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Physica A 261 (1998) 534–
The role of constraints within generalized
nonextensive statistics
Constantino Tsallisa;b;∗, Renio S. Mendesc^ , A.R. Plastino a;d;e
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 Maringa, Avenida Colombo 5790, 87020-900 – Maringa – 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
Abstract
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) S 1 = −k
i pi^ ln^ p^ i^ , (ii)^
i pi^ = 1, and (iii)^
i pi^ ^ i^ = U 1 ({ (^) i } ≡ eigenvalues of the Hamiltonian; U 1 ≡ internal energy). Equilibrium consists in optimizing S 1 with regard to {p (^) i } in the presence of constraints (ii) and (iii). Within the recently introduced nonextensive statistics, (i) is generalized into S (^) q = k[1 −
i p^
qi ]=[q − 1]
(q → 1 reproduces S 1 ), (ii) is maintained, and (iii) is generalized in a manner which might involve p qi. In the present e ort, we analyze the consequences of some special choices for (iii), 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.
- 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 stan- dard thermodynamics present serious diculties 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 Fsicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil.; e-mail: [email protected].
0378-4371/98/$ – see front matter ©c 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 4 3 7 - 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], Levy 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 diculties a proposal has been advanced, one decade ago, by one of us [11] (see also [12–14]), which is based on a generalized entropic form, namely
Sq = k
∑W
i=1 p^
q i q − 1
( W
i=
pi = 1; q ∈ R
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
∑W
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:
S (^) q(A + B)=k = [Sq(A)=k] + [Sq(B)=k] + (1 − q)[Sq(A)=k][Sq(B)=k] ; (2)
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 S (^) q¿ 0 ; q ¡ 1 ; q = 1 and q ¿ 1 respectively correspond to superadditivity (superexten- sivity), additivity (extensivity) and subadditivity (subextensivity). Another important (since it eloquently exhibits the surprising e ects of nonexten- sivity) 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 ). We de ne p (^) L ≡
∑WL
i=1 pi^ and^ pM^ ≡^
∑W
i=W (^) L+1 pi, hence^ pL^ +^ pM^ = 1. It can then be straightforwardly established that [12]
S (^) q({pi}) = S (^) q(pL ; pM ) + p qL Sq({pi =p (^) L}) + p qM Sq({pi =p (^) M }) ; (3)
where the sets {p (^) i =p (^) L} and {pi =p (^) M } 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 p qL and p qM instead of p (^) L 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 {pi} are generically numbers between zero and unity, p qi ¿ pi for q ¡ 1 and p qi ¡ 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],
plasma [24], cosmology [25,58], long-range magnetic and uid-like systems [26,59– 64], low-dimensional (at the edge of chaos) [27,65,66] 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.
- 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 {p (^) i}, and minimize for q ¡ 0, since in this case it is convex instead; see [11]) Sq in the presence of appropriate constraints. For the microcanonical ensemble (isolated system) there will be only one constraint (already mentioned), namely
∑^ W
i=
pi = 1 : (7)
Consequently, the optimization of Sq immediately yields equiprobability, i.e., pi = 1 =W ∀i, hence [11]
S (^) q = k lnq W (8)
with [32,75]
lnq x ≡
x^1 −q^ − 1 1 − q
(ln 1 x = ln x) (9)
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
e (^) qx ≡ expq(x) ≡ [1 + (1 − q)x]^1 =(1−q)^ (e 1 x = e x) : (10) 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 [11] (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: ∑^ W
i=
pi (^) i = U (1)^ ; (11)
where the superindex (1) stands for rst choice and the {i} 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 [11]
p(1) i =
[1 − (q − 1) ∗ (^) i] 1 =(q−1) ∑W j=1 [1^ −^ (q^ −^ 1)^ ∗^ j^ ]^
1 =(q−1) ;^ (12)
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^ ) in the q → 1 limit, and (ii) 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 diculties (undesirable divergences) present in a variety of anomalous systems such as the Levy anomalous superdi usion (we remind that the second moment associated with Levy distributions diverges). Then the second choice (introduced in [11], rst worked out in [12], and since then inten- sively studied and used: see bibliography indicated in [13]) became the natural way out of the diculties. It goes as follows. The internal energy constraint is postulated to be
∑^ W
i=
p qi i = U (^) q(2) ; (13)
where the superindex (2) stands for second choice. The extremization of Sq now yields [12]
p(2) i =
[1 − (1 − q) i]^1 =(1−q) Z(2) q
with the generalized partition function given by
Z q(2) ( ) ≡
∑^ W
j=
[1 − (1 − q) j ] 1 =(1−q)^ (15)
