Boltzmann-Gibbs Entropy Research Articles

Exploring Tsallis thermodynamics for boundary conformal field theories in gauge/gravity duality

Inspired by AdS/CFT interpretation Karch and Robinson (2015), we analyze thermodynamics for thermal boundary conformal field theories (CFT) that are dual to four-dimensional charged anti-de Sitter (AdS) black holes embedded in 11-dimensional M-theory-inspired models with AdS4×S7 space–time, employing the framework of Tsallis entropy. The latter is recognized as a nonadditive extension of the Boltzmann–Gibbs entropy that can satisfy the thermodynamic extensivity for black holes. Specifically, we consider AdS black holes in AdS4×S7 spacetime, interpreting the number of M2-branes as a thermodynamic variable. Using the Tsallis entropy enables us to highlight peculiar thermodynamic features, resorting to essential tools such as the chemical potential and Gibbs energy while examining phase transitions along iso-charge partitions. Then, we leveraged a class of geometrothermodynamic formalisms, including Weinhold, Ruppeiner, Quevedo I, and II metrics. Quevedo’s formulations provide richer information about phase transitions than the first two methods. Our study sheds new light on AdS black holes in a thermodynamically proper context, deepening our understanding of the role of non-extensivity in the critical behavior of such complex systems.

Tracing the classical molecular nature of entropy generation

The second law of thermodynamics and the concept of positive entropy generation are analyzed following classical statistical mechanics methods. Previously, using the generalized Boltzmann–Gibbs entropy and its associated general entropy conservation relation, positive entropy generation expressions were obtained in agreement with phenomenological results and the work of Boltzmann and Gibbs. In this study, using the general approach, we formally and explicitly trace the specific entropy generation expressions to truncations in the full N-body description of the entropy state to a lower s-body description. Using higher-order superposition approximations, it is formally shown that the generalized Boltzmann–Gibbs entropy in the s-order state is always less than the corresponding Boltzmann–Gibbs entropy in the lower (s − 1)-order state. Using the general form of the entropy conservation equation, entropy generation is shown to be a required compensatory effect to ensure that all physical variables and physical processes associated with heat, work, temperature, etc., are independent of the particular entropy definition state.

A mechanical derivation of the evolution equation for scintillating crystals: Recombination–diffusion–drift equations, gradient flows and Wasserstein measures

In a series of previous papers we obtained, by the means of the mechanics of continua with microstructure, the Reaction–Diffusion–Drift equation which describes the evolution of charge carriers in scintillators. Here we deal, first of all, with the consequences of constitutive assumptions for the entropic and dissipative terms. In the case of Boltzmann–Gibbs entropy, we show that the equation admits a gradient flows structure: moreover, we show that the drift–diffusion part is a Wasserstein gradient flow and we show how the energy dissipation is correlated with an appropriate Wasserstein distance.

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.

Quintessence reconstruction through new Tsallis holographic dark energy model

In statistical theory, the Tsallis entropy is an extended form of the Boltzmann–Gibbs entropy. The dimensionless parameter [Formula: see text] is employed to state the quantitative difference from the standard scenario. The concepts of Tsallis entropy and the future event horizon are employed in formulating the present new Tsallis holographic dark energy (NTHDE) model. The model attempts to explain the properties of dark energy using the foundation of quantum gravity. The differential equation characterizing the evolution of the NTHDE density parameter is obtained. Expressions stating the dynamic behavior such as equation of state (EoS), deceleration and jerk parameters are obtained in terms of the NTHDE density parameter. For [Formula: see text], the quintessence nature of scalar field could completely characterize the NTHDE. A reconstruction of the scalar field’s dynamics and quintessence potential is attempted. We demonstrate that the diagnosis made by statefinder is adaptive enough to distinguish between quintessence and cosmological constant-based dark energy models. Additionally, observational data obtained from CC[Formula: see text] SNIa [Formula: see text] union 2.1 sources are used to evaluate the model’s effectiveness.

Equivalence of information production and generalised entropies in complex processes.

Complex systems with strong correlations and fat-tailed distribution functions have been argued to be incompatible with the Boltzmann-Gibbs entropy framework and alternatives, so-called generalised entropies, were proposed and studied. Here we show, that this perceived incompatibility is actually a misconception. For a broad class of processes, Boltzmann entropy -the log multiplicity- remains the valid entropy concept. However, for non-i.i.d. processes, Boltzmann entropy is not of Shannon form, -k∑ipi log pi, but takes the shape of generalised entropies. We derive this result for all processes that can be asymptotically mapped to adjoint representations reversibly where processes are i.i.d. In these representations the information production is given by the Shannon entropy. Over the original sampling space this yields functionals identical to generalised entropies. The problem of constructing adequate context-sensitive entropy functionals therefore can be translated into the much simpler problem of finding adjoint representations. The method provides a comprehensive framework for a statistical physics of strongly correlated systems and complex processes.

