presence of long-range correlations among its constituents; in the limiting case of zero correlations, the entropy defect vanishes and the entropy becomes extensive and expressed by the BG formulation.

Non-additive entropies and statistical mechanics at the edge of chaos: a bridge between natural and social sciences.

The Boltzmann-Gibbs (BG) statistical mechanics constitutes one of the pillars of contemporary theoretical physics. It is constructed upon the other pillars-classical, quantum, relativistic mechanics and Maxwell equations for electromagnetism-and its foundations are grounded on the optimization of the BG (additive) entropic functional [Formula: see text]. Its use in the realm of classical mechanics is legitimate for vast classes of nonlinear dynamical systems under the assumption that the maximal Lyapunov exponent is positive (currently referred to as strong chaos), and its validity has been experimentally verified in countless situations. It fails however when the maximal Lyapunov exponent vanishes (referred to as weak chaos), which is virtually always the case with complex natural, artificial and social systems. To overcome this type of weakness of the BG theory, a generalization was proposed in 1988 grounded on the non-additive entropic functional [Formula: see text]. The index [Formula: see text] and related ones are to be calculated, whenever mathematically tractable, from first principles and reflect the specific class of weak chaos. We review here the basics of this generalization and illustrate its validity with selected examples aiming to bridge natural and social sciences. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)'.

Neural complexity through a nonextensive statistical–mechanical approach of human electroencephalograms

The brain is a complex system whose understanding enables potentially deeper approaches to mental phenomena. Dynamics of wide classes of complex systems have been satisfactorily described within q-statistics, a current generalization of Boltzmann-Gibbs (BG) statistics. Here, we study human electroencephalograms of typical human adults (EEG), very specifically their inter-occurrence times across an arbitrarily chosen threshold of the signal (observed, for instance, at the midparietal location in scalp). The distributions of these inter-occurrence times differ from those usually emerging within BG statistical mechanics. They are instead well approached within the q-statistical theory, based on non-additive entropies characterized by the index q. The present method points towards a suitable tool for quantitatively accessing brain complexity, thus potentially opening useful studies of the properties of both typical and altered brain physiology.

Thermodynamics and phase transition of Bardeen black hole via Rényi statistics in grand canonical ensemble and canonical ensemble

The thermodynamics of the Bardeen black hole in asymptotically flat space is investigated with the corrected first law of thermodynamics via Rényi statistics. The nonextensive parameter lambda gives the possibility to the thermal stability of Bardeen black hole, and there is a Hawking–Page phase transition in the grand canonical ensemble (fixed the potential), which is similar to the cases of Bardeen black hole and corrected Bardeen black hole in asymptotically anti-de Sitter (AdS) space via standard Gibbs–Boltzmann (GB) statistics. By introducing the general Smarr formula via Rényi statistics, the thermodynamic pressure P is defined with the parameter lambda and its conjugate quantity V is the thermodynamic volume (not a geometric spherical volume with horizon radius r_{h}). The thermodynamics of the asymptotically flat Bardeen black hole via Rényi statistics in the canonical ensemble (fixed the charge q) behaves like the van der Waals system, which is also same as the asymptotically Bardeen-AdS black hole via GB statistics. The analogy between the thermodynamics of the asymptotically flat Bardeen black hole from Rényi statistics and the Bardeen-AdS black hole from GB statistics makes us to consider what is the relation between the nonextensive parameter lambda and the cosmological constant Lambda .

Bending fluctuations in semiflexible, inextensible, slender filaments in Stokes flow: Toward a spectral discretization.

Semiflexible slender filaments are ubiquitous in nature and cell biology, including in the cytoskeleton, where reorganization of actin filaments allows the cell to move and divide. Most methods for simulating semiflexible inextensible fibers/polymers are based on discrete (bead-link or blob-link) models, which become prohibitively expensive in the slender limit when hydrodynamics is accounted for. In this paper, we develop a novel coarse-grained approach for simulating fluctuating slender filaments with hydrodynamic interactions. Our approach is tailored to relatively stiff fibers whose persistence length is comparable to or larger than their length and is based on three major contributions. First, we discretize the filament centerline using a coarse non-uniform Chebyshev grid, on which we formulate a discrete constrained Gibbs-Boltzmann (GB) equilibrium distribution and overdamped Langevin equation for the evolution of unit-length tangent vectors. Second, we define the hydrodynamic mobility at each point on the filament as an integral of the Rotne-Prager-Yamakawa kernel along the centerline and apply a spectrally accurate "slender-body" quadrature to accurately resolve the hydrodynamics. Third, we propose a novel midpoint temporal integrator, which can correctly capture the Ito drift terms that arise in the overdamped Langevin equation. For two separate examples, we verify that the equilibrium distribution for the Chebyshev grid is a good approximation of the blob-link one and that our temporal integrator for overdamped Langevin dynamics samples the equilibrium GB distribution for sufficiently small time step sizes. We also study the dynamics of relaxation of an initially straight filament and find that as few as 12 Chebyshev nodes provide a good approximation to the dynamics while allowing a time step size two orders of magnitude larger than a resolved blob-link simulation. We conclude by applying our approach to a suspension of cross-linked semiflexible fibers (neglecting hydrodynamic interactions between fibers), where we study how semiflexible fluctuations affect bundling dynamics. We find that semiflexible filaments bundle faster than rigid filaments even when the persistence length is large, but show that semiflexible bending fluctuations only further accelerate agglomeration when the persistence length and fiber length are of the same order.