which coincides with the result produced by the rst choice excepting for the fact that now (1−q) plays the role that was before played by (q−1) (i.e., [p(2) i ( )]q=[p(1) i ( ∗^ → )] (^2) −q, ∀i). 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 e (^) q 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
p(2) i = e
−q (^) i
Z q(2)
Z q(2) ≡
∑^ W
j=
eq−^ ^ j
Although it is clear that by no means the de nition of these quantities violates theory of probabilities (indeed, 〈Oi〉q = 〈p q i −^1 Oi〉, hence the q-expectation values of familiar quantities are nothing but standard mean values of say unfamiliar quantities), the math- ematical fact that 〈 1 〉q 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 〈Oi〉q 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 p Aij+ B= p Ai pj B as well as Aij +B= Ai + (^) jB , then [34]
U (^) q(2) (A + B) = U (^) q(2) (A) + U (^) q(2) (B) + (1 − q)[U (^) q(2) (A)S (^) q(B)=k
+U (^) q(2) (B)Sq(A)=k] ; (21)
which generically di ers from U (^) q(2) (A) + U (^) q(2) (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. Indeed, one way to avoid (simultaneously) the three unfamiliar consequences we just discussed is to introduce the internal energy constraint as follows:
∑W i=1 p^
q ∑ i^ ^ i W i=1 p^
q i
= U (^) q(3) ; (22)
i.e., we weigh the Hamiltonian eigenvalues with the set of probabilities p qi =
∑W
j=1 p^
q j (sometimes referred to as the escort probabilities [35]); the superindex (3) stands for third choice. The optimization of Sq now yields
p(3) i =
[1 − (1 − q) (i − U (^) q(3) )=
∑W
j=1 (p
(3) j )q]
1 =(1−q)
Z
(3) q
expq[ − (i − U (^) q(3) )=
∑W
j=1 (p
(3) j )q] Z (^) q(3)
with
Z (^) q(3) ( ) ≡
∑^ W
i=
1 − (1 − q) (i − U (^) q(3) )=
∑W
j=
(p j(3) )q
1 =(1−q)
∑^ W
i=
expq
− (i − U (^) q(3) )=
∑W
j=
(p j(3) )q
It can be shown [33] that
1 T =^
@Sq @U (^) q(3)
(T ≡ 1 =(k )) (25)
and
F q(3) ≡ U (^) q(3) − TSq = U (^) q(3) − 1 lnq Z (^) q(3) (26)
hence
S (^) q = k lnq Z (^) q(3) : (27)
Also
∑^ W
i=
(p(3) i )q^ = (Z (^) q(3) ) 1 −q^ ; (28)
hence Z(3) 0 = W for all temperatures such that all states have a nonvanishing associated
probability, limT →∞ Z(3) q = W (∀q) and, using Eq. (27),
p(3) i =
expq[( =(Z (^) q(3) ) 1 −q)( (^) i − U (^) q(3) )] Z (^) q(3)
Moreover, the use of relation (26), as well as the fact that @U (^) q(3) =@ = @(lnq Z (^) q(3) )=@ , yields an interesting relation, namely
U (^) q(3) =
( F q(3) ) (∀q) : (30)
We notice that Z (^) q(3) refers to the energy levels {i} with regard to U (^) q(3). In order to
use, instead, zero as the energy reference, we can de ne Z q(3) through
lnq Z q(3) = lnq Z (^) q(3) − U (^) q(3) : (31)
Consequently, Eqs. (26) and (30) can be rewritten in the familiar forms, namely
F q(3) = −
lnq Z q(3) (32)
and
U (^) q(3) = −
lnq Z q(3) : (33)
hence, by de ning T ′^ ≡ 1 =(k ′) and using Eq. (28),
T ′(T ) = T [Z q(3) ( (T ))]^1 −q^ + (1 − q)U (^) q(3) ( (T ))=k : (41)
We can immediately check that p(3) i ( ) = p(2) i ( ′) and Z(3)
′ q (^ ) =^ Z q(2) (^ ′). In other words, the equilibrium probabilities associated with the third choice coincide with those associated with the second choice but with a renormalized temperature given by Eq. (41). This is the reason for which all the theorems which do not explicitly use the speci c temperature dependence of the involved thermostatistical quantities (but rather only use that the system is at some xed arbitrary nite temperature) remain valid. Moreover, all the systems for which the second-choice formalism was successful in providing satisfactory theoretical and=or experimental results (e.g., self-gravitating sys- tems, turbulence, anomalous di usions, peculiar velocities of galaxies, solar neutrinos, bremsstrahlung, etc.), are also successful in the third-choice formalism because they do not involve speci c thermal dependences. The invariance of the probabilities given by Eqs. (23) and (24) with regard to uniform translation of the energy spectrum makes of course that the function T ′(T ) given by Eq. (41) is not invariant. When T varies from zero to in nity, T ′^ varies from (1 − q)U (^) q(3) (T = 0)=k (which vanishes if we choose the energy of the fundamental level to be zero) to in nity. Although we have not succeeded to prove it in general, for all the examples for which we have checked the function T ′(T ) appears to be (after elimination of possible quantum-originated spurious oscillations which disappear at high temperatures; details are given in Section 5) monotonically increasing with T , which is desirable (see Fig. 2). At this point let us make some observations about the set of escort probabilities {P i( q)} de ned through
P i( q)≡
p qi ∑W j=1 p^
q j
with
∑^ W
i=
P i( q)= 1 (43)
from which follows the dual relation
p (^) i = [P
(q) i ]^
1 =q ∑W j=1 [P
(q) j ]^
1 =q (44)
as well as
∑^ W
i=
p qi =
∑W
i=1 [P
(q) i ]^
1 =q}q :^ (45)
Let us rst comment that Eqs. (42) and (44) have, within the present formalism, a role analogous to the direct and inverse Lorentz transformations in Special Relativity (see [36] and references therein).
