The role of constraints within generalized

The role of constraints within generalized

(Parte 1 de 4)

The role of constraints within generalized nonextensive statistics a Centro Brasileiro de Pesquisas F sicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil b Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DK-2100 Denmark c Departamento de F sica, Universidade Estadual de Maring a, Avenida Colombo 5790, 87020-900 { Maring a { PR, Brazil d Facultad de Ciencias Astronomicas y Geof sicas, Universidad Nacional de La Plata, C.C. 727, 1900 La Plata, Argentina e Comision Nacional de Investigaciones Cient cas y Tecnicas, Conicet, Argentina

Received 31 August 1998


The Gibbs{Jaynes path for introducing statistical mechanics is based on the adoption of a speci c entropic form S and of physically appropriate constraints. For instance, for the usual canonical ensemble, one adopts (i) S1 = −k P i pi ln pi, (i) P i pi = 1, and (i) P i pi i =

U1 (f ig eigenvalues of the Hamiltonian; U1 internal energy). Equilibrium consists in optimizing S1 with regard to fpig in the presence of constraints (i) and (ii). Within the

(q ! 1 reproduces S1), (i) is maintained, and (ii) is generalized in a manner which might involve pqi . In the present e ort, we analyze the consequences of some special choices for (i), and their formal and practical implications for the various physical systems that have been studied in the literature. To illustrate some mathematically relevant points, we calculate the speci c heat respectively associated with a nondegenerate two-level system as well as with the classical and quantum harmonic oscillators. c© 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

It is nowadays quite well known that a variety of physical systems exists for which the powerful (and beautiful!) Boltzmann{Gibbs (BG) statistical mechanics and standard thermodynamics present serious di culties or anomalies, which can occasionally achieve the status of just plain failures. Within a long list, we may mention systems involving long-range interactions (e.g., d=3 gravitation) [1,40], long-range microscopic

Correspondence address: Centro Brasileiro de Pesquisas F sicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil.; e-mail:

0378-4371/98/$ { see front matter c© 1998 Elsevier Science B.V. All rights reserved. PII: S0378-4371(98)00437-3 memory (e.g., non-Markovian stochastic processes, on which quite little is known, in fact) [2], and, generally speaking, conservative (Hamiltonian) or dissipative systems which in one way or another involve a relevant space{time (hence, a relevant phase space) which has a (multi)fractal-like structure. For instance, pure-electron plasma two-dimensional turbulence [3], L evy anomalous di usion [4,41], granular systems [5], phonon{electron anomalous thermalization in ion-bombarded solids ([6] and references therein), solar neutrinos [7], peculiar velocities of galaxies [8], inverse bremsstrahlung in plasma [9] and black holes [10] clearly appear to be (in some cases), or could possibly be (in others), concrete examples.

To overcome at least some of these di culties a proposal has been advanced, one decade ago, by one of us [1] (see also [12{14]), which is based on a generalized entropic form, namely

Sq = k i=1 pq i where k is a positive constant and W is the total number of microscopic possibilities of the system (for the q< 0 case, care must be taken to exclude all those possibilities whose probability is not strictly positive, otherwise Sq would diverge; such care is not necessary for q> 0). This expression recovers the usual BG entropy (−k PW i=1 pi ln pi) in the limit q ! 1. The entropic index q (intimately related to and determined by the microscopic dynamics, as we shall mention later on) characterizes the degree of nonextensivity re ected in the following pseudo-additivity entropy rule:

where A and B are two independent systems in the sense that the probabilities of A + B factorize into those of A and of B. We immediately see that, since in all cases

Sq>0;q <1;q =1 and q> 1 respectively correspond to superadditivity (superextensivity), additivity (extensivity) and subadditivity (subextensivity).

Another important (since it eloquently exhibits the surprising e ects of nonextensivity) property is the following. Suppose that the set of W possibilities is arbitrarily separated into two subsets having respectively WL and WM possibilities (WL+WM=W).

i=1pi and pM PW i=W +1 pi, hence pL + pM = 1. It can then be straightforwardly established that [12]

Sq(fpig)= Sq(pL;pM)+ pqL Sq(fpi=pLg)+ pqM Sq(fpi=pMg) ; (3) where the sets fpi=pLg and fpi=pMg are the conditional probabilities. This would precisely be the famous Shannon’s property were it not for the fact that, in front of the entropies associated with the conditional probabilities, appear pqL and pqM instead of pL and pM. This fact will play, as we shall see later on, a central role in the whole generalization of thermostatistics. Indeed, since the probabilities fpig are generically numbers between zero and unity, pqi >pi for q< 1 and pqi <pi for q> 1, hence q< 1 and q> 1 will respectively privilegiate the rare and the frequent events. This simple property lies at the heart of the whole proposal. Santos has recently shown [14], strictly following along the lines of Shannon himself, that, if we assume (i) continuity of the entropy, (i) increasing monotonicity of the entropy as a function of W in the case of equiprobability, (i) property (2), and (iv) property (3), then only one entropic form exists, namely that given in de nition (1).