Nonextensive effects on QCD chiral phase diagram and baryon-number fluctuations within Polyakov-Nambu-Jona-Lasinio model* *Supported by the National Natural Science Foundation of China (12005192) and the Project funded by China Postdoctoral Science Foundation (2020M672255, 2020TQ0287)

In this paper, a version of the Polyakov-Nambu-Jona-Lasinio (PNJL) model based on nonextensive statistical mechanics is presented. This new statistics summarizes all possible factors that violate the assumptions of the Boltzmann-Gibbs (BG) statistics to a dimensionless nonextensivity parameter q. Thus, when q tends to 1, it returns to the BG case. Within the nonextensive PNJL model, we found that as q increases, the location of the critical end point (CEP) exhibits non-monotonic behavior. That is, for , CEP moves in the direction of lower temperature and larger quark chemical potential. However, for , CEP turns to move in the direction of lower temperature and lower quark chemical potential. In addition, we studied the moments of the net-baryon number distribution, that is, variance (), skewness (S), and kurtosis (κ). Our results are generally consistent with the latest experimental data reported, especially for , when q is set to .

Brownian particles in periodic potentials: Coarse-graining versus fine structure.

We study the motion of an overdamped particle connected to a thermal heat bath in the presence of an external periodic potential in one dimension. When we coarse-grain, i.e., bin the particle positions using bin sizes that are larger than the periodicity of the potential, the packet of spreading particles, all starting from a common origin, converges to a normal distribution centered at the origin with a mean-squared displacement that grows as 2D^{*}t, with an effective diffusion constant that is smaller than that of a freely diffusing particle. We examine the interplay between this coarse-grained description and the fine structure of the density, which is given by the Boltzmann-Gibbs (BG) factor e^{-V(x)/k_{B}T}, the latter being nonnormalizable. We explain this result and construct a theory of observables using the Fokker-Planck equation. These observables are classified as those that are related to the BG fine structure, like the energy or occupation times, while others, like the positional moments, for long times, converge to those of the large-scale description. Entropy falls into a special category as it has a coarse-grained and a fine structure description. The basic thermodynamic formula F=TS-E is extended to this far-from-equilibrium system. The ergodic properties are also studied using tools from infinite ergodic theory.

Nonextensive Footprints in Dissipative and Conservative Dynamical Systems

Despite its centennial successes in describing physical systems at thermal equilibrium, Boltzmann–Gibbs (BG) statistical mechanics have exhibited, in the last several decades, several flaws in addressing out-of-equilibrium dynamics of many nonlinear complex systems. In such circumstances, it has been shown that an appropriate generalization of the BG theory, known as nonextensive statistical mechanics and based on nonadditive entropies, is able to satisfactorily handle wide classes of anomalous emerging features and violations of standard equilibrium prescriptions, such as ergodicity, mixing, breakdown of the symmetry of homogeneous occupancy of phase space, and related features. In the present study, we review various important results of nonextensive statistical mechanics for dissipative and conservative dynamical systems. In particular, we discuss applications to both discrete-time systems with a few degrees of freedom and continuous-time ones with many degrees of freedom, as well as to asymptotically scale-free networks and systems with diverse dimensionalities and ranges of interactions, of either classical or quantum nature.

Tsallis’ Analysis of the Horizontal Component of the Earth’s Magnetic Field over India during 2002

A widely used measure of entropy to quantify uncertainity in an open system is the Boltzmann-Gibbs (B-G) entropy. It, however, cannot describe non-equilibrium systems with large variability and multi-fractality. A generalisation of the B - G entropy is the Tsallis’ entropy. The values of the horizontal components of the Earth’s magnetic field, observed at various stations in India in 2002 were used. During the years 2000 – 2002, solar cycle 23 reached its maximum. Data from Ettaiyapuram (ETT, latitude = 90 10’ N, longitude = 780 01’ E, geomagnetic latitude = 0.130 N), Visakhapatnam (VIS, 170 41’ N, 830 19’ E, 8.170 N), Hyderabad (HYD, 170 25’ N, 780 33’ E, 8.170 N), Alibag (ABG, 180 37’ N, 720 52’ E, 10.020 N) and Tirunelveli (TVI, 80 42’ N, 770 48’ E, 0.320 S) were used. Using these values as inputs, we demonstrate that Tsallis’ entropy can be used to detect minor differences in the horizontal components of the geomagnetic field observed between different pairs of stations. The method is a very simple and elegant one to detect minor variations between pairs of similar signals.