The isothermal Boltzmann–Gibbs entropy reduction affects survival of the fruit fly Drosophila melanogaster

To the best of our knowledge, this is the first experimental evidence of the effect of isothermal changes in entropy on a living organism. In greater detail, the effect of the reduction of the total Boltzmann–Gibbs entropy (S) of the aquatic environment on the survival rate and body mass of the fruit fly Drosophila melanogaster was investigated. The tests were carried out in standard thermodynamic states at room temperature of 296.15 K and ambient atmospheric pressure of 1 bar. Two variants of entropy reduction (ΔS) were tested for ΔS = 28.49 and 51.14 J K−1 mol−1 compared to the blind and control samples. The entropy level was experimentally changed, using the quantum system for isothermal entropy reduction. This system is based on quantum bound entanglement of phonons and the phenomenon of phonon resonance (interference of phonon modes) in condensed matter (Silicon dioxide (SiO2) and single crystals of Silicon (Si0), Aluminum (Al0) plates (“chips”), glass, and water). All studied organisms were of the same age (1 day). Mortality was observed daily until the natural death of the organisms. The investigations showed that changes in the Boltzmann–Gibbs entropy affected the survival and body mass of the fruit flies. On the one hand, the reduction in entropy under isothermal conditions in the aquatic environment for ΔS = 28.49 J K−1 mol−1 resulted in an extension of the lifespan and an increase in the body mass of female fruit flies. On the other hand, the almost twofold reduction in this entropy for ΔS = 51.14 J K−1 mol−1 shortened the lives of the males. Thus, the lifespan and body mass of flies turned out to be a specific reaction of metabolism related to changes in the entropy of the aquatic environment.

Black hole geometrothermodynamics and critical phenomena: A look from Tsallis entropy-based perspective

We analyze geometrothermodynamics of charged anti-de Sitter (AdS) black holes with a global monopole in the framework of Tsallis statistics. The latter arises from a non-additive generalization of Boltzmann–Gibbs entropy, which is still recovered as a particular sub-case. We examine the effects of Tsallis entropy on small-large black holes phase-transitions and critical exponents, comparing the results with the liquid–gas change of phase occurring in van der Waals fluids. We also discuss some physical constraints on Tsallis parameter and the impact on sparsity of BH radiation. Thermodynamic curvature behavior and the nature of microscopic interactions are then explored from a geometric perspective within Ruppeiner formalism on the S−P coordinate space. Similar considerations are extended to charged (2+1)-dimensional Banados–Teitelboim–Zanelli (BTZ) black holes by computing the entropy-pressure and entropy-volume corrected curvatures. Our analysis shows that Tsallis prescription has non-trivial implications on black hole geometrothermodynamics, emphasizing the role of non-additive entropy in the correspondence between black holes and van der Waals systems.

Entropy evolution at generic power-law edge of chaos

For strongly chaotic classical systems, a basic statistical–mechanical connection is provided by the averaged Pesin-like identity (the production rate of the Boltzmann–Gibbs entropy SBG=−∑i=1Wpilnpi equals the sum of the positive Lyapunov exponents). In contrast, at a generic edge of chaos (vanishing maximal Lyapunov exponent) we have a subexponential divergence with time of initially close orbits. This typically occurs in complex natural, artificial and social systems and, for a wide class of them, the appropriate entropy is the nonadditive one Sqe=1−∑i=1Wpiqeqe−1(S1=SBG) with qe≤1. For such weakly chaotic systems, power-law divergences emerge involving a set of microscopic indices {qk}’s and the associated generalized Lyapunov coefficients. We establish the connection between these quantities and (qe,Kqe), where Kqe is the Sqe entropy production rate.

Tsallis Entropy applied to microfluidic channels analysis

This work investigated the possibility to describe the fluid flow in a microchannel from a thermodynamic point of view, exploring the possibility to evaluate the presence of obstacles (or, more in general, geometry imperfection) and their influence on the fluid. Tsallis entropy concept was employed. This form of entropy was introduced in 1988 by Costantino Tsallis as a basis for generalizing the standard statistical mechanics and as a generalization of the standard Boltzmann-Gibbs entropy. Inspired by nature, where storing information is an intrinsic ability of natural systems, here we investigate the capability of interacting systems to transport/store the information generated/exchanged in the interaction process in the form of energy or matter, preserving it over time. In detail, here we test the possibility to consider a fluid as a carrier of information, speculating about how to use such information. The final goal is to demonstrate that information theory can be used to illuminate physical observations, even in those cases where the equations describing the phenomenon under investigation are intractable, are affected by a budget of uncertainty that makes their solution not affordable or may not even be known. In this exploratory work, an information theory-based approach is applied to microfluidic data. In detail, the classical study of the fluid flow in a microchannel with an obstacle of a specific geometry is faced by integrating fluid mechanics theory with Tsallis entropy. Technically, computational fluid dynamics simulations at Reynolds’ numbers (Re) equal to 1 were carried out in fluidic channels presenting a rectangular obstacle and on the simulated flow fields.

