DOCSLIB.ORG

Supercanonical Probability Distributions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Supercanonical Probability Distributions Portland State University PDXScholar Physics Faculty Publications and Presentations Physics 8-8-2018 Supercanonical Probability Distributions John D. Ramshaw Portland State University, [email protected] Follow this and additional works at: https://pdxscholar.library.pdx.edu/phy_fac Part of the Physics Commons Let us know how access to this document benefits ou.y Citation Details Ramshaw, J. D. (2018). Supercanonical probability distributions. Physical Review E, 98(2), 020103. This Article is brought to you for free and open access. It has been accepted for inclusion in Physics Faculty Publications and Presentations by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. PHYSICAL REVIEW E 98, 020103(R) (2018) Rapid Communications Supercanonical probability distributions John D. Ramshaw Department of Physics, Portland State University, Portland, Oregon 97207, USA (Received 19 February 2018; published 8 August 2018) The canonical probability distribution describes a system in thermal equilibrium with an infinite heat bath. When the bath is finite the distribution is modified. These modifications can be derived by truncating a Taylor-series expansion of the entropy of the heat bath, but their form depends on the expansion parameter chosen. We consider two such expansions, which yield supercanonical (i.e., higher-order canonical) distributions of exponential and power-law form. The latter is identical in form to the “Tsallis distribution,” which is therefore a valid asymptotic approximation for an arbitrary finite heat bath, but bears no intrinsic relation to Tsallis entropy. DOI: 10.1103/PhysRevE.98.020103 −1 The probability that a system in thermal equilibrium with an and Cb(E) ≡ (∂Tb/∂E) . Note that entropy is dimensionless infinite heat bath occupies microstate i is given by the canonical when temperature is expressed in energy units, which is for- distribution mally equivalent to replacing Boltzmann’s constant by unity. −1 pi = Z exp{−βEi }, (1) If the heat bath is a proper thermodynamic system, then as it becomes infinitely large Tb(E) will approach a finite limiting where Ei is the energy of microstate i, β = 1/Tb, Tb is the temperature of the heat bath in energy units, and Z(β) ≡ value, while E itself and Cb(E) become infinite. The latter i divergence reflects the fact that it requires infinite energy to exp{−βEi } is the canonical partition function. Equation (1) is widely regarded as the single most important formula produce a finite change in the temperature of an infinite system. in statistical physics. Our purpose here is to explore the The standard derivation of the canonical distribution from corrections or modifications to it that arise when the heat bath Eq. (2) is based on the assumption that the heat bath is much larger than the system, so that Ei E0 and a Taylor series is finite. Such corrections represent an important aspect of the − = − thermodynamics and statistical mechanics of small systems expansion of Sb(E0 Ei ) log Wb(E0 Ei )inpowersof [1–3], which are experiencing a lively renaissance (e.g., [4–8]) Ei can reasonably be presumed to converge rapidly [13]. stimulated in part by rapid advances in nanotechnology. Much Equation (1) is then an immediate consequence of truncating of the literature on finite heat baths has been narrowly focused this expansion at the linear term [13–15]. However, this on the observation that simple ideal heat baths imply power-law procedure further implies that Eq. (1) is not exact but is in probability distributions of the same form as those obtained by principle modified by the higher-order terms in the expansion, extremizing the Tsallis entropy [9–11]. This correspondence at least for finite values of E0. It is instructive and enlightening has often been interpreted as evidence in support of the Tsallis to examine the form of those modifications when the quadratic E2 entropy, but that inference is unjustified [12]. term of order i [13–16] is retained. Thus we proceed to The probability distribution for a system in thermal equilib- explore the implications of the approximation rium with a finite heat bath is fundamentally given by [13–15] − = − − 2 Sb(E0 Ei ) Sb(E0 ) β1Ei β2Ei , (5) W (E − E ) p = b 0 i , i − (2) where j Wb(E0 Ej ) where E is the total energy of the system and bath together, and ∂Sb 1 0 β ≡ = , (6) W (E) is the number of bath states with energies in the interval 1 0 b ∂E 0 Tb (E,E+E), where