Thermodynamics of asymptotically de Sitter black hole in dRGT massive gravity from Rényi entropy

The thermodynamic properties of the de Rham–Gabadadze–Tolley (dRGT) black hole in the asymptotically de Sitter (dS) spacetime are investigated by using Rényi entropy. It has been found that the black hole with asymptotically dS spacetime described by the standard Gibbs–Boltzmann statistics cannot be thermodynamically stable. Moreover, there generically exist two horizons corresponding to two thermodynamic systems with different temperatures, leading to a nonequilibrium state. Therefore, in order to obtain the stable dRGT black hole, we use the alternative Rényi statistics to analyze the thermodynamic properties in both the separated system approach and the effective system approach. Interestingly, we found that it is possible concurrently obtain positive pressure and volume for the dRGT black hole while it is not for the Schwarzschild-de Sitter (Sch-dS) black hole. Furthermore, the bounds on the nonextensive parameter for which the black hole being thermodynamically stable are determined. In addition, the key differences between the systems described by different approaches, e.g., temperature profiles and types of the Hawking–Page phase transition are pointed out.

Quasi-static Decomposition and the Gibbs Factorial in Small Thermodynamic Systems

For a classical system consisting of N-interacting identical particles in contact with a heat bath, we define the free energy from thermodynamic relations in equilibrium statistical mechanics. Concretely, the temperature dependence of the free energy is determined from the Gibbs-Helmholtz relation, and its volume dependence is determined from the condition that the quasi-static work in a volume change is equal to the free energy change. Now, we argue the free energy difference in a quasi-static decomposition of small thermodynamic systems. We can then determine the N dependence of the free energy, which includes the Gibbs factorial N! in addition to the phase space integration of the Gibbs–Boltzmann factor.

Emergent phase, thermodynamic geometry, and criticality of charged black holes from Rényi statistics

Recently, a novel emergent phase can occur from thermodynamic consideration of the asymptotically flat Reissner-Nordstr\"om black hole (RN-AF) using R\'enyi statistics. We present an analysis of the thermodynamical and mechanical stabilities of the RN-AF in both the Gibbs-Boltzmann (GB) and the alternative R\'enyi statistics when charge $q$ and electrostatic potential $\phi$ are treated as pressure and volume, respectively. Interestingly, the emergent phase of the RN-AF can be both thermodynamically and mechanically stable in some range of parameters in the framework of R\'enyi thermodynamics. With the construction of the Maxwell equal area law in $q-\phi$ plane, the coexistence line between the near-extremal black hole phase and the emergent phase can be found in some values of charge which can be associated as the vapor pressure at which the liquid and gas phases coexist. In the aspect of thermodynamic geometry, the microscopic interaction between the black hole microstructures can be repulsive in the R\'enyi description. This implies that a novel correlation between the microstates of a self-gravitating system could be emerged via the nonextensive nature of long-range interaction systems. Finally, we also investigate the critical phenomena of the RN-AF in R\'enyi statistics compared to that of the van der Waals (vdW) fluid and find that the critical exponents of the relevant physical quantities of both systems are identical. This implies that both systems are in the same universality class of the phase transition.

Nonextensive effects on QCD chiral phase transition with a chiral chemical potential* *Supported by the National Natural Science Foundation of China (12005192) and the Project funded by China Postdoctoral Science Foundation (2020M672255, 2020TQ0287)

In this study, we investigate the QCD chiral phase diagram in the presence of a chiral chemical potential based on nonextensive statistical mechanics. A feature of this new statistic is a dimensionless nonextensivity parameter q, which summarizes all possible effects violating the assumptions of Boltzmann-Gibbs (BG) statistics (when , it returns to the BG case). Within the nonextensive Polyakov-Nambu-Jona-Lasinio model, we find that as increases, the critical end point (CEP) in the plane continues to in the plane, and nonextensive effects have a significant impact on the evolution from the CEP to . Generally, with an increase in q, both the CEP and move in the direction of a lower temperature T and larger chemical potential μ (). In addition, we find that chiral charge density generally increases with T, μ, , and q. Our study may provide useful hints about lattice QCD and relativistic heavy-ion collision experiments.