Nonextensive physics: a possible connection between generalized statistical mechanics and quantum groups

12 December 1994

PHYSICS LETTERS A

ELSEVIER

Physics Letters A 195 (1994) 329-334

Nonextensive physics: a possible connection between generalized statistical mechanics and quantum groups Constantino Tsallis Centro Brasileirode PesquisasFisicas, Rua Dr. XavierSigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil Received 2 June 1994;revised manuscript received 24 August 1994;accepted for publication 30 August 1994 Communicated by A.R. Bishop

Abstract

Two different formalisms have been recently developed for nonextensive physics, namely the generalized statistical mechanics and thermodynamics (characterized by q # l ) and the quantum groups (characterized by qc # 1 ). Through the discussion of the mean values of observables, we propose a (temperature dependent) connection between q and qG, and illustrate with bosonic oscillators. An extremely interesting tendency towards nonextensive physics keeps growing along recent years. This fact appears in at least two different areas, namely in statistical mechanics and quantum groups, through apparently independent paths. We establish here that these two nonextensivities can exactly compensate each other in such a way as to yield extensive mean values of the observables. Such a connection may play a relevant role in many (maybe all) physical systems where (spatial a n d / o r temporal) long-range interactions are present in such a relevant way as to lead to nonextensive behavior. This circumstance might indeed occur in very large systems (galaxies, globular clusters, Universe) where long-range interactions (like the gravitational forces ) are present, or in small "droplets" (in condensed matter, or nuclear, or elementary particle physics) whenever the range of the interactions is comparable to or larger than the (linear) size o f the system, or in irreversible processes related to microscopic long-memory functions (power-like, instead o f the exponential-like typical of short-memory), or in problems with (multi-) fractally structured space-time. Let us now briefly pres-

ent some crucial aspects o f the two nonextensive formalisms mentioned aboved. In what generalized statistical mechanics is concerned, a generalized entropy has been proposed as follows [ 1 ]

Sq=k 1-Z~p7 q-1

(qegq)

(1)

where p, is the probability associated with the ith microscopic state o f the system and k a conventional positive constant. The q--, l limit of Sq yields the wellknown Shannon expression - kB57~Pi In Pi (where we have used p7-~ ~ 1 + ( q - 1 ) In Pi), So satisfies, for all q > 0, the standard properties o f non-negativity, equiprobability, expansibility, concavity (which guarantees thermodynamic stability for the system), Htheorem [ 2-4 ], among others. However, if we have two independent systems r and Z" (i.e., ~6xw, =/~x® /~s,, where/~ denotes the density operator, whose eigenvalues are the {p~};Psvx, acts on the tensor product of the Hilbert spaces respectively associated with _r and Z"), we immediately verify pseudo-additivity, more precisely

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330

C Tsallis / Physics Letters A 195 (1994) 329-334

S~qVr' - S--~zq+ k k

+ ( 1 - q) S~ SZq ' k k "

(2)

In other words, Sq is generically extensive if and only if q= 1; otherwise, it is nonextensive (Sq should be clearly distinguished from the well known-Renyi entropy ( 1 - q ) - ~ In ~ p q , which is extensive but not necessarily concave; this lack of concavity makes the use of the Renyi entropy uncertain in physics, since the thermodynamic stability of the system would not be guaranteed). The connection to a consistently generalized equilibrium thermodynamics is established by extremizing Sq with the constraints Tr/~ = 1 and (for the canonical ensemble) [ 5 ] ( ~)q-=Tr~q ~ = Uo,

(3)

where 3f' is the Hamiltonian and Uo the generalized internal energy. This optimization yields the distribution [ 1,5 ] /~=

[l--fl(1--q) ~ ] l/(l--q) ,

z~

(4)

where Z q - T r [ 1 - f l ( 1 - q ) ~ ] '/(~-q) ,

(5)

with f l - 1/kTbeing a Lagrange parameter. It can be shown [ 5 ] that

1 OSq T-OUq' O

Fq=-Uv-TSa=-fl

1 Z~ - ¢ - 1 1-q '

Z~-~- 1

u~=- O p -1-- q In the limit q--, l, Eq. (4) recovers the well-known Boltzmann-Gibbs distribution/~ ocexp ( - fl ~:). Also, if we introduce the entropy operator ~ q - k('~ - ~ 1-q) / (l-q) (so denominated because it satisfies ( $ q ) q - T r ~ q $ q =Sq for arbitrary/~), we straightforwardly establish that

