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Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 1
Nonextensive Statistics:
Theoretical, Exp erimental and Computational
Evidences and Connections
Constantino Tsallis Centro Brasileiro de Pesquisas Fsicas Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil e-mail: tsal [email protected]
Received 07 Decemb er, 1998
The domain of validity of standard thermo dynamics and Boltzmann-Gibbs statistical me- chanics is discussed and then formally enlarged in order to hop efully cover a variety of anomalous systems. The generalization concerns nonextensive systems, where nonextensiv- ity is understo o d in the thermo dynamical sense. This generalization was rst prop osed in 1988 inspired by the probabilistic description of multifractal geometries, and has b een in- tensively studied during this decade. In the present e ort, after intro ducing some historical background, we brie y describ e the formalism, and then exhibit the present status in what concerns theoretical, exp erimental and computational evidences and connections, as well as some p ersp ectives for the future. In addition to these, here and there we p oint out various (p ossibly) relevant questions, whose answer would certainly clarify our current understand- ing of the foundations of statistical mechanics and its thermo dynamical implications.
I Intro duction
A di use b elief exists, among many physicists as well as other scientists, that Boltzmann-Gibbs (BG) statistical mechanics and standard thermo dynamics are eternal and universal. It is certainly fair to say that \eter- nal", in precisely the same sense that Newtonian me- chanics is \eternal", they indeed are. But, again in complete analogy with Newtonian mechanics, we can by no means consider them as universal. Indeed, we all know that, when the involved velo cities approach that of light, Newtonian mechanics b ecomes only an approximation (an increasingly bad one) and reality is b etter describ ed by sp ecial relativity. Analogously, when the involved masses are as small as say the elec- tron mass, once again Newtonian mechanics b ecomes but a (bad) approximation, and quantum mechanics b e- comes necessary to understand nature. Also, if the in- volved masses are very large, Newtonian mechanics has to b e extended into general relativity. In these senses we certainly cannot consider Newtonian mechanics as
b eing universal. I b elieve that the same typ e of con- siderations apply to standard statistical mechanics and thermo dynamics. Indeed, after more than one century highly successful applications of the magni cent Boltz- mann's connection of Clausius macroscopic entropy to the theory of probabilities applied to the microscopic world , BG thermal statistics can (and should) eas- ily b e considered as one of the pillars of mo dern sci- ence. However, it is unavoidable to think that, like all other pro ducts of human mind, this formalism must have physical restrictions, i.e., domains of applicabil- ity, out of which it can at b est b e but an approxi- mation. It seems that BG statistics satisfactorily de- scrib es nature if the e ective microscopic interactions are short-ranged (i.e., close spatial connections) and the e ective microscopic memory is short-ranged (i.e., close time connections) and the b oundary conditions are non(multi)fractal. Roughly sp eaking, the standard formalisms are applicable whenever (and probably only whenever) the relevant space-time (hence the relevant phase space) is non(multi)fractal. If this is not the
2 Constantino Tsallis
case, some kind of extension app ears to b ecome nec- essary. Indeed, an everyday increasing list of physical anomalies are, here and there, b eing p ointed out which defy (not to say that plainly violates) the standard BG prescriptions. A nonextensive thermostatistics, which recovers the extensive, BG one as particular case, was prop osed in 1988 [1, 2] which might correctly cover at least some of the known anomalies. Although a fair amount of what legitimately lo oks like b eing successful applications is nowadays accumulating, further veri ca- tions and deep er understanding is needed and welcome. Computational work is highly desired since, on various grounds, the analytic discussion frankly app ears to b e untractable. Needless, of course, to say that more ex- p erimental and theoretical work is absolutely relevant to exhibit the applicability and robustness of the ideas I intend to present herein. In the present contribution, I prop ose some (hop efully relevant) questions that are right now op en to such theoretical, exp erimental and computational contributions.