E is a macroscopically small but finite 1 ∂2S β energy tolerance. Equation (2) is based on the hypothesis of β ≡− b = 1 , (7) 2 2 0 0 equal aprioriprobabilities, according to which those bath 2 ∂E 0 2Cb Tb states are presumed to be equally probable, and the further = 0 ≡ subscript zero denotes evaluation at E E0, Tb Tb(E0 ), presumption that the heat bath is sufficiently large and complex 0 ≡ =∼ and Cb Cb(E0 ). Combining Eqs. (2), (3), and (5), we obtain that Wb(E) ωb(E) E, where ωb(E) is a differentiable continuous approximation to the true discontinuous density = −1 − − 2 pi Z2 exp β1Ei β2Ei , (8) of bath states [15]. If the heat bath were isolated with energy E, its entropy S , temperature T , and heat capacity C would where b b b be given by ≡ − − 2 Z2(β1,β2 ) exp β1Ei β2Ei . (9) i Sb(E) = log Wb(E), (3) It follows from Eq. (7) that β2 vanishes for an infinite heat bath, 1 ∂Sb = , (4) whereupon Z2(β1,β2 ) reduces to Z(β) and Eq. (8) reduces to Tb(E) ∂E Eq. (1). If higher-order terms had been retained in Eq. (5), 2470-0045/2018/98(2)/020103(3) 020103-1 ©2018 American Physical Society JOHN D. RAMSHAW PHYSICAL REVIEW E 98, 020103(R) (2018) they too would clearly vanish in the same limit. The canonical Wb(E) in finite ideal systems, such as ideal gases or harmonic distribution therefore becomes exact when the heat bath is solids, is typically proportional to Eα, where the exponent α is infinite, but not otherwise. Its universality derives from the fact on the order of the number of particles or degrees of freedom that the only bath parameter it depends upon is the temperature. of the system [9,13–15]. This suggests that faster convergence According to Eq. (8), the probabilities pi are unequal, so the might be obtained by regarding Sb(E) = log Wb(E)asa entropy S of the system must be evaluated as the Boltzmann- function of log E rather than E itself. We therefore proceed Gibbs-Shannon (BGS) entropy to approximate Sb(E) − Sb(E0 ) by a truncated Taylor series expansion in powers of log E − log E . Since E will remain =− 0 0 S pi log pi , (10) large but finite, the lowest-order correction to the canonical i distribution is obtained by truncating the expansion at the linear which contrary to a common misconception is not uniquely or term. Thus we write inherently associated with a canonical (or any other) probabil- S (E) = S (E ) + α(log E − log E ) ity distribution [15]. Combining Eqs. (8) and (10), we obtain b b 0 0 α = + E = + 2 + Sb(E0 ) log , (17) S β1E β2E log Z2, (11) E 0 ≡ ≡ = where Q Qi i pi Qi . When β2 0, Eq. (11) reduces where to the standard canonical expression for the entropy S.The ∂S ∂S E quadratic term in Eq. (5) therefore introduces an additional ≡ b = b = = 0 α E0 E0β1 0 (18) 2 ∂ log E ∂E T term β2E into the entropy of the system. It follows from 0 0 b Eq. (9) that and use has been made of Eq. (6). Combining Eqs. (2), (3), (17), and (18), we obtain ∂ log Z2 =−E, (12) α ∂β − β1Ei 1 p = Z 1 1 − , (19) ∂ log Z i α 2 =−E2, (13) ∂β2 where which combine with Eq. (11) to imply that α β1Ej Z(α, β1) ≡ 1 − . (20) = + 2 α dS β1 dE β2 dE . (14) j Note, however, that E2 is not an independent variable, because Aside from minor notational differences, Eq. (19)isof 2 2 E and E are both functions of E0,sodE and dE are both precisely the same form as the so-called Tsallis distribu- simply proportional to dE0 and therefore to each other. Thus tion previously obtained by extremizing the Tsallis entropy dS is simply proportional to dE, so that the second law of [17], and subsequently legitimized by the observation that it thermodynamics remains valid in the form dE = TdS, which describes a system in equilibrium with an ideal finite heat defines and determines the temperature T of the system. Thus bath for which Wb(E) has the same power-law form [9]. The present derivation shows that this distribution is actually 1 = dS = + = 1 + R much more general: it is not restricted to ideal heat baths β1 Rβ2 0 1 0 0 , (15) T dE Tb 2Cb Tb but rather represents a first-order asymptotic approximation to S (E) for an arbitrary finite heat bath with an arbitrary where b quasicontinuous density of states. This derivation also provides dE2 dE2/dE exact expressions for the parameters α and β1 in terms of R ≡ = 0 . (16) the caloric equation of state Tb(E) of the heat bath and the dE dE/dE0 total energy E0. Moreover, the systematic expansion procedure The parameters β1 and β2 are functions of E0 via Eqs. (6) on which this approximation is based could in principle be and (7), so T can be evaluated by substituting Eq.
