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Thermodynamics from First Principles: Correlations and Nonextensivity

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Thermodynamics from First Principles: Correlations and Nonextensivity Thermodynamics from first principles: correlations and nonextensivity S. N. Saadatmand,1, ∗ Tim Gould,2 E. G. Cavalcanti,3 and J. A. Vaccaro1 1Centre for Quantum Dynamics, Griffith University, Nathan, QLD 4111, Australia. 2Qld Micro- and Nanotechnology Centre, Griffith University, Nathan, QLD 4111, Australia. 3Centre for Quantum Dynamics, Griffith University, Gold Coast, QLD 4222, Australia. (Dated: March 17, 2020) The standard formulation of thermostatistics, being based on the Boltzmann-Gibbs distribution and logarithmic Shannon entropy, describes idealized uncorrelated systems with extensive energies and short-range interactions. In this letter, we use the fundamental principles of ergodicity (via Liouville's theorem), the self-similarity of correlations, and the existence of the thermodynamic limit to derive generalized forms of the equilibrium distribution for long-range-interacting systems. Significantly, our formalism provides a justification for the well-studied nonextensive thermostatistics characterized by the Tsallis distribution, which it includes as a special case. We also give the complementary maximum entropy derivation of the same distributions by constrained maximization of the Boltzmann-Gibbs-Shannon entropy. The consistency between the ergodic and maximum entropy approaches clarifies the use of the latter in the study of correlations and nonextensive thermodynamics. Introduction. The ability to describe the statistical tion [30] or the structure of the microstates [27]. Another state of a macroscopic system is central to many ar- widely-used approach is to generalize the MaxEnt prin- eas of physics [1{4]. In thermostatistics, the statistical ciple to apply to a different entropy functional in place of BGS state of a system of N particles in equilibrium is de- S (fwzg). At the forefront of this effort is the so-called Ts scribed by the distribution function wz over z where q-thermostatistics based on Tsallis' entropy Sq (fwzg) / P q z = (fq1; ··· ; qN g; fp1; ··· ; pN g) defines a point (the (1− z wz)=(q −1) and expressing the constraints as av- q microstate) in the concomitant 6N-dimensional phase erages with respect to escort probabilities fwzg [4,8]. space. The central question addressed in this letter is, q-thermostatistics is known to describe a wide range of what is the generalized form of wz for a composite sys- physical scenarios [3, 13, 20, 23, 31{46], including high-Tc tem at thermodynamic equilibrium that features corre- superconductivity, long-range-interacting Ising magnets, lated subsystems? turbulent pure-electron plasmas, N-body self-gravitating This question has been the subject of intense research stellar systems, high-energy hadronic collisions, and low- for more than a century [3{14, 16{23]. Correlations dimensional chaotic maps. The approach has also been and nonextensive energies are associated with long-range- refined and extended [3, 47{50]. interacting systems, which are at the focus of much of A contentious issue, however, is that the Tsallis en- the effort (in particular see [13, 18{20, 22]). The most tropy does not satisfy Shore and Johnson's system- widespread approach to finding wz is the maximum en- independence axiom [26{29]. Although Jizba et al. [22] tropy (MaxEnt) principle introduced by Jaynes [6,7] recently made some headway towards a resolution, objec- on the basis of information theory. The principle en- tions remain [51], and the generalization of the MaxEnt tails making the least-biased statistical inferences about principle continues to be controversial. This brings into a physical system consistent with prior expected values focus the need for an independent approach to our central of a set of its quantities ff¯(1); f¯(2);:::; f¯(l)g. It requires question. the distribution wz to maximize the Gibbs-Shannon We propose an answer by introducing a general for- GS (GS) logarithmic entropy functional S (fwzg) subject malism based on ergodicity [1] for deriving equilibrium P (i) ¯(i) GS to constraints fz wz = f . Here, S (fwzg) = distributions, including ones for correlated systems. Pre- P z −k z wz ln(wz) for constant k > 0. Despite outstanding viously this derivation was thought impossible as corre- success [24, 25] in capturing thermodynamics of weakly- lations have been linked with nonergodicity (see e.g. [4], interacting gases, the principle { in its original form { p. 68 and p. 320). However, we circumvent these diffi- does not describe correlated systems. Attempts have culties by showing how the self-similarity of correlations been made to generalize the principle, however, as there can be invoked to derive the distribution wz under well- is no accepted method for doing so, controversy has en- defined criteria. Then we show how to employ the Max- sued [26{28]. Ent principle consistently