PHY$1CA ELSEVIER
Physica A 221 (1995) 277290
Nonextensive thermostatistics: brief review and comments Constantino
Tsallis
Department of Chemistry, Baker Laboratory, and Materials Science Center, Cornell University, Ithaca, NY 148531301, USA and Centro Brasileiro de Pesquisas Fisicas  CBPE Rua Dr. Xavier Sigaud, 150, 22290180 Rio de Janeiro, RJ, Brazil
Abstract We briefly review, with regard to physical applications, the present status of the recently introduced nonextensive thermostatistics characterized by the entropic index q (q = 1 corresponds to standard, extensive, B oltzmannGibbs thermostatistics). In addition to that, we comment on (i) how metaequilibriumequilibfium crossovers may occur as a function of time, (ii) spin glasses and the replica trick, and (iii) the theory of perceptions.
1. Introduction and brief review BoitzmannGibbs ( B G ) thermostatistics and the usual thermodynamics constitute complete and powerful theoretical physics tools in the standard situations, more precisely whenever thermodynamic extensivity (additivity) holds (i.e. when the effective microscopic interactions are shortranged or inexistent, and the microscopic memory is shortranged or inexistent, and if the system evolves, in some relevant sense, in a euclideanlike spacetime. By "euclideanlike" we basically refer, in the present context, to a continuous and sufficiently differentiable variety, either curved or not). But these f o r m a l i s m s f a i l whenever the physical system includes longrange forces or longmemory effects or if it evolves in a noneuclideanlike spacetime (e.g., a (multi)fractal spacetime). By "failure" we refer to the fact that standard sums (or integrals) that appear in the calculation of the relevant thermostatistical quantities (e.g., partition function, internal energy, entropy, average square displacement) diverge. Consequently, we are left without wellbehaved mathematical prescriptions for calculating the quantities (e.g., specific heat, susceptibility, diffusivity) which are normally used for characterizing a system, and which enable meaningful comparisons with experimental data ( a l w a y s f i n i t e / ) . 03784371/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 03784371 (95)002367
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These difficulties are since long well known in gravitational [1,2] and magnetic [3] systems, Lfivylike anomalous diffusion [4], some surface tension problems [5], among others. Although not yet clearly identified (to the best of our knowledge), the same or analogous type of difficulties might be present in longranged Casimirlike systems (e.g., small inert uncharged spheres in a fluid at criticality [6] ), in granular materials such as sandpiles [7] (whose mechanical and acoustic properties are basically determined by the formation of a fractallike forcechain arching network), twodimensional turbulence [ 8 ], and others such as glassy materials (presenting a relevant set of complex metastable states). Generally speaking, systems which, either in their direct spacetime description or in their phase space evolution [9], present a (multi)fractallike or unconventional structure can exhibit serious mathematical untractability within standard thermostatistical formalisms or, at least, unfamiliar scalings with size for large sizes (or with time for long times); see Refs. [1012] and references therein. As a possible theoretical path for discussing this type of anomalies (or at least some of them), a formalism has been proposed [ 13,14] that generalizes conventional thermostatistics and thermodynamics. It basically relies upon two postulates, the validity of which is naturally to be confirmed or rejected through their consequences. The first postulate consists in assuming the following generalized entropy: 1  Eip?
Sqk
(qE]R; Z
q1
i
p/=l)
(1)
or, in general,
Sq : k
1  Tr/$q , q1
(2)
where k is a positive constant, {Pi} are the probabilities of the microscopic states, and /~ the corresponding density operator. With the following definition for the qexpectation value of the arbitrary observable 0:
(O)q = Tr~qO = (~qlO)l ,
(3)
we can rewrite Eq. (2) as follows:
Sq =  k (
/~1q _ l  q 1 )q.