Load more

Recommended publications
  • Quantum Entanglement Inferred by the Principle of Maximum Tsallis Entropy
    Quantum entanglement inferred by the principle of maximum Tsallis entropy Sumiyoshi Abe1 and A. K. Rajagopal2 1College of Science and Technology, Nihon University, Funabashi, Chiba 274-8501, Japan 2Naval Research Laboratory, Washington D.C., 20375-5320, U.S.A. The problem of quantum state inference and the concept of quantum entanglement are studied using a non-additive measure in the form of Tsallis’ entropy indexed by the positive parameter q. The maximum entropy principle associated with this entropy along with its thermodynamic interpretation are discussed in detail for the Einstein- Podolsky-Rosen pair of two spin-1/2 particles. Given the data on the Bell-Clauser- Horne-Shimony-Holt observable, the analytic expression is given for the inferred quantum entangled state. It is shown that for q greater than unity, indicating the sub- additive feature of the Tsallis entropy, the entangled region is small and enlarges as one goes into the super-additive regime where q is less than unity. It is also shown that quantum entanglement can be quantified by the generalized Kullback-Leibler entropy. quant-ph/9904088 26 Apr 1999 PACS numbers: 03.67.-a, 03.65.Bz I. INTRODUCTION Entanglement is a fundamental concept highlighting non-locality of quantum mechanics. It is well known [1] that the Bell inequalities, which any local hidden variable theories should satisfy, can be violated by entangled states of a composite system described by quantum mechanics. Experimental results [2] suggest that naive local realism à la Einstein, Podolsky, and Rosen [3] may not actually hold. Related issues arising out of the concept of quantum entanglement are quantum cryptography, quantum teleportation, and quantum computation.
    [Show full text]
  • On the Exact Variance of Tsallis Entanglement Entropy in a Random Pure State
    entropy Article On the Exact Variance of Tsallis Entanglement Entropy in a Random Pure State Lu Wei Department of Electrical and Computer Engineering, University of Michigan, Dearborn, MI 48128, USA; [email protected] Received: 26 April 2019; Accepted: 25 May 2019; Published: 27 May 2019 Abstract: The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy. Keywords: entanglement entropy; quantum information theory; random matrix theory; variance 1. Introduction Classical information theory is the theory behind the modern development of computing, communication, data compression, and other fields. As its classical counterpart, quantum information theory aims at understanding the theoretical underpinnings of quantum science that will enable future quantum technologies. One of the most fundamental features of quantum science is the phenomenon of quantum entanglement. Quantum states that are highly entangled contain more information about different parts of the composite system. As a step to understand quantum entanglement, we choose to study the entanglement property of quantum bipartite systems. The quantum bipartite model, proposed in the seminal work of Page [1], is a standard model for describing the interaction of a physical object with its environment for various quantum systems.
    [Show full text]
  • Arxiv:1707.03526V1 [Cond-Mat.Stat-Mech] 12 Jul 2017 Eq
    Generalized Ensemble Theory with Non-extensive Statistics Ke-Ming Shen,∗ Ben-Wei Zhang,y and En-Ke Wang Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China (Dated: October 17, 2018) The non-extensive canonical ensemble theory is reconsidered with the method of Lagrange multipliers by maximizing Tsallis entropy, with the constraint that the normalized term of P q Tsallis' q−average of physical quantities, the sum pj , is independent of the probability pi for Tsallis parameter q. The self-referential problem in the deduced probability and thermal quantities in non-extensive statistics is thus avoided, and thermodynamical relationships are obtained in a consistent and natural way. We also extend the study to the non-extensive grand canonical ensemble theory and obtain the q-deformed Bose-Einstein distribution as well as the q-deformed Fermi-Dirac distribution. The theory is further applied to the general- ized Planck law to demonstrate the distinct behaviors of the various generalized q-distribution functions discussed in literature. I. INTRODUCTION In the last thirty years the non-extensive statistical mechanics, based on Tsallis entropy [1,2] and the corresponding deformed exponential function, has been developed and attracted a lot of attentions with a large amount of applications in rather diversified fields [3]. Tsallis non- extensive statistical mechanics is a generalization of the common Boltzmann-Gibbs (BG) statistical PW mechanics by postulating a generalized entropy of the classical one, S = −k i=1 pi ln pi: W X q Sq = −k pi lnq pi ; (1) i=1 where k is a positive constant and denotes Boltzmann constant in BG statistical mechanics.