~"~' k

- - k~ +

+(q-l)

~' --k--"

(6)

The application of Tr/~:®/)~,, on both sides of this equality naturally recovers Eq. (2) (note that q - 1 in one equation becomes 1 - q in the other! ). Furthermore, this generalized statistics has been shown to satisfy appropriate forms of the Ehrenfest theorem (form-invariant, Vq) [6], yon Neumann

equation (form-invariant, Vq) [ 7 ], Jaynes information theory duality relations [6], fluctuation-dissipation theorem [ 8,9 ], Bogolyubov inequality [ 10 ], Langevin and Fokker-Planck equation [ 11 ], Callen's identity [ 12 ], quantum statistics [ 13 ], Onsager reciprocity theorem (form-invariant, Vq) [ 14], zeroth principle of thermodynamics (form-invariant, ¥q) [ 15 ], classical equipartition principle [ 16 ], among others. All these remarkable mathematical properties are verified under the simultaneous assumptions of entropy being generalized as given by Eq. ( 1 ), and the q-expectation values as given by Eq. (3). As its first physical application, this generalized scheme has enabled [17] to overcome the Boltzmann-Gibbs inability to provide finite mass for astrophysical systems within the polytropic model as studied by Chandrasekhar et al. (Balian and many others [ 18 ] have already pointed the possible need for a nonextensive entropy in astropysics; this is a very natural thing to happen whenever the long-range gravitational forces are essentially involved in the problem; indeed, in such a case, the range of the forces is at least comparable to the linear size of the system itself). Plastino and Plastino [ 17] have shown (by establishing a relationship between q and the polytropic index n) that the mass becomes finite if q sufficiently differs from unity. The same result was recently found, under quite general conditions (much larger than the polytropic model itself), by Aly [ 17 ]. As its second physical application, this generalization has recently enabled [ 19 ] to derive L6vy flights (relevant for a great variety of physical systems [20]; among them we have CTAB micelles dissolved in salted water [21 ] as well as heartbeat histograms [22] ) from an entropic optimization using physically acceptable a priori constraints, q being directly related to the fractal dimension y of the random motion (q= ( 3 + y ) / ( 1 +y)). Again, this overcomes a well-known inability [20] of q= 1 statistics. This particular example illustrates very transparently the usefulness of the q-expectation value. Indeed, let us for instance consider y= 1 (Cauchy-Lorentz distributions ) hence q = 2: ( x 2) l diverges, whereas ( x 2) 2 converges, thus being appropriate as an auxiliary condition. The same benefit is obtained for all long-tail distributions. After these first two, many other applications followed, such as a possible use for discussing self-orga-

C. Tsallis/ Physics LettersA 195 (1994) 329-334

nization in biological systems [23], a very performant simulated annealing optimization algorithm [24], learning in simple perceptrons [25], calculation of the nonionized hydrogen atom specific heat (not computable within Boltzmann-Gibbs statistics since quantities such as the partition function diverge, essentially due to the long-range Coulombian force) [26]. Let us now focus, on the other hand, on quantum groups (qG-deformations, qG-oscillators, qQ-calculus, where we use qG, instead of the traditional notation q, in order to avoid confusion with the present entropy index q). These are generalizations of Lie groups and algebras, which are recovered for qG--"1. They have provided applications in as varied areas as (see, e.g., Refs. [27-32] and references therein) inverse scattering method, vertex models, anisotropic spin chains Hamiltonians, knot theory, conformal field theory, heuristic phenomenology of deformed molecules and nuclei, noncommutative approach to quantum gravity and anyon physics. They have enabled, in particular, a formulation of quantum mechanics [ 28 ] in a discontinuous spacetime (where qG-- 1 plays the role of minimal lattice step, and (qG-1 )2 that of minimal time step). To illustrate the nonextensivity associated with quantum groups let us consider a bosonic q6-oscillator. Its Hamiltonian is given (see, e.g., Ref. [32] and references therein ) by Ae = ~oJA +A = ~co [~ b ,

(7)

(8)

being the qc-generalized number operator, 09> 0 is a characteristic frequency, and A + and A are, respectively, the creation and annihilation operators satisfying AA+-qgA+A='I,