Kaniadakis agegraphic dark energy

In this manuscript, we present a novel dark energy scenario based on Kaniadakis entropy, which is a generalization of Boltzmann–Gibbs entropy that arises from relativistic statistical theory and is characterized by a parameter K which quantifies the deviations from standard expressions, and we use the age of the Universe as infrared cutoff. The cosmological characteristics of the proposed dark energy model, as well as the evolution of the cosmos filled with pressure-free matter and the ensuing dark energy candidates, are investigated. The interaction as well as non-interaction among the two sectors will also be considered. The differential equation for the dark energy density parameter, including the expression of the equation of state and deceleration parameters, are derived. The analysis of deceleration parameter clearly shows the universe to transit from decelerated to accelerated phase around z≈0.6. The most recent 30 H(z) data samples and the 580 Union 2.1 SNIa data set were also used to limit the theoretical results. The squared sound speed is also plotted against redshift z to check the stability behavior of the model for both the cases.

Entropy defect in thermodynamics

This paper describes the physical foundations of the newly discovered “entropy defect” as a basic concept of thermodynamics. The entropy defect quantifies the change in entropy caused by the order induced in a system through the additional correlations among its constituents when two or more subsystems are assembled. This defect is closely analogous to the mass defect that arises when nuclear particle systems are assembled. The entropy defect determines how the entropy of the system compares to its constituent’s entropies and stands on three fundamental properties: each constituent’s entropy must be (i) separable, (ii) symmetric, and (iii) bounded. We show that these properties provide a solid foundation for the entropy defect and for generalizing thermodynamics to describe systems residing out of the classical thermal equilibrium, both in stationary and nonstationary states. In stationary states, the consequent thermodynamics generalizes the classical framework, which was based on the Boltzmann–Gibbs entropy and Maxwell–Boltzmann canonical distribution of particle velocities, into the respective entropy and canonical distribution associated with kappa distributions. In nonstationary states, the entropy defect similarly acts as a negative feedback, or reduction of the increase of entropy, preventing its unbounded growth toward infinity.

Brief Review on the Connection between the Micro-Canonical Ensemble and the Sq-Canonical Probability Distribution.

Non-standard thermostatistical formalisms derived from generalizations of the Boltzmann-Gibbs entropy have attracted considerable attention recently. Among the various proposals, the one that has been most intensively studied, and most successfully applied to concrete problems in physics and other areas, is the one associated with the Sq non-additive entropies. The Sq-based thermostatistics exhibits a number of peculiar features that distinguish it from other generalizations of the Boltzmann-Gibbs theory. In particular, there is a close connection between the Sq-canonical distributions and the micro-canonical ensemble. The connection, first pointed out in 1994, has been subsequently explored by several researchers, who elaborated this facet of the Sq-thermo-statistics in a number of interesting directions. In the present work, we provide a brief review of some highlights within this line of inquiry, focusing on micro-canonical scenarios leading to Sq-canonical distributions. We consider works on the micro-canonical ensemble, including historical ones, where the Sq-canonical distributions, although present, were not identified as such, and also more resent works by researchers who explicitly investigated the Sq-micro-canonical connection.

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.

Information Shift Dynamics Described by Tsallis q = 3 Entropy on a Compact Phase Space.

Recent mathematical investigations have shown that under very general conditions, exponential mixing implies the Bernoulli property. As a concrete example of statistical mechanics that are exponentially mixing we consider the Bernoulli shift dynamics by Chebyshev maps of arbitrary order N≥2, which maximizes Tsallis q=3 entropy rather than the ordinary q=1 Boltzmann-Gibbs entropy. Such an information shift dynamics may be relevant in a pre-universe before ordinary space-time is created. We discuss symmetry properties of the coupled Chebyshev systems, which are different for even and odd N. We show that the value of the fine structure constant αel=1/137 is distinguished as a coupling constant in this context, leading to uncorrelated behaviour in the spatial direction of the corresponding coupled map lattice for N=3.

Kaniadakis holographic dark energy in Brans–Dicke cosmology

By using the holographic hypothesis and Kaniadakis generalized entropy, which is based on relativistic statistical theory and modified Boltzmann–Gibbs entropy, we build Kaniadakis holographic dark energy (DE) model in the Brans–Dicke framework. We derive cosmological parameters of the Kaniadakis holographic DE model, with IR cutoff as the Hubble horizon, in order to investigate its cosmological consequences. Our study shows that even in the absence of an interaction between the dark sectors of the cosmos, the Kaniadakis holographic dark energy model with the Hubble radius as IR cutoff can explain the present accelerated phase of the universe expansion in the Brans–Dicke theory. The stability of the model, using the squared of sound speed, has been checked and it is found that the model is unstable in non-interacting case and can be stable for some range of model parameters within the interacting case.