    [Show full text]
  • Grand Canonical Ensemble of the Extended Two-Site Hubbard Model Via a Nonextensive Distribution
    Grand canonical ensemble of the extended two-site Hubbard model via a nonextensive distribution Felipe Américo Reyes Navarro1;2∗ Email: [email protected] Eusebio Castor Torres-Tapia2 Email: [email protected] Pedro Pacheco Peña3 Email: [email protected] 1Facultad de Ciencias Naturales y Matemática, Universidad Nacional del Callao (UNAC) Av. Juan Pablo II 306, Bellavista, Callao, Peru 2Facultad de Ciencias Físicas, Universidad Nacional Mayor de San Marcos (UNMSM) Av. Venezuela s/n Cdra. 34, Apartado Postal 14-0149, Lima 14, Peru 3Universidad Nacional Tecnológica del Cono Sur (UNTECS), Av. Revolución s/n, Sector 3, Grupo 10, Mz. M Lt. 17, Villa El Salvador, Lima, Peru ∗Corresponding author. Facultad de Ciencias Físicas, Universidad Nacional Mayor de San Marcos (UNMSM), Av. Venezuela s/n Cdra. 34, Apartado Postal 14-0149, Lima 14, Peru Abstract We hereby introduce a research about a grand canonical ensemble for the extended two-site Hubbard model, that is, we consider the intersite interaction term in addition to those of the simple Hubbard model. To calculate the thermodynamical parameters, we utilize the nonextensive statistical mechan- ics; specifically, we perform the simulations of magnetic internal energy, specific heat, susceptibility, and thermal mean value of the particle number operator. We found out that the addition of the inter- site interaction term provokes a shifting in all the simulated curves. Furthermore, for some values of the on-site Coulombian potential, we realize that, near absolute zero, the consideration of a chemical potential varying with temperature causes a nonzero entropy. Keywords Extended Hubbard model,Archive Quantum statistical mechanics, Thermalof properties SID of small particles PACS 75.10.Jm, 05.30.-d,65.80.+n Introduction Currently, several researches exist on the subject of the application of a generalized statistics for mag- netic systems in the literature [1-3].
    [Show full text]
  • Generalized Molecular Chaos Hypothesis and the H-Theorem: Problem of Constraints and Amendment of Nonextensive Statistical Mechanics
    Generalized molecular chaos hypothesis and the H-theorem: Problem of constraints and amendment of nonextensive statistical mechanics Sumiyoshi Abe1,2,3 1Department of Physical Engineering, Mie University, Mie 514-8507, Japan*) 2Institut Supérieur des Matériaux et Mécaniques Avancés, 44 F. A. Bartholdi, 72000 Le Mans, France 3Inspire Institute Inc., McLean, Virginia 22101, USA Abstract Quite unexpectedly, kinetic theory is found to specify the correct definition of average value to be employed in nonextensive statistical mechanics. It is shown that the normal average is consistent with the generalized Stosszahlansatz (i.e., molecular chaos hypothesis) and the associated H-theorem, whereas the q-average widely used in the relevant literature is not. In the course of the analysis, the distributions with finite cut-off factors are rigorously treated. Accordingly, the formulation of nonextensive statistical mechanics is amended based on the normal average. In addition, the Shore- Johnson theorem, which supports the use of the q-average, is carefully reexamined, and it is found that one of the axioms may not be appropriate for systems to be treated within the framework of nonextensive statistical mechanics. PACS number(s): 05.20.Dd, 05.20.Gg, 05.90.+m ____________________________ *) Permanent address 1 I. INTRODUCTION There exist a number of physical systems that possess exotic properties in view of traditional Boltzmann-Gibbs statistical mechanics. They are said to be statistical- mechanically anomalous, since they often exhibit and realize broken ergodicity, strong correlation between elements, (multi)fractality of phase-space/configuration-space portraits, and long-range interactions, for example. In the past decade, nonextensive statistical mechanics [1,2], which is a generalization of the Boltzmann-Gibbs theory, has been drawing continuous attention as a possible theoretical framework for describing these systems.