(4)
This entropy is nonnegative, concave (convex) in the {Pi} for q > 0 (q < 0), reproduces the BoltzmannGibbsShannon one ($I = kB~]~iPilnpi) in the q ~ 1 limit and, among other properties [ 1521 ], satisfies pseudoadditivity. More specifically, if A and B are two independent systems (i.e. pA+B,j= papB), then
Sq(A kB) k
Sq(A) 
k
Sq(n) +~+ (1q)
Sq(A) Sq(n) k
k
(5)
which shows that ( 1  q) is a measure of the nonextensivity of the system. The second postulate consists in assuming the following generalized internal energy:
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(6) i
or, in general,
Uq  Tr/3q~  (7/)q,
(7)
where {ei} is the energy spectrum (i.e. the set of eigenvalues of the Hamiltonian 7~). This postulate deserves a comment. In spite of its unfamiliar aspect, by no means it violates the theory of probabilities. Indeed, as shown in Eq. (3), the qexpectation value of any operator 0 is nothing but the standard mean value of fiq10. In the sense that it averages an operator which depends on t3, it is very similar to the standard entropy Sj = kBTr~61n~ = kB(ln¢3)l. In particular, if we choose 0 = i, we have (])q = Tr/) q = ~]~ipq. This quantity has already been considered in various contexts [22], and generically equals unity if and only if q = 1 (V/~) or ~ is a pure (Vq) [23]. The optimization of Sq (given by Eq. ( 2 ) ) with the constraints Tr/~ = 1 and Tr pq~~ = Uq < ~ straightforwardly yields [ 14] the canonical ensemble equilibrium distribution [ ]  ( 1  q)1372/] l/(1q) p =
,
z~
(8)
where 13  1/kT is the Lagrange parameter associated with the thermostat and the generalized partition function is given by
Zq  T r [ ]  (1  q)13~]l/(1q)
(9)
In the q * ! limit we recover the BoltzmannGibbs distribution/3 o( exp(137~). It can be shown [ 14] that, for all q, 1 _ osq
T Uq
(lO)
OUq ' 0 Zl  q  1 013 1  q
,
(11)
1 ZlqqI Fq = Uq  VSq 
13
1q
,
(12)
and
c~ =_ r os. OUq =  r °2v~ 07" = 07" OT2 ,
(13)
i.e. the entire Legendretransform structure of thermodynamics is preserved in spite of the system being nonextensive. Moreover, thermodynamic stability can be proved to hold for all q [24], based upon the fact that always Cq/q >>.0 [18]. Also, it can be shown [ 16] that the Ehrenfest theorem is qinvariant; more precisely,
O(O)q i at   ~ ([O, 7"~l)q.
(14)
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Consequently, [7~, O] = 0 implies that (O)q (and generically not (0)1) is a constant of motion. Another important point that deserves clarification is the physical meaning of the index q, the way we presently understand it. This index is meant to characterize universality classes with the appropriate scalings of thermostatistical quantities with the number N of elements of the system. If the system is a conventional one in the sense we have described in the introduction, then q = 1. If this is not the case, then q might differ from unity and, although the general answer is yet unknown, it is expected to be uniquely determined by the microscopic dynamics in a way which we now illustrate through five recent applications in physics, namely L6vylike anomalous diffusion [25], correlated anomalous diffusion [ 2629], selfgravitating systems [ 30,31 ], longrange "ferrofluids" [ 3 2 ] and twodimensional turbulence [31]. In any case, it is important to remark that q < 1 (q > 1) privileges the rare (frequent) events.