(9)

(lO)

The eigenvalues of [~r]A are given by [ 32 ]

In the qG--' 1 limit, Eq. (12) recovers the well-known extensive expression [ n ]A = n. If we have two independent systems 27 and 27', we have ~ z u r , = ~ z + ~ z , , hence Eq. (8) implies a very suggestive relationship, namely

+ (q2 _ 1)[2Qz]a [NZ']A,

(1 3)

which presents a striking analogy with Eq. (6) ! Let us now address the main aim of the present paper, namely the basic question of a possible connection between q and qc. We shall use as a guideline the following trivial observation: within the generalized statistical mechanics, the q-expectation value of any observable 0 is given by [ 5-9,12 ] (())q-~TrfiqO=TrtO(tOq-lO)=-(~q-lO)

,

(14)

i.e., it can be thought as nonextensive statistics (:q) on an extensive operator ( 0 ) , or as extensive statistics (~) on a nonextensive operator (:q-10). We consider now a q~-deformed arbitrary extensive observable 0 (e.g., number of particles, energy, or any other observable which, were it not for the qodeformation, would be extensive) associated with two independent systems 27 and Z ' (/~vr, =/~r®/~r, with Trfizvz, =Tr/~z=Tr/~z, = 1 ). We have (15)

where 0 ~ ' is, by definition, the nonextensive correction associated with qG (although qG might be a complex number, we restrict our discussion to q G ~ , as in Ref. [28] ). Eq. (15) implies

( O ( Z U S ' ) )q = Tr (/~: ®/~,) [0(27) + O ( Z ' ) +(q2_l

-zz' ] )qqo

= [ T r / ~ 0 ( S ) ] [Tr/~q,] + [ T r / ~ , 0 ( Z ' ) ] [Trfi q] (16)

hence, by using (1) (i.e., Trbq= 1 + (1 -q)Sq), (0(27u27'))~= ( O ( s ) )q + ( 0 ( 2 7 ' ) ) q

and

[&g]=-g.

(12)

+ ( q ~ - 1 ) ( q~z~, q~)q,

as well as

[&A+]=2 +

( n = 0 , 1 , 2,...)

O(ZUS') = 0(27) + 0 ( Z ' ) + (qo -- 1 )Y]qG -zz, ,

where q~--I [N]A= q ~ - - I '

[n]A -- q~n-1 q~ _ 1

331

(11)

"J[-( 1 --q) [ ( O( S ) >qSft'~ - ( O( S t) ) qS~q]/k + ( q 2 - I) (0q~or')¢.

(17)

332

(7. Tsallis / Physics Letters A 195 (1994) 329-334

If we impose now that the mean values of the observable must be extensive whenever these are measurable quantities (i.e., that the q ~ 1 effect is exactly compensated by the qG ~: 1 e f f e c t ) we obtain

and

q- l =k(q~- l )

q~nX,__ 1 [nz,].4 = q ~ _ 1

( Off°~'>q qS~q'-~ <0 ( ~ ' ) >qa~q ' (18)

which yields the connection we were looking for. Genetically, q= 1 if and only ifq~ = 1. In the qG--' 1 limit, Eq. (18) becomes 1) < O ( Z ) > S { ' + < 0 ( Z ' ) > S f '

(nz, =0, 1, 2, ...),

(20)

(21)

hence [nzvr,]~ -

q--~'~-f

•

nqo _ [ R ~ I A [ ~ ' h (19)

i.e. q-locga-1

( n z = 0 , 1,2 .... )

(22)

Consequently, using Eq. (12), we have

2

q- 1~kn(qo-

q~"~- 1 [nz]a-- q ~ _ l

(19')

.