    [Show full text]
  • 0802.3424 Property of Tsallis Entropy and Principle of Entropy
    arXiv: 0802.3424 Property of Tsallis entropy and principle of entropy increase Du Jiulin Department of Physics, School of Science, Tianjin University, Tianjin 300072, China E-mail: [email protected] Abstract The property of Tsallis entropy is examined when considering tow systems with different temperatures to be in contact with each other and to reach the thermal equilibrium. It is verified that the total Tsallis entropy of the two systems cannot decrease after the contact of the systems. We derived an inequality for the change of Tsallis entropy in such an example, which leads to a generalization of the principle of entropy increase in the framework of nonextensive statistical mechanics. Key Words: Nonextensive system; Principle of entropy increase PACS number: 05.20.-y; 05.20.Dd 1 1. Introduction In recent years, a generalization of Bltzmann-Gibbs(B-G) statistical mechanics initiated by Tsallis has focused a great deal of attention, the results from the assumption of nonadditive statistical entropies and the maximum statistical entropy principle, which has been known as “Tsallis statistics” or nonextensive statistical mechanics(NSM) (Abe and Okamoto, 2001). This generalization of B-G statistics was proposed firstly by introducing the mathematical expression of Tsallis entropy (Tsallis, 1988) as follows, k S = ( ρ q dΩ −1) (1) q 1− q ∫ where k is the Boltzmann’s constant. For a classical Hamiltonian system, ρ is the phase space probability distribution of the system under consideration that satisfies the normalization ∫ ρ dΩ =1 and dΩ stands for the phase space volume element. The entropy index q is a positive parameter whose deviation from unity is thought to describe the degree of nonextensivity.
    [Show full text]
  • Generalized Simulated Annealing Algorithms Using Tsallis Statistics: Application to Conformational Optimization of a Tetrapeptide
    PHYSICAL REVIEW E VOLUME 53, NUMBER 4 APRIL 1996 Generalized simulated annealing algorithms using Tsallis statistics: Application to conformational optimization of a tetrapeptide Ioan Andricioaei and John E. Straub Department of Chemistry, Boston University, Boston, Massachusetts 02215 ~Received 18 December 1995! A Monte Carlo simulated annealing algorithm based on the generalized entropy of Tsallis is presented. The algorithm obeys detailed balance and reduces to a steepest descent algorithm at low temperatures. Application to the conformational optimization of a tetrapeptide demonstrates that the algorithm is more effective in locating low energy minima than standard simulated annealing based on molecular dynamics or Monte Carlo methods. PACS number~s!: 02.70.2c, 02.60.Pn, 02.50.Ey Finding the ground state conformation of biologically im- portant molecules has an obvious importance, both from the S52k( pilnpi ~3! academic and pragmatic points of view @1#. The problem is hard for biomolecules, such as proteins, because of the rug- when q 1. Maximizing the Tsallis entropy with the con- gedness of the energy landscape which is characterized by an straints → immense number of local minima separated by a broad dis- tribution of barrier heights @2,3#. Algorithms to find the glo- bal minimum of an empirical potential energy function for q ( pi51 and ( pi ei5const, ~4! molecules have been devised, among which a central role is played by the simulated annealing methods @4#. Once a cool- ing schedule is chosen, representative configurations of the where ei is the energy spectrum, the generalized probability allowed microstates are generated by methods either of the distribution is found to be molecular dynamics ~MD! or Monte Carlo ~MC! types.