1.1. L~vylike anomalous diffusion [25] The optimization of Sq = k
1  f dx[p(x)]q qI
(15)
with the constraints
d x p ( x ) = 1,
(X2)q ~ /
(16)
dxx2[p(x)] q =
0 2 < ~
(17)
(x being dimensionless) straightforwardly yields
pq(X) =
[ 1  13(1  q ) x 2 ] l / ( l  q )
(18)
with
 fax[l 13(1
q)x2] l/(1q) ,
(19)
where 13 is the Lagrange parameter associated with the constraint (17), and q < 3 in order to be possible to satisfy (16) (if q ~> 3, Zq diverges). It can be easily verified t h a t ( X 2 ) l ( l ) ~ f d x x 2 p q ( x ) isfinite for q < 5/3 and diverges for 5/3 ~< q < 3, while (X2)q(1)  f dxx2[pq(X)] q is finite for all q ~< 3. The remarkable mathematical convenience of the qexpectation value is well illustrated with the Lorentzian (or Cauchy) distribution, i.e. q = 2. Indeed, (x 2) 1( 1 ) cx f~oo dxx2 ( 1 + 13x2 )  1 diverges (hence, it is unacceptable as a constraint in any mathematically wellposed optimization procedure), while (X2)2( 1 ) OCf~oo dxx2( 1 + 13x2)2 is finite.
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The distribution (18) corresponds to one diffusive step. The Nstep distribution is given by the convolution product
p q ( x , N ) =pp(X) *pq(X) * . . . *pq(X)
( N factors)
(20)
which, in the N ~ cxD limit, yields
pq(X, N) ~ N1/VL~(x/N 1/?') ,
(21)
where L~(y) is the L6vy distribution associated with the fractal dimension y given by 2 Y=
if q ~< 5/3
3 ______qq if 5/3 < q < 3 q1
(22)
By recalling that L~(y) c( l/lyll+r(lyl~ c~; y < 2), the connection between q and the spatial range of the distribution has been exhibited. The ddimensional extension of the present results is straightforward. Summarizing, the present formalism provides a solution for an old problem [4], namely how to obtain L&y distributions from an entropy principle with simple auxiliary conditions. Its applications concern a variety of physical situations which includes micelles of amphiphilic CTAB molecules [33], heartbeats [34], nonlinear Hamiltonian dynamics [9], chaotic transport in a laminar fluid flow of waterglycerol [35], subrecoil laser cooling [36], among others. From the mathematical point of view, it is interesting to remark [37] that: (i) If we make in Eqs. (18) and (19) the transformations ( 2tim "~ 1/2 z \l+m j x,
3__+m q= l + m
we obtain, for 1 < q < 3 (hence, 0 < m < oc), the Student tdistribution with m degrees of freedom; (ii) If we make, in the same equations, the transformations rz
( n 2 f 1 4 ) 1/2
x,
n6 q~ n4
we obtain, for  ~ < q < 1 (hence, 4 < n < ~ ) , the rdistribution with n  2 degrees of freedom. In other words, the present formalism enables the derivation of two important statistical distributions from a simple entropic principle.
1.2. Correlated anomalous diffusion [2629] A variety of diffusive physical situations are described by the equation
3 [ p ( x , t ) ]~ 32[p(x,t) ] ~  D 3t cgx2
(D > 0(/z,v)
E R 2) .
(23)
The particular case /z = v = 1 of course corresponds to normal diffusion. The case ( # = 1, v > 0) has been discussed by Spohn [26] (in particular, v = 3 corresponds to
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a specific solidonsolid model for surface growth [38] ). The general (/.t, u) case has been recently solved by Duxbury [27], and we briefly review it here. The following ansatz is proposed: p ( x , t) = t  r ~ p ( x t r2 , 1).