Second, we notice that O(3) q becomes an usual mean value when expressed in terms of probabilities {P i( q)}, i.e.,
O(3) q ≡
∑W
i=1 p^
q ∑ i^ Oi W j=1 p^
q j
∑^ W
i=
P i( q)Oi : (46)
Third, the entropy (1) can be written as
S (^) q = k 1 − {
∑W
i=1 [P
(q) i ]^
1 =q}−q q − 1
Consequently, the equilibrium escort probabilities (associated with Eq. (23)) can be alternatively found by optimizing Sq as given by Eq. (47) with the constraints (43) and
∑^ W
i=
P i( q) (^) i = U (^) q(3) ≡ Uq : (48)
This procedure yields
P i( q)=
{ 1 − (1 − q) (i − Uq)[
∑W
j=1 (P
(q) j )^1 =q]
q}q=(1−q) ∑W k=1{^1 −^ (1^ −^ q)^ (k^ −^ Uq)[^
∑W
j=1 (P
(q) j )^1 =q]
q}q=(1−q) :^ (49)
By using the same trick we used before (see Eq. (40)), these probabilities can be written as
P i( q)=
[1 − (1 − q) ′ (^) i]q=(1−q) ∑W k=1 [1^ −^ (1^ −^ q)^ ′^ k^ ]
q=(1−q) (50)
or even
P i( q)=
[1 − (1=q − 1) ′′ (^) i]^1 =(1=q−1) ∑W k=1 [1^ −^ (1=q^ −^ 1)^
′′ (^) k ] 1 =(1=q−1) (51)
with
′′ ≡ q ′. Comparison with Eq. (12) yields an interesting connection, namely P i( q)( ) = p(1) i ( ∗^ →
′′ ; q → 1 =q) : (52)
Let us summarize and comment the main results of the present section. The present second and third choices for the q-generalized internal energy (Eqs. (13) and (22), respectively) are very similar in nature. They both provide a satisfactory solution for the known unpleasant divergences which appear in many anomalous systems, and are both consistent with a whole set of relevant theorems and properties. Remarkable sim- pli cations which emerge within the third choice, as opposed to the second one, are that all equilibrium thermostatistical quantities are independent of the choice of the energy spectrum zero point, and that the normalized q-expectation value 〈〈Oi〉〉q to be associated with an arbitrary observable O satis es 〈〈 1 〉〉q = 1. Moreover, if it turns out to be legitimate to consider two q-systems A and B as independent in the sense that the probabilities satisfy p (^) ij (A + B) = pi(A)pj (B) and the Hamiltonians satisfy H(A+B)=H(A)+H(B), a fact which is not obvious [37,38] by just looking (without further considerations such as the thermodynamic limit) at the equilibrium probability
di erentiating with respect to ′. It is important to stress that all these derivatives can be computed once again explicitly, without solving implicit equations on the proba- bilities p j(3). Moreover, it is interesting to realize that, from a computational point of view, the present prescription is very close to the procedure used within the second choice. Indeed, it only requires a few additional computational steps. This means that all the computational developments already done with the nonextensive formalism us- ing the second choice of generalized mean values can be easily adapted to incorporate the third choice. In a similar way, the calculations already available within the present rst choice can be transformed into those corresponding to the third choice, through use of Eq. (52). Typical operational situations are illustrated in Sections 4 and 5 with three academic examples, namely the classical harmonic oscillator, the nondegenerate two-level system, and the quantum harmonic oscillator. We have used the word “academic” to stress the following fact. Unless proved otherwise, one-body problems (as well as many-body problems with short-range interactions, like say the square-lattice rst-neighbor Ising ferromagnet, or with no interactions at all, such as the ideal gas) are physically mean- ingful only for q = 1. It has been occasionally argued in the literature that q 6 = 1 calculations of such systems could make physical sense if the system was evolving in a (multifractal) “medium”. Although we cannot de nitively rule out such possibility, one should keep aware of the fact that, at the present stage of understanding, this is only speculative on physical grounds. Nevertheless, on mathematical grounds, such q 6 = 1 discussions are certainly helpful in order to operationally clarify the details of the implementation of the present generalized thermostatistical formalism.