The situation is schematically indicated in Fig. 1. Before going on let us remark that the entropy operator ~q is not included among the operators 0 to which Eq. (18) refers. Indeed, it satisfies Eq. (16), consequently it is not extensive (unless q= 1 ). Let us now illustrate the present calculation with two bosonic oscillators of the type described by Hamiltonian (7). Let the observable 0 be [N]A. According to Eq. ( 12 ) we have

(23)

This is a good point for commenting the generic form we expect for OqZoS'.As illustrated in Eq. (23) (and in what follows from it), we expect to be t/qG .zr, f ( O ( Z ) ; q G ) f ( O ( Z ' ) ; q~), where f ( x ; q~) is analytic in qo at qG = 1, being genericallyf(x; 1 ) # 0. By denoting now t/q~' the eigenvalues of r~rq~ r ' we have that, in the qG--*1 limit, tlZlz'~nzns,

(nz, nz,=0, 1, 2, ...) ,

(24)

hence, using Eq. (19), q - 1 ~ ( q~ - 1 )<~>kr~/S~,

(25)

where we have used (~rrNz, > = (~rz) (Nz, > and By using the well-known quantum harmonic oscillator results ( ~ r ) = ( e ~ ° ' - l ) - ~ and S~/kB= flhmet~°'/(e ~ ' ° - 1 ) - l n ( e ~ ' ° - 1 ), we finally obtain qc-1 q - 1 ~ flhme~O~_ (e~,O_ 1 ) ln(e t~°~- 1 ) '

oo

.............. /

~Extensivephysics

00

qG (Ouo~um Groups)

Fig. 1. Typical (finite temperature) relation between q and qoBy "extensive physics" we mean Shannon entropy, BoltzmannGibbs statistics, continuous space-time, differential-equations physics, etc.

(26)

which is represented in Fig. 2 (where the kBT/hto--,O and k B T / h w - - , ~ asymptotic behaviors are indicated). We remark: (i) a temperature exists (kBT*/ ho9 ~- 2.31 ) for which (Oq/Oqo)r= 1; this, together with the fact that q= 1 for qo= 1, implies q - 1 ~ q c - 1, i.e., q can (asymptotically) equal q~ and they could be merged into a single parameter; (ii) at very low temperatures, i.e., T<< T* (where the system is practically not excited) q~ can vary a lot without making the thermodynamics appreciably nonextensive; (iii) at very high temperatures, i.e., T>> T* (where the system is highly excited), the slightest departure of qc from unity yields a highly nonextensive

C. Tsallis ~Physics Letters A 195 (1994) 329-334

that nonextensive statistics can exactly compensate nonextensive mechanics in such a way as to provide extensive mean values of the observables. This fact might (together, for instance, with the re-analysis of the available data on the universe background radiation and/or other astrophysical black-bodies, as well as the re-analysis of the data relative to the existence of dark matter) help the understanding of one among the most puzzling problems of contemporary science, namely the deep nature of space-time.

20

IE~

15

I0 05

I10.00

I

k.T "it'/"

I

",,i"'" 2

3

333

•

4

5

keT/h~ Fig. 2. Temperature dependent relation between q and qo in the region q~-qa~- 1 for bosonic oscillators (kBT*/h~o~-2.31). The asymptotic behaviors for T<< T* and T>> T* are, respectively, given by kBT/hto and ( kBT/hog ) /In ( kBT/hto ).

thermodynamics; (iv) as we can see in Eqs. (6) and (13) both q and qG act in a monotonic way; furthermore, both can vary between zero and infinity; these facts make plausible that, for any finite temperature, q monotonically increases from zero to infinity when qo increases from zero to infinity. The Plastino and Plastino discussion [ 17 ] of the q= 1 paradox of the polytropic model for stellar systems was done in the classical limit ( h ~ 0 ) , where it seems now reasonable to expect that a value of qG slightly different from unity would imply a value of q quite different from unity, as they indeed found. Since it seems possible to interpret qÙ > Fas a discontinuous (or, perhaps, continuous but not differentiable) space-time [28] (see also Ref. [33] ), we should certainly not exclude the possibility for astrophysical systems being privileged candidates for exploring the deepest effects of gravitation in nature, including nonextensivity of the entropy. Also, along similar lines, Bacry has recently argued [ 34 ] that q6 slightly different from one could be enough for explaining the frequently discussed "discrepancy" at the origin of the so-called dark matter. Let us now synthesize the present work. Although the connection between generalized statistical mechanics and quantum groups was done on effective or phenomenological grounds (i.e., the relation between q and qG depends on temperature for a system in thermal equilibrium), it was established through a remarkably simple and generic assumption, namely

The author is very indebted to E.M.F. Curado, M.R-Monteiro and I. Roditi for stressing his attention onto quantum groups as well as for very valuable discussions. He also acknowledges useful remarks from A.C.N. Magalh~es.

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