Another interesting property is the following. The Boltzmann{Gibbs entropy S1 satis es the following relation:

d WX

Moreover, Jackson introduced in 1909 [15,42] the following generalized di erential operator (applied to an arbitrary function f(x)):

i=1 pq i

This property provides some insight into the generalized entropic form Sq. Indeed, the inspiration for its use in order to generalize the usual thermal statistics came from multifractals [1], and its applications concern, in one way or another, systems which exhibit scale invariance. Therefore, its connection with Jackson’s di erential operator appears to be kind of natural. Indeed, this operator \tests" the function f(x) under dilatation of x, in contrast to the usual derivative, which \tests" it under translation of x.

Finally, let us close the present set of properties reminding that Sq has, with regard to fpig, a de nite concavity for all values of q (Sq is always concave for q> 0

values of q.

A wealth of works has shown that the above nonextensive entropic form leads, through further assumptions on the constraints (which we shall detail later on), to a statistical mechanics which retains much of the formal structure of the standard theory. Indeed, important properties such as the Legendre thermodynamic structure, the H-theorem (macroscopic time irreversibility), the Ehrenfest theorem (correspondence principle), the Onsager reciprocity theorem (microscopic time reversibility), the Kramers and Wannier relations (causality), Einstein’ 1910 reversal of Boltzmann’s formula (factorization of the likelihood function), Bogolyubov inequality and thermodynamic stability (e.g., a de nite sign for the speci c heat), among others, are q-invariant [1{13,17,43{47]. Further, it has been applied to a series of anomalous systems, including those cited above (self-gravitating astrophysical systems [18,48], turbulence [19,49], L evy type [20,50{52] as well as correlated-type [21,53{56] anomalous di usions, solar neutrinos [2,57], peculiar velocities of galaxies [23], bremsstrahlung in plasma [24], cosmology [25,58], long-range magnetic and uid-like systems [26,59{ 64], low-dimensional (at the edge of chaos) [27,65,6] as well as high-dimensional (self-organized criticality [28]) [29,67] (see also [68]) nonlinear dissipative systems, Hamiltonian (conservative) nonlinear dynamical systems [30], and also to optimization techniques (generalized simulated annealing) [31,69{75], among others.

2. Choices of the internal energy constraint

Let us now right away address the constraints we have referred to earlier in order to establish an equilibrium thermal statistics which generalizes the Boltzmann{Gibbs one.

To do this we will extremize (maximize for q> 0, since in this case Sq is concave in the fpig, and minimize for q< 0, since in this case it is convex instead; see [1]) Sq in the presence of appropriate constraints. For the microcanonical ensemble (isolated system) there will be only one constraint (already mentioned), namely

Consequently, the optimization of Sq immediately yields equiprobability, i.e., pi = 1=W 8i, hence [1]

Sq = k lnq W (8) with [32,75]

thus generalizing the celebrated Boltzmann’s formula. Since it will be useful later on, let us right now mention that the inverse function of lnq x is

The subtleties of the formalism we are describing concern the case in which the system is in thermal contact with a reservoir, i.e., the canonical ensemble (and also, of course, the grand-canonical ones, which we shall not address herein because their discussion is analogous to the case of the canonical ensemble, to which the present e ort is dedicated). Three plausible choices will be considered herein, namely referred to as the rst, second and third choices.

The rst choice was introduced and worked out in [1] (and has been used, for some speci c systems, only a couple of times since then, basically in the early times just after the proposal). It consists in using, besides the norm constraint (7), the following one: WX

where the superindex (1) stands for rst choice and the f ig are the eigenvalues of the (quantum) Hamiltonian of the system (with the chosen boundary conditions). In other words, the standard de nition of internal energy has been maintained within the generalization. By recourse to the usual extremalization technique, we obtain [1]

where it must be stressed that here, in spite of its appearence, is not the Lagrange multiplier associated to the internal energy constraint (it is in fact for this reason that we note it with instead of the usual notation ). This expression (i) of course recovers the usual Boltzmann{Gibbs statistics (pi _ e− )i nt he q ! 1 limit, and (i) depends on the microscopic energy as a power law instead of the familiar exponential.