    [Show full text]
  • Electron Power-Law Spectra in Solar and Space Plasmas
    Space Science Reviews manuscript No. (will be inserted by the editor) Electron Power-Law Spectra in Solar and Space Plasmas M. Oka · J. Birn · M. Battaglia · C. C. Chaston · S. M. Hatch · G. Livadiotis · S. Imada · Y. Miyoshi · M. Kuhar · F. Effenberger · E. Eriksson · Y. V. Khotyaintsev · A. Retino` Received: date / Accepted: date Abstract Particles are accelerated to very high, non-thermal energies in solar and space plasma environments. While energy spectra of accelerated electrons often exhibit a power law, it remains unclear how electrons are accelerated to high energies and what processes determine the power-law index d. Here, we review previous observations of the power-law index d in a variety of different plasma environments with a particular focus on sub-relativistic electrons. It appears that in regions more closely related to magnetic reconnection (such as the ‘above-the-looptop’ solar hard X-ray source and the plasma sheet in Earth’s magnetotail), M. Oka · C. C. Chaston Space Sciences Laboratory, University of California Berkeley 7 Gauss Way, Berkeley, CA 94720 Tel.: +1-510-642-1350 E-mail: [email protected] J. Birn Space Science Institute, Boulder, Colorado, USA Los Alamos National Laboratory, Los Alamos, NewMexico, USA M. Battaglia · M. Kuhar Institute of 4D Technologies, School of Engineering, University of Applied Sciences and Arts Northwestern Switzerland, CH-5210 Windisch,Switzerland S. M. Hatch Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire, USA G. Livadiotis Southwest Research Institute, San Antonio, TX-78238, USA S. Imada · Y. Miyoshi Institute for Space-Earth Environmental Research, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464- 8601 Aichi, Japan F.
    [Show full text]
  • Information Theoretical Properties of Tsallis Entropies
    Information theoretical properties of Tsallis entropies Shigeru FURUICHI1∗ 1Department of Electronics and Computer Science, Tokyo University of Science, Yamaguchi, 756-0884, Japan Abstract. A chain rule and a subadditivity for the entropy of type β, which is one of the nonadditive entropies, were derived by Z.Dar´oczy. In this paper, we study the further relations among Tsallis type entropies which are typical nonadditive entropies. The chain rule is generalized by showing it for Tsallis relative entropy and the nonadditive entropy. We show some inequalities related to Tsallis entropies, especially the strong subadditivity for Tsallis type entropies and the subadditivity for the nonadditive entropies. The subadditivity and the strong subadditivity naturally lead to define Tsallis mutual entropy and Tsallis conditional mutual entropy, respectively, and then we show again chain rules for Tsallis mutual entropies. We give properties of entropic distances in terms of Tsallis entropies. Finally we show parametrically extended results based on information theory. Keywords : Chain rule, subadditivity, strong subadditivity, entropic distance, Fano’s inequality, entropy rate, Tsallis mutual entropy and nonadditive entropy 2000 Mathematics Subject Classification : 94A17, 46N55, 26D15 PACS number: 65.40.Gr,02.50.-r,89.70.+c 1 Introduction For the probability distribution p(x) ≡ p(X = x) of the random variable X, Tsallis entropy was defined in [1] : q Sq(X) ≡− p(x) lnq p(x), (q 6= 1) (1) x X with one parameter q as an extension of Shannon entropy, where q-logarithm function is defined x1−q−1 by lnq(x) ≡ 1−q for any nonnegative real number x and q.
    [Show full text]
  • Generalized Maxwell Relations in Thermodynamics with Metric Derivatives
    entropy Article Generalized Maxwell Relations in Thermodynamics with Metric Derivatives José Weberszpil 1 ID and Wen Chen 2,* 1 Instituto Multidisciplinar–Departamento de Tecnologias e Linguagens, Universidade Federal Rural do Rio de Janeiro, UFRRJ-IM/DTL, Governador Roberto Silveira n/n, Nova Iguaçú, Rio de Janeiro 23890-000, Brazil; [email protected] or [email protected] 2 College of Mechanics and Materials, Hohai University, Jiangning District, Fuchengxi Road 8, Nanjing 211100, China * Correspondence: [email protected] Received: 5 July 2017; Accepted: 27 July 2017; Published: 7 August 2017 Abstract: In this contribution, we develop the Maxwell generalized thermodynamical relations via the metric derivative model upon the mapping to a continuous fractal space. This study also introduces the total q-derivative expressions depending on two variables, to describe nonextensive statistical mechanics and also the a-total differentiation with conformable derivatives. Some results in the literature are re-obtained, such as the physical temperature defined by Sumiyoshi Abe. Keywords: deformed maxwell relations; metric derivatives; fractal continuum; generalized statistical mechanics; Legendre structure 1. Introduction Maxwell relations are thermodynamic equations which establish the relations between various thermodynamic quantities (e.g., pressure, P, volume, V, Entropy, S, and temperature, T) in equilibrium thermodynamics via other fundamental quantities known as thermodynamical potentials—the most important being internal energy, U, Helmholtz free energy, F, enthalpy, H, and Gibbs free energy, G. A Legendre transform converts from a function of one set of variables to another function of a conjugate set of variables. With the use of Legendre transforms, we can study different equations of state. For example, with the adequate change in variables, U(S, V) transforms to F = F(V, T).