Let us b e more sp eci c. As mentioned ab ove, it is nowadays quite well known that a variety of physical systems exist for which the p owerful (and b eautiful) BG statistical mechanics and standard ther- mo dynamics present serious diculties or anomalies, which can o ccasionally achieve the status of just plain failures. Within a long list that will b e sys- tematically fo cused on later on, we may mention at this p oint systems involving long-range interactions (e.g., d = 3 gravitation)[3], long-range microscopic memory (e.g., nonmarkovian sto chastic pro cesses, on which much remains to b e known, in fact)[4, 5], and, generally sp eaking, conservative (e.g., Hamilto- nian) or dissipative systems which in one way or an- other 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[6 ], Levy anomalous di usion[7], granular systems[8], phonon-electron anomalous thermalization in ion-b ombarded solids ([9] and references therein), so- lar neutrinos[10], p eculiar velo cities of galaxies[11], in- verse bremsstrahlung in plasma[12] and black holes[13], to cite a few, clearly app ear to b e (in some cases), or could p ossibly b e (in others), concrete examples. The
present status of these and others will b e discussed in Sections I I I, IV and V.
I I Formalism
I I.1 Entropy As an attempt to overcome at least some of these diculties a prop osal has b een advanced, one decade ago[1], (see also [14, 15 ]), which is based on a general- ized entropic form, namely
Sq = k 1 ^
PW
i=1 p
q i q 1
X^ W
i=
pi = 1; q 2 R
where k is a p ositive constant and W is the total num- b er of microscopic p ossibilities of the system (for the q < 0 case, care must b e taken to exclude all those p ossibilities whose probability is not strictly p ositive, otherwise Sq would diverge; such care is not necessary for q > 0; due to this prop erty, the entropy is said to b e expansible for q > 0). This expression recovers the usual
BG entropy ( k PWi=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
Sq (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 (i.e., pij (A + B ) = pi (A)pj (B )). We im- mediately see that, since in all cases Sq 0 (nonneg- ativity prop erty), q < 1 ; q = 1 and q > 1 resp ec- tively corresp ond to superadditivity (superextensivity), additivity (extensivity) and subadditivity (subextensiv- ity). Eq. (2) exhibits a prop erty which has appar- ently never b een fo cused b efore, and which we shall from now on refer to as the composability prop erty. It concerns the nontrivial fact that the entropy S (A + B ) of a system comp osed of two indep endent subsystems A and B can b e calculated from the entropies S (A) and S (B ) of the subsystems, without any need of mi- croscopic know ledge about A and B , other than the know ledge of some generic universality class, herein the
4 Constantino Tsallis
Moreover, Jackson intro duced in 1909[20] the general- ized di erential op erator (applied to an arbitrary func- tion f (x))
Dq f (x) f^ (q q^ x x)^ ^ fx^ ( x); (8)
which satis es D 1 l imq! 1 Dq = (^) dxd. Ab e[21] recently remarked that
k
Dq
XW
i=
pi
=
= k 1 ^
PW
i=1 p
q i q 1 ^ Sq^ (9)
This prop erty provides some insight into the generalized entropic form Sq. Indeed, the inspiration for its use in order to generalize the usual thermal statistics came[1] from multifractals, and its applications concern, in one way or another, systems which exhibit scale invariance. Therefore, its connection with Jackson's di erential op- erator app ears to b e rather natural. Indeed, this op er- ator \tests" the function f (x) under dilatation of x, in contrast to the usual derivative, which \tests" it under translation of x. Another prop erty which no doubt must b e men- tioned in the present intro duction is that Sq is consis- tent with Laplace's maximum ignorance principle, i.e., it is extremum at equiprobability (pi = 1 =W 8 i). This extremum is given by
Sq = k W^