- Heat and work
Let us now address heat and work within the third choice. If we use Eqs. (27) we obtain
dS (^) q = k
dZ (^) q(3) (Z (^) q(3) )q
which, performing the di erential operation on Eq. (24), yields
dU q (3) = TdSq +
∑^ W
j=
(p j(3) )q ∑W i=1 (p
(3) i )q^
dj =
∑^ W
j=
P j(q )d (^) j : (58)
We can consequently identify (for a quasi-static process) the heat tranfer as
Q(3) q = TdS (^) q (59)
and the work performed as
W q(3) =
r
F r(3) dr ; (60)
where the generalized force F r(3) is given by
F r(3) ≡ −
∑W
j=1 (p
(3) j )q^ @j^ =@^ r ∑W j=1 (p
(3) j )q^
∑^ W
i=
P(q)j
@ (^) j @r
the set { (^) r } being the external parameters on which the energy spectrum depends. In other words, we have that
dU q (3) = Q(3) q − W q(3) ; (62)
i.e., as we already saw previously, the rst principle of Thermodynamics holds as usual (∀q) (see [33]).
- The classical harmonic oscillator
To illustrate typical calculations associated with the present third choice, let us dis- cuss the classical harmonic oscillator, characterized by the Hamiltonian H=p^2 =(2m)+ m!^2 x^2 =2. It is known that the analytical discussion of this classical system can be done in a manner analogous to the one used for the quantum system. More precisely, we can associate with the classical oscillator a continuous (instead of discrete) energy spec- trum (n) = n, where is an arbitrary positive constant and n runs over all positive real numbers. So, we have
p(3) q (n; t) = 1 Z (^) q(3) (t)
[
1 − 1 −^ q t
(n − uq(t)) (Z (^) q(3) (t)) 1 −q
] 1 =(1−q) (63)
with
Z (^) q(3) =
∫ (^) nmax
0
dn
[
1 − q t
(n − uq(t)) (Z (^) q(3) (t))^1 −q
] 1 =(1−q) (64)
and
u (^) q(t) =
Z
(3) q (t)
∫ (^) nmax
0
dn n
[
1 − q t
(n − uq(t)) (Z
(3) q (t))^1 −q
]q=(1−q) ; (65)
where we have used Eq. (28), introduced t ≡ kT= and uq ≡ U (^) q(3) =, and where nmax = ∞ if q¿1 and is determined by 1 − (1 − q)(1=t)(nmax − u (^) q(t))¿0 (with 1 − (1 − q)(1=t)(nmax + 1 − uq(t)) ¡ 0) if q ¡ 1. For xed t, Eqs. (64) and (65) implicitly
provide Z (^) q(3) (t) and uq(t), which, replaced into Eq. (63), provide p(3) q (n; t), thus solving the problem. By performing the integrals of Eqs. (64) and (65) we obtain, for both q¿1 and q ¡ 1,
Z (^) q(3) =
t(Z(3) q ) 1 −q (2 − q)
[
(1 − q)uq t(Z(3) q )^1 −q
](2−q)=(1−q) (66)
- The two-level system and the quantum harmonic oscillator
Let us consider here the spectrum n = n with ¿ 0 and n = 0; 1 ; 2 ; : : : ; N. The case N = 1 corresponds to the case of two nondegenerate levels, and the case N → ∞ corresponds to the quantum harmonic oscillator. To calculate the thermal dependence of the associated (dimensionless) speci c heat C(3) q =k, it is enough to know uq(t) (indeed, C(3) q =k = duq(t)=dt). To achieve this, we need to solve the following system of equations:
p(3) n =
[
1 − (1−q)(n−u^ q^ ) t
∑N
m=0 (p
(3) m )q
] 1 =(1−q)
∑N
l=
[
1 − (1−q)(l−u^ q^ ) t ∑Nm=0 (p(3) m )q
] 1 =(1−q) (n^ = 0;^1 ;^2 ; : : : ; N^ )^ (77)
and
u (^) q =
∑N
n=0 (p
(3) n )q (^) n ∑N n=0 (p
(3) n )q^
For xed (N; q; t), this system of (N + 2) implicit equations can be solved numerically for the unknown quantities (p(3) 0 ; p(3) 1 ; : : : ; p(3) N ; uq). For typical values of (N; q) we have calculated C q(3) =k. The results are exhibited in Fig. 1. To perform these calculations, two di erent procedures have been used, namely by BG-based iterations, and through − ′^ transformation. First procedure: The q = 1 (BG) solution of Eqs. (77) and (78) is straightforwardly given by
p BGn =
e−n=t ∑N m=0 e−m=t^
e−n=t^ (1 − e−^1 =t^ ) 1 − e−(N^ +1)=t^
(n = 0; 1 ; 2 ; : : : ; N ) (79)
and
u 1 =
∑^ N
n=
p BGn n =
e^1 =t^ − 1
N + 1
e(N^ +1)=t^ − 1