These two important features will be present in all three choices we are discussing in this paper.

It quickly became evident, however, that this choice for the internal energy was inadequate for handling the serious mathematical di culties (undesirable divergences) present in a variety of anomalous systems such as the L evy anomalous superdi usion (we remind that the second moment associated with L evy distributions diverges). Then the second choice (introduced in [1], rst worked out in [12], and since then intensively studied and used: see bibliography indicated in [13]) became the natural way out of the di culties. It goes as follows. The internal energy constraint is postulated to be

where the superindex (2) stands for second choice. The extremization of Sq now yields [12]

with the generalized partition function given by

which coincides with the result produced by the rst choice excepting for the fact that

)]2−q, 8i). Now and from now on is, as usual and in contrast with , the Lagrange parameter associated with the internal energy. This distribution presents a cut-o (i.e., vanishing probabilities for energy levels high enough to produce a negative value for the argument of the eq function) for all values of q< 1, whereas this phenomenon occurred, in the rst choice, for q> 1. The present equilibrium distribution can be conveniently written as

which formally resembles the Boltzmann{Gibbs result (in fact, this Boltzmann{Gibbslike appearance will be present along the entire formalism). By introducing T 1=(k ) it is straightforward to prove [12]

T = @Sq


As we see, the usual thermodynamics Legendre structure remains valid for all values of q. This is still true, in fact, for quite general forms for the entropy and the constraints (see [3,7]). In other words, although de nitively welcome for our present, nonextensive case, the validity of the Legendre structure cannot be considered as a severe criterion of physical plausibility. However, it is also within this second choice (Eq. (13)) that the long list of important q-invariant properties mentioned earlier (going from the H-theorem to the thermodynamic stability) has been proved. And, since the validity of all those properties is far from being a trivial mathematical feature, the fact that they indeed hold certainly is reassuring in terms of theoretical physics. These properties, together with the success for handling a variety of anomalies in speci c physical systems, probably are at the basis of the popularity the present formalism exhibits in the literature. Nevertheless, this second choice implies three consequences which, although in principle possible, are no doubt strange in terms of the theoretical physical grounds we are familiar with. The rst unfamiliar consequence is that the distribution given by Eqs. (14) and (15) is not invariant through uniform translation of the energy spectrum f ig, i.e., the thermodynamical results depend on the choice of the origin of energies. In practice, this has not been a great problem because quite systematically authors have chosen the ground energy (lowest eigenvalue of the Hamiltonian) as the zero point for the energies, which no doubt makes sense. But, on theoretical grounds, we believe we all agree that this feature is in some sense disturbing. The second unfamiliar consequence is that the manner in which the generalized inter- nal energy U(2)q is de ned suggests that all observables in the theory should appear in the same way, this is to say through the so-called q-expectation values O(2)q h Oiiq PW i=1 pqi Oi where fOig are the eigenvalues associated with an arbitrary observable O

(here and later on we are assuming, for simplicity, that the observable commutes with the density operator; if it does not, the full quantum formalism has to be used).

Although it is clear that by no means the de nition of these quantities violates theory of probabilities (indeed, hOiiq = hpq−1i Oii, hence the q-expectation values of familiar quantities are nothing but standard mean values of say unfamiliar quantities), the math- ematical fact that h1iq is generically not equal to unity is not easy to interpret (unless one is ready to accept that entropic nonextensivity results in an \e ective" gain or loss of norm). In spite of the fact that the q-generalized Ehrenfest theorem (mentioned earlier) guarantees that, whenever the observable O commutes with the Hamiltonian, the q-expectation value hOiiq is a constant of motion, this kind of non-preservation of the norm is again something to worry about.

Finally, the third unfamiliar consequence is that, if two systems A and B satisfy

which generically di ers from U(2)q (A)+ U(2)q (B). In other words, the rst principle of thermodynamics (energy conservation) does not preserve macroscopically the same form it has microscopically. One can of course object that, if we are willing to consider nonadditivity of the entropy (see Eq. (2)), why is it so strange to accept the same for the energy? The point is that entropy is an informational quantity whereas energy is a mechanical one. Since the present formalism does not at all alter things at the level of the dynamics, it is kind of against intuition the above-mentioned nonadditive composition of internal energies.

We are now in position to describe the third choice for the internal energy constraint.

(Parte 1 de 4)