Non-thermal particle acceleration from maximum entropy in collisionless plasmas

Dissipative processes cause collisionless plasmas in many systems to develop non-thermal particle distributions with broad power-law tails. The prevalence of power-law energy distributions in space/astrophysical observations and kinetic simulations of systems with a variety of acceleration and trapping (or escape) mechanisms poses a deep mystery. We consider the possibility that such distributions can be modelled from maximum-entropy principles, when accounting for generalizations beyond the Boltzmann–Gibbs entropy. Using a dimensional representation of entropy (related to the Renyi and Tsallis entropies), we derive generalized maximum-entropy distributions with a power-law tail determined by the characteristic energy scale at which irreversible dissipation occurs. By assuming that particles are typically energized by an amount comparable to the free energy (per particle) before equilibrating, we derive a formula for the power-law index as a function of plasma parameters for magnetic dissipation in systems with sufficiently complex topologies. The model reproduces several results from kinetic simulations of relativistic turbulence and magnetic reconnection.

Kaniadakis holographic dark energy and cosmology

We construct a holographic dark energy scenario based on Kaniadakis entropy, which is a generalization of Boltzmann-Gibbs entropy that arises from relativistic statistical theory and is characterized by a single parameter K which quantifies the deviations from standard expressions, and we use the future event horizon as the Infrared cutoff. We extract the differential equation that determines the evolution of the effective dark energy density parameter, and we provide analytical expressions for the corresponding equation-of-state and deceleration parameters. We show that the universe exhibits the standard thermal history, with the sequence of matter and dark-energy eras, while the transition to acceleration takes place at zapprox 0.6. Concerning the dark-energy equation-of-state parameter we show that it can have a rich behavior, being quintessence-like, phantom-like, or experience the phantom-divide crossing in the past or in the future. Finally, in the far future dark energy dominates completely, and the asymptotic value of its equation of state depends on the values of the two model parameters.

The Principle of Maximum Entropy and the Distribution of Mass in Galaxies

We do not have a final answer to the question of why galaxies choose a particular internal mass distribution. Here we examine whether the distribution is set by thermodynamic equilibrium (TE). Traditionally, TE is discarded for a number of reasons including the inefficiency of two-body collisions to thermalize the mass distribution in a Hubble time, and the fact that the mass distribution maximizing the classical Boltzmann–Gibbs entropy is unphysical. These arguments are questionable. In particular, when the Tsallis entropy that describes self-gravitating systems is used to define TE, the mass distributions that result (i.e., the polytropes) are physically sensible. This work spells out this and other arguments for TE and presents the polytropes and their properties. It puts forward empirical evidence for the mass distribution observed in galaxies to be consistent with polytropes. It compares polytropes with Sérsic functions and it shows how the DM halos resulting from cosmological numerical simulations become polytropes when efficient collisions are allowed. It also discusses pathways to thermalization bypassing two-body collisions. It finally outlines future developments including deciphering whether or not DM particles collide efficiently.

Analysis of temporal variations of seismicity through non-extensive statistical physics

SUMMARYIn recent years, there has been increasing interest in theoretical descriptions of seismicity in terms of statistical physics. Most aspects of these studies are encompassed by the concept of ‘intermittent criticality’, in which a region alternately approaches and retreats from a critical point. In this study, we analyse a descriptor of seismic activity that acts as a measure of the criticality of a system, such that its variations can be associated with changes in the state of the system. As some classical methods of analysis are not suitable for dealing with some of the features of complex systems such as the Earth’s crust, we derive the probability distribution of the magnitude by maximizing a non-extensive generalization of the Boltzmann–Gibbs entropy given by the Tsallis entropy. In particular, the shape parameter q of this distribution, called the entropic index, characterizes the subadditive q > 1 and superadditive q < 1 regimes. Following the Bayesian approach for parameter estimation, we examine the seismic activity that has affected two seismogenic areas in central Italy that were hit recently by destructive earthquakes: L’Aquila in 2009, and Amatrice–Norcia in 2016. To analyse in detail the variations of the q index and the entropy, we estimate these for time windows of a fixed number of events that shift at each new event. Both the q index and the Tsallis entropy show significant and lasting decreases before the first strong earthquake in the sequences, and sudden increases after them. This indicates that these quantities can be considered as indicators of the level of concentration of energy, and hence of the activation state of the systems. More reliable results need to come from further studies of different cases in different seismotectonic settings.