For arbitrary q, we introduce this initial proposal in the right hands of Eqs. (77) and (78) and then iterate until satisfactory precision is obtained. This method converges quickly for q ¡ 1 but slower and slower for q increasing above unity. In any case, the convergence can be accelerated for given values of (q; N ) by starting the iterations from the (previously established) results corresponding to close values of q, or of N , or of both. Second procedure: We implement the method based on Eqs. (55) and (56). Typical t(t′) functions (with t ≡ 1 =( ) and t′^ ≡ 1 =( ′)) are indicated in Fig. 2. This procedure converges quickly for all values of q but, for q suciently below unity (q ¡ qc ' 0 :56), spurious peaks appear as illustrated in Fig. 2. These peaks are a mathematical artifact (related to the cut-o ) and are to be eliminated as indicated in that gure. The nal
Fig. 1. Thermal dependence of C q(3) =k for typical values of (N; q) (dashed lines: classical limit for the harmonic oscillator). N = 1 and N → ∞ respectively correspond to the two-level system and the quantum harmonic oscillator. In the q¿1 case, C(3) q remains nite for all values of q if N is nite, but, in the N → ∞ limit, is not de ned unless q ¡ 2: (a) q = 1 and typical values of N ; (b) N → ∞ and typical values of q; (c) q ¿ 1 example for typical values of N ; (d) q ¡ 1 example for typical values of N.
Fig. 2. t ≡ kT= versus t′^ ≡ kT ′= for N = 8 and typical values of q (dashed lines: classical limit for the harmonic oscillator). For q ¡ q (^) c ' 0 :56, N spurious peaks (dotted lines) appear.
physical reason for the generalization (q 6 = 1) was not transparent. But, in the light of recent developments [27,29,30] showing the relationship of the formalism with a pos- sible violation of the usual ergodic mixing hypothesis, features like the q-expectation value of unity not being equal to one became clearly unacceptable. Then naturally emerged the third choice, which we believe to be fully satisfactory. All the important theorems (macroscopic irreversibility, microscopic reversibility, causality, correspon- dence principle, factorization of the likelihood function, etc.) and properties (energy cuto for q ¡ 1 and long power-law tails for q ¿ 1, in the equilibrium energy dis- tribution) that have been established at xed temperature within the second choice, remain valid for the third one. The speci c thermal dependences are di erent, but all the previous calculations (done either within the rst or the second choice) can be transformed, with no heavy e ort (as shown in detail in the present paper), into those associated with the third choice. Let us nally remind, in order to avoid every ambiguity, that the present formalism does not focus on noninteracting or short-range interacting many-body Hamiltonian systems (which, as long as they follow the usual mechanics laws, certainly belong to the q = 1 universality class). It essentially focuses on long-range interacting Hamiltonian systems (like that discussed in [30]) and similar ones (possibly some slowly dissipating systems, for instance) which, in one way or another, exhibit nontrivial anomalies in their ergodicity and mixing properties.
Acknowledgements
We acknowledge fruitful remarks from E.M.F. Curado, J.E. Straub, A.K. Rajagopal, S. Abe and L. Borland. We also are grateful to E.K. Lenzi who communicated to us Eq. (30) which he established. One of us (C.T.) also acknowledges P. Bak for warm hospitality at the Niels Bohr Institute, where part of this work was done. This work was partially supported by CNPq and PRONEX (Brazilian Agencies).
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