    [Show full text]
  • Physica a X-Ray Binary Systems and Nonextensivity
    Physica A 389 (2010) 854–858 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa X-ray binary systems and nonextensivity Marcelo A. Moret a,b,∗, Valter de Senna a, Gilney F. Zebende a,b, Pablo Vaveliuk a a Programa de Modelagem Computational - SENAI - CIMATEC 41650-010 Salvador, Bahia, Brazil b Departamento de Física, Universidade Estadual de Feira de Santana, CEP 44031-460, Feira de Santana, Bahia, Brazil article info a b s t r a c t Article history: We study the x-ray intensity variations obtained from the time series of 155 light curves Received 9 April 2009 of x-ray binary systems collected by the instrument All Sky Monitor on board the satel- Received in revised form 14 August 2009 lite Rossi X-Ray Timing Explorer. These intensity distributions are adequately fitted by q- Available online 20 October 2009 Gaussian distributions which maximize the Tsallis entropy and in turn satisfy a nonlinear Fokker–Planck equation, indicating their nonextensive and nonequilibrium behavior. From PACS: the values of the entropic index q obtained, we give a physical interpretation of the dynam- 05.10.Gg ics in x-ray binary systems based on the kinetic foundation of generalized thermostatistics. 05.90.+m 89.75.Da The present findings indicate that the binary systems display a nonextensive and turbulent 97.80.-d behavior. ' 2009 Elsevier B.V. All rights reserved. Keywords: Astrophysical sources Tsallis statistics In recent years, the generalized thermostatistical formalism (GTS) proposed by Tsallis [1] has received increasing attention due its success in the description of certain phenomena exhibiting atypical thermodynamical features.
    [Show full text]
  • A Maximum Entropy Framework for Nonexponential Distributions
    A maximum entropy framework for nonexponential distributions Jack Petersona,b, Purushottam D. Dixitc, and Ken A. Dillb,1 aDepartment of Mathematics, Oregon State University, Corvallis, OR 97331; bLaufer Center for Physical and Quantitative Biology, Departments of Physics and Chemistry, State University of New York, Stony Brook, NY 11794; and cDepartment of Systems Biology, Columbia University, New York, NY 10032 Contributed by Ken A. Dill, November 7, 2013 (sent for review June 26, 2013) P Probability distributions having power-law tails are observed in functional S½fpkg = − kpklog pk subject to constraints, such as a broad range of social, economic, and biological systems. We the known value of the average energy hEi. This procedure gives −βE describe here a potentially useful common framework. We derive the exponential (Boltzmann) distribution, pk ∝ e k ,whereβ distribution functions {pk} for situations in which a “joiner particle” is the Lagrange multiplier that enforces the constraint. This var- k pays some form of price to enter a community of size k − 1, where iational principle has been the subject of various historical justifi- costs are subject to economies of scale. Maximizing the Boltzmann– cations. It is now commonly understood as the approach that Gibbs–Shannon entropy subject to this energy-like constraint chooses the least-biased model that is consistent with the known predicts a distribution having a power-law tail; it reduces to the constraint(s) (39). Boltzmann distribution in the absence of economies of scale. We Is there an equally compelling principle that would select fat- show that the predicted function gives excellent fits to 13 different tailed distributions, given limited information? There is a large distribution functions, ranging from friendship links in social net- literature that explores this.
    [Show full text]