(24)
If we define R ( y ) = p ( x t ~2, l) and replace the proposal (24) into Eq. (23) we obtain  I z R IzI
[fiR +
r2yR'] = D v [ ( v  1)R '2 + R R ' I ] R v2 ,
(25)
whose fundamental solution is R(y) =
[1  (1  q ) f l y 2 ] U ( l  q )
Zq
(26)
with q =l+/zv,
(27)
/.t ~'2 =/z~'l , /z+v
(28)
2/z fl
l~(lz k l J ) O Z l  q "
(29)
We verify that (/x, v) = (1,3) (solidonsolid model) yields q =  1 , which coincides with the value obtained by Plastino and Plastino [30] for selfgravitating systems! We ignore whether this is a numerical coincidence, or if there is some physical reason for that. The solution (26) found by Duxbury recovers the Barenblatt solution discussed by Spohn (Eq. (2.13) of Ref. [26]) as the/.t = 1 particular case. Very recently, Plastino and Plastino [28] have extended the/z = 1 case of Eq. (23) in order to discuss what happens in the presence of an external force F ( x ) (details have been given for the harmonic oscillator particular case F ( x ) oz x, socalled UhlenbeckOrnstein process). Finally, for arbitrary (/z, v) and F ( x ) = kl  k2x (Vkl, and k2/> 0), the problem has just been worked out [29]. In all these cases, the exact timedependent solution is given, through the appropriate spacetime scaling, by Eq. (18). 1.3. N e w t o n i a n gravitation [30]
The exact timedependent solutions of the Vlasov equations associated with two galaxy models, namely the generalized Freeman disk (planar galaxy) [39] and Kalnajs' finite amplitude homologous oscillation of a homogeneous slab of stars [40], have been shown to precisely correspond to q =  1 . The fact that q =  1 is consistent with previous considerations [41], which (after a due q ~~ 1 / q transformation [12,31]) showed that Newtonian gravitation is compatible with simultaneously finite mass, energy and entropy if and only if q < 7/9. Let us stress that to obtain the q =  1 result, Plastino and Plastino [30] explicitly used the qexpectation value appearing in Eq. (7).
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1.4. Longrange "ferrofluid" [32] Rings (d = 1 topology) of N permanent dipoles have been recently considered including, besides the dipoledipole interaction, a nonmagnetic potential U=e
Z

(0 ~< a < p ~< o~; e > 0; o  > 0 ) .
(30)
i
The cases (p, a ) = ( 12, 6), (c~, ce) and (p, 0) correspond to LennardJones potential, hard core model and (essentially) mean field approximation, respectively. For N >> 1, the p = 12 numerical for the energy E ( N ) of the classical ground state are precisely fitted by the d = 1 case of
E ( N ) ~ C(ce, p , d ) N N *
(0 < C ( a , p , d ) < c~)
(31)
with N* z
N 1~/a  1 1 a/d
(32)
Hence
E ( N ) cx
NlnN
if d/ot < 1 if d/oz = 1 if d / a > 1
(33)
For this system, the precise connection with the qextended thermostatistics is yet unknown, in spite of the obvious manifestation of nonextensive behavior. On the one hand, the discussion of full equilibrium of meanfieldlike systems ( a ~ 0 in the present model) suggests that, after the appropriate scaling with size (Eqs. ( 3 1 )  ( 3 3 ) ; see also Ref. [42] ), the BG scheme (q = 1) is applicable; since the problem is easily remedied through a simple rescaling, we can consider this as a weak failure of BG thermostatistics. On the other hand, the analysis of some slowly varying equilibriumlike states (see, for instance, Refs. [ 30,31 ] ) suggests that a strong failure of BG thermostatistics occurs, and the entire q = 1 formalism should have to be replaced by the q 4= 1 one. In this case, it seems physically plausible that q should depend on ( a , d) through the ratio o~/d. The simplest conjecture of course is
q=
1 ce/d
if d/o~ ~< 1 if d / a > 1
(34)
But, if we wish to incorporate also the q =  1 result by Plastino and Plastino [30] for Newtonian gravitation ( ( d , ce) = (3, 1)), we are rather led towards another conjecture, namely [32]
q=
l 2d/a
if d/ot ~< 1 if d / a > 1
(35)
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This type of connection (say Eq. (34) or (35)) should also apply to longrange magnetism. For example, for the Ising ferromagnet with Hamiltonian
7t =  J E (1/rij)s~
(Si = +I,Vi; a / > 0; J > 0)
i ~.j
it is known [3] that, within BoltzmannGibbs statistics, kBTc/J is finite for d/~ < 1 and diverges for d/a ~> 1 (in which case, further qualification of the system is obviously needed). The proof (or disproof) of Eq. (34) or (35) would be very welcome. Such a proof should of course be based (as in Refs. [30,31] ) on the appropriate identification of the physical quantities which reflect the microscopic dynamics of the system.