with correlations encoded as a self-similarity constraint function. After comparing our arXiv:1907.01855v4 [cond-mat.stat-mech] 16 Mar 2020 One approach is based on the extension of the MaxEnt principle by Shore and Johnson [29], and entails gener- results with previous works, we present a numerical ex- alizing the way knowledge of the system is represented ample for completeness and end with a conclusion. by constraints [25, 29]. Information about correlations Key ideas. Our approach rests on two key ideas. are incorporated, e.g. by modifying the partition func- (i) Liouville's theorem for equilibrium systems. Con- 2 sider a generic, classical, dynamical system described by B, where wA and wB are not the marginals of wAB for zA zB zAB Hamiltonian H and phase-space distribution w(z; t). Be- q 6= 1. As this result has previously been regarded [4] as ing a Hamiltonian system guarantees the incompressibil- incompatible with Eq. (2), it shows that Liouville's the- ity of phase-space flows, which is represented via Liou- orem has an underappreciated application for describing ville's equation [1,5] by w being a constant of motion highly correlated systems. along a trajectory, i.e. Finding a generalized distribution. With these ideas in mind, we derive our main results for a composite, @w @w dw + z_ · rw = + fw; Hg = = 0; (1) self-similar, classical Hamiltonian system in thermody- @t @t dt namic equilibrium. For brevity, we explicitly treat a com- where f ; g denotes the Poisson bracket. Imposing the posite system AB composed of two subsystems A and equilibrium condition @w=@t = 0 implies B, although our results are easily extendable to com- positions involving an arbitrary number of macroscopic @w=@t = −{w; Hg = 0; (2) subsystems. Let the tuples (wAB ;HAB ), (wA ;HA ), zAB zAB zA zA (wB ;HB ) denote the composite and isolated equilib- zB zB i.e. the existence of a steady-state, w(z; t) = wz 8 t. Any rium distributions and Hamiltonians of the composite Hamiltonian ergodic system at equilibrium will obey this AB, and separate A, B subsystems, respectively; wX zX condition. A possible solution of Eq. (2) is given by is the equilibrium probability that system X is in phase w = af(bH + c), where f(·) is any differentiable func- z z space point zX . The following criteria encapsulate prop- tion, for macrostate-defining and normalisation constants erties of the system required for subsequent work. They a, b and c (see e.g. [1, 56]). We only consider solutions of immediately lead to two key Theorems, which generalize this form, which is equivalent to invoking the fundamen- thermostatistics. tal postulate of equal a priori probabilities for accessible Criterion I { Thermodynamic limit: Consider a se- microstates [1]. For brevity, we shall write the solution quence of systems A1;A2; ··· for which the solution as (n) Eq. (3) for the nth term is given by wAn = G (HAn ). zAn An zAn A sequence that increases in size is said to have a ther- wz = GX (Hz) (3) modynamic limit if G(n) attains a limiting parametrized An and leave the dependence on the parameters a, b and c form as A becomes macroscopic, i.e. if G(n) !G as n An A as being implicit in the label X. n ! 1. The distribution wA = G (HA ), where the de- zA A zA (ii) Deriving equilibrium distributions. Consider the pendence on system, macrostate, and normalisation con- equilibrium distributions wA, wB and Hamiltonians HA, B stants is implicit in the label A on GA, is said to represent H of two isolated, conservative, short-range-interacting the thermostatistical properties of the physical material systems labelled A and B where comprising A in the thermodynamic limit. AB A B Examples of limiting forms include the BG distri- Hz = Hz + Hz ; (4) AB A B bution G(H ) = ae−bHz and the Tsallis distribution AB A B z w = w w (5) −bHz zAB zA zB G(Hz) = aeq for macrostate-dependent parameter b and normalization constant a. are the total Hamiltonian and joint distributions, respec- Criterion II { Compositional self-similarity: We define tively, for the isolated, composite system AB at equi- a system as having compositional self-similarity if there librium, and z ≡ (z ; z ). From (3), each distribu- AB A B exist mapping functions and such that the composite tion is a function of its respective Hamiltonian. Taken C H equilibrium distribution and energy of macroscopic AB together, Eqs. (3)-(5) imply the general solution wX zX are related to the isolated equilibrium distribution and is the Boltzmann-Gibbs (BG) distribution G (HX ) = X zX bH energy of macroscopic A and B by the following relations ae zX for macrostate-dependent constants a, b and X = A; B and AB. This well-known result can easily be generalised. For AB A B Hz = H(Hz ;Hz ) (7a) example, replacing Eq. (5) with AB A B wAB = (wA ; wB ); 0 ≤ ≤ 1 ; (7b) zAB C zA zB C wAB = wA ⊗ wB ; (6) zAB zA q zB for all zAB, where H embodies the nature of the interac- where ⊗q is the q-product [3], correspondingly implies tions, and C embodies the nature of the correlations.
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