1.5. Twodimensional turbulence [31] Huang and Driscoll [31] have recently exhibited a neat failure of the BG scheme for explaining the experimentally quite well defined state (socalled "metaequilibrium" state), at intermediate times, of the relaxation of twodimensional Eulerlike turbulence of electron columns bounded by conducting cylindrical walls in a 10 1° torr vacuum in the presence of a 507 G magnetic field. Indeed, the vorticity radial profile showed a clear departure of the profile (much flatter) predicted through optimization of the BG entropy S1 under appropriate constraints. They also showed that the experimental data are satisfactorily fitted through an alternative variational procedure, namely the socalled "Restricted Minimum Enstrophy" (RME) one. Very recently, Boghosian showed [31] that the RME profile is precisely the one that the present generalized thermostatistical formalism provides for q = 1/2. This fact can be considered as constituting a quite strong evidence of the experimental validity of the present formalism. Let us also mention that, besides the above five physical applications, the present formalism (i) has been used to sensibly improve [43] a standard optimization technique, namely simulated annealing; (ii) has been considered [44] as possibly applicable to selforganized biological systems; (iii) has enabled a possible connection with quantum groups [45] ; (iv) has enabled a test of the blackbody radiation Planck law with regard to the cosmic microwave background radiation detected by the COBE/FIRAS experiment [46]; and (v) is capable of yielding a finite specific heat for the nonionized hydrogen atom in free space [47]. The difficulties that appear in the BoltzmannGibbs approach of the last problem are essentially quite analogous (though mathematically much more elementary) to those appearing in the path integral discussion of the same system. By "analogous" we refer to the fact that both formalisms are based on exponential distribution laws which cause, whenever longrange interactions are involved, undesirable divergences. At this point, it is worthwhile remembering that these difficulties are of such a nature that eventually made R.P. Feynman to quit teaching the entire subject of path integrals in his course on quantum mechanics! [48]. Some (effective or possible) applications within areas outside physics deserve to be mentioned, such as in the theory of financial decisions [ 11 ], and in the theory of probabilities and information (the socalled "envelope game" [49]).
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To conclude this section, let us mention why we use k (instead of kB) whenever q 4~ 1. It is clear that k = f ( q ) k B where f ( q ) is a pure number which in principle could depend on q (with f ( l ) = 1). To fix f ( q ) one should make a thermometric identification of T = 1/ilk. Since the qgeneralizations of heat transfer 6Q = TdSl and work performed (say 6W = pdV) are not yet established, such an identification cannot be done on a solid basis at the present time. Consistently, in the meanwhile, it seems preferable to just keep k without further specification.
2. C o m m e n t on metaequilibriumequilibrium crossovers along time A very natural question can puzzle our mind, namely how so diverse and complex phenomena involving longrange interactions, longrange microscopic memory, fractai spacetime, could possibly be taken into account by a formalism based on only one extra parameter, namely q? Moreover, how a terminal equilibrium behavior, which would be essentially well described by a q = 1 thermostatistics (with appropriate sizescaling of all thermodynamic quantities [42] ), could be compatible with some slowly varying longliving q ~ 1 equilibriumlike state ("metaequilibrium" as referred to in [31])? The deep answer to these questions is not yet available (though some indications do exist which make us to believe that the "key" regards the conditions of validity of the (quasi)ergodic hypothesis, or, equivalently, under what circumstances and how an ensemble description "h la Gibbs" (with a generalized entropy) can be compatible with a microdynamical picture "h la Boltzmann"). Hence, we restrict ourselves here to the more modest task to show that this is mathematically possible. Let us illustrate this by focusing on a system of N "particles" interacting through twobody interactions which are repulsive at very short distances and which decay, at long distances, as 1/r~ (for the potential) in a euclidean ddimensional space (notice that fc~toff dr r dr r  , is finite for a > d, and diverges for 0 ~< a ~< d); exponentially decaying interactions, or no interactions at all, correspond to the a >> d region. Let us further assume that the microscopic dynamics are also characterized by a "memory function" which is nonsingular (integrable) at the origin, and decays for long past times as Ir/tl d with r > 0 and a ~ >~ 0 (notice that fcut°ffdt (  t )  d is finite for o/ > 1, and diverges for 0 <~ ce~ ~< 1; an exponentially decaying memory, or no memory at all, correspond to the cd > > 1 region. A dimensionless function f ( a , d , a ' ) is assumed to exist which is uniquely determined by the microscopic dynamics, generically differs from unity, and satisfies f ( d , d, 1) = 1. We can imagine for an effective index qeff the following (N, t/7") ~ (c~, ~ ) behavior:
A + N I~/a ( t / r ) t  d f ( o ~ , d , a ') qeff ~
A + N 1a/d ( t / r ) l  , '
'
(36)
A being a constant pure number (e.g., A = 1). Let us check, in the (N,t/'r) ~ ( ~ , ~ ) limit, the four main cases (related to ct/d > 1 or < 1, and o / > I or < 1).
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If
aid
> 1 (shortrange forces) and a ' > 1 (shortrange memory), then (37)
qeff = 1 ;
if
aid
< 1 (longrange forces) and a ' < 1 (longrange memory), then
(38)
qeff = f ( a , d , d ) ; if
a/d
< I (longrange forces) and a ' > 1 (shortrange memory), then
I qeff =
N 1a/d (t/r)a,_ l
1
if
A+Kf(a,d,d) _ df ,~+ K  d
if
Nla/d (t/r) ''I
f(o:,d,a')
if
(t/r)~,_l
~ (X)
K E (0,
(39)
N 1 a / d
if
a/d
0;
> 1 (shortrange forces) and a' < 1 (longrange memory), then
I
N~/d 1 +(30
1 qeff=
if ( tN/~)dl__l,~, d, a') _ d f ,,~+t¢  d
A JrK f ( o l ,
if
f(a,d,a')
if
(t/r) l~'
+ K C (0, oo)
(40)
Na/d1
(t/r)l_~,
40.
Putting everything together, a physically quite interesting picture emerges, of which some consequences follow: (i) If the relevant spacetime is euclideanlike, and both the forces and the memory are shortranged, then the familiar BG formalism fully applies, i.e. q = 1 is the correct microscopic description and conventional thermodynamics holds (U, F, G, S, M . . . . are extensive variables, and T,p, tz, H... are intensive ones). (ii) If the relevant spacetime is euclideanlike, and either the forces or the memory (or both) are longranged, but we are only interested in the ultimate equilibrium state (i.e. limN,~ l i m t _ ~ ) , the BG description would be weakly violated in the sense that the q = 1 formalism can be used only after approximate scalings have been introduced such as those indicated in Eqs. ( 3 1 )  ( 3 3 ) and Refs. [32,42] ( U , F , G . . . scale with NN*; S, M... scale with N; and T, p,/z, H . . . scale with N*). (iii) If the relevant spacetime is euclideanlike, and either the forces or the memory (or both) are longranged, and we are interested in the slowly varying intermediatetime equilibriumlike state (i.e. limt~oo limN.oo; metaequilibrium of [31] ), the BG description is strongly violated, and q # 1 is needed. (iv) If the relevant spacetime is (multi)fractal, then the BG description is once again strongly violated, and q = df/d is needed since, in this case, all three space, time and size, anomalously scale with each other [42,46].
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It is clear that expressions like "relevant" spacetime and "effective" index qeff need further qualification, but this cannot yet be done since the formal connection of the present ideas with quantities such as Zq, Uq, Sq, F q . . . is not available at the present moment. However, although expressions like Eq. (36) have been herein considered only as mathematical possibilities, the picture that emerges is physically so appealing that it makes one believe that something of this sort indeed occurs in Nature.
3. Comment on spin glasses Let us make a suggestive comment. The present formalism contains ingredients (Eqs. (4), (11), (12), (32)) that greatly remind us of the replica trick used in the discussion of spin glasses. One would therefore not be surprised if some deep connection appeared to exist with those systems. Some recent progress provides, in fact, plausibility to this conjecture. Indeed, Parisi [50] just established a connection between spin glasses and quantum groups; simultaneously, quantum groups were tentatively connected [ 45] with the present qformalism. These facts seem to consolidate the suspicion of a relationship between spin glasses and the present generalized statistical mechanics. Remains, nevertheless, open the following question. Although exhibiting unusual pathologies, BoltzmannGibbs statistics (q = 1) seem to work satisfactorily for standard spin glasses. If so, the only way for these systems to have some connection with the q :~ I formalism is to lie precisely on the frontier between extensive and nonextensive thermodynamics. According to Eq. (33) (and also, for instance, to Ref. [3]) this occurs tbr d / a = 1. This is precisely the case for the permanent dipoledipole interaction (d = cr = 3), and generally speaking whenever conditionally convergent sums are involved. An important class of real spin glasses (d = 3) are thought to be so because the RKKY interaction ( a = 3) is involved. But why the same type of behavior could be related to other spin glasses, and particularly to their mean field approach (where the replica trick is known to be relevant), remains to be clarified (or rejected). To make the puzzle even harder, Parisi connection [50] of spin glasses with quantum groups occurs far from (his) q = 1.
4. Comment on the theory of perceptions Let us comment on an important phenomenon studied in physiology and which is common to a variety of problems related to learning, unconscious inference and the task of perception by the neural networks of higher mammals (humans, for instance). Indeed, in their theoretical approach it is badly needed averages in which the standard weight p~ of a given configuration s is replaced by (  lnps), i.e. a weight which enhances the role of the rare events. The point is lengthily discussed by Barlow (see Ref. [51] and references therein). For instance, commenting the pictorial Selfridge's Pandemonium (in which the outworld information is transmitted into the brain by the image demon to the
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feature demons, from these to the cognitive demons who, in turn, influence the decision demon), Barlow writes: "In a probabilistic pandemonium the shrieks [of the cognitive demons] would be proportional to log P, where P is the probability of occurrence of the feature the demon detects',".., so the perceptual output neuron shrieks loudly when its feature is unexpectedly present, softly when it is present but in circumstances such that it is not surprising.", and also "If one considers a plausible physiological realization, the requirement that the demons shriek with loudness proportional to log P would not be too hard to approximate, for any mechanism of adaptation or habituation discounts frequently repeated events and thus leads to something like the desired "unexpectedness" signal.". The present formalism straightforwardly yields, in the q * 0 limit, Sq T~(w1)q
w
E(
In Ps)
(41)
s=l
and w
w
(O)q==Z O s P q s ~ Z ~I
s=l
w
OsqE
O s (  l n Ps),
(42)
s=l
where W is the total number of configurations and 0 is an arbitrary observable. In other words, except for unimportant constants, the averages are performed with weights proportional to  I n Ps, as desired!
Acknowledgements I acknowledge with great pleasure useful remarks from T.A. Kaplan, S.D. Mahanti, P.M. Duxbury and E. Brezin. Also, I am indebted to P.M. Duxbury [27] and to J.R. Grigera [32] for communicating to me their unpublished results. Finally, I am pleased to acknowledge warm hospitality, at the Baker Laboratory, by B. Widom, in whose research group this work was concluded.
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