1 q (^) 1 1 q (W^ ^ 1)^ (10) which, in the limit q! 1, repro duces Boltzmann's cel- ebrated formula S = k ln W (carved on his marble grave in the Central Cemetery of Vienna). In the limit W! 1 , Sq diverges if q 1, and saturates at k =(q 1) if q > 1. Finally, let us close the present set of prop erties by reminding that Sq has, with regard to fpi g, a def- inite concavity for al l values of q (Sq is always con- cave for q > 0 and always convex for q < 0). In this sense, it contrasts with Renyi's entropy S Rq
(ln PWi=1 pqi )=(1 q ) = fln [1 + (1 q )Sq =k ]g=(1 q ),
which do es not have this prop erty for al l values of q. Before addressing other relevant quantities, let us intro duce the following convenient functions[22]:
exq [1 + (1 q ) x]^1 =(1 q^ )^ ; 8 (x; q ) (11)
(hence, ex 1 = ex^ ) with the de nition supplement, for q < 1, that exq = 0 if 1 + (1 q ) x 0, (and analo- gously, for q > 1, exq diverges at x = 1 =(q 1)) and
lnq x [x^1 q^ 1]=[1 q ]; 8 (x; q ) (12)
(hence, ln 1 x = ln x). We can easily verify that
e (^) qln q^ x= lnq exq = x; 8 (x; q ): (13)
For instance, Eq. (10) can b e rewritten in the following Boltzmann-like form:
Sq = k lnq W (14)
Let us also intro duce the following unnormalized q - expectation value:
hAiq
X^ W
i=
pqi Ai (15)
hence hAi 1 corresp onds to the standard mean value of a physical quantity A. If our system is a generic quantum one, its proba- bilistic description is given by the density op erator , whose eigenvalues are the fpi g. Then, the generalized entropy is given by
Sq = k 1 ^ T^ r^
q q 1 (T^ r^ ^ =^ 1)^ (16) and the unnormalized q -exp ectation value of an observ- able A which do es not necessarily commute with is given by hAiq T r q^ A : (17)
Eq. (16) can b e rewritten as
Sq = k hlnq iq ; (18)
and also as Sq = k hln 2 q i 1 : (19)
If our system is a generic classical one, the rele- vant variables are typically continuous variables, and its probabilistic description is given by a distribution of probabilities p(r), where r is a dimensionless variable in a many-b o dy phase space. Then, the generalized entropy is given by
Sq = k 1 ^
R
dr [p(r)]q q 1 (
Z
dr p(r) = 1) (20)
and the unnormalized q -exp ectation value of an observ- able A(r) is given by
hAiq
Z
dr [p(r)]q^ A(r) : (21)
Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 5
Although we shall, in what follows, b e illustrating the present formalism with the case of W discrete micro- scopic p ossibilities, the generic quantum and classical discussions follow along the same lines, mutatis mutan- dis.
I I.2 Canonical ensemble
Once we have a generalized entropic form, as given in Eq. (1) (or an even more general one, or a di erent one), we can use it in a variety of ways. For instance, if we are interested in information theory, some opti- mization algorithms, image pro cessing, among others, we can take advantage of a particular form in di erent ways. See, for instance, [17, 19, 23, 24 ] and references therein, where it can b e veri ed that not less than 25 (!) di erent entropic forms have received, along the years, a great variety of technological and mathematical ap- plications. For instance, the Renyi entropy mentioned ab ove has b een quite useful in the geometrical charac- terization of strange attractors and similar multifractal structures (see [25] and references therein). However, if our primary interest is Physics, this is to say the (qualitative and quantitative) description and p ossible understanding of phenomena o ccurring in Na- ture, then we are naturally led to use the available gen- eralized entropy in order to generalize statistical me- chanics itself and, if unavoidable, even thermo dynam- ics. It is along this line that we shall pro ceed from now on (see also [26]). To do so, the rst nontrivial (and quite ubiquitous) physical situation is that in which a given system is in contact with a thermostat at tem- p erature T. To study this, we shall follow along Gibbs' path and fo cus the so called canonical ensemble. More precisely, to obtain the thermal equilibrium distribution asso ciated with a conservative (Hamiltonian) physical system in contact with the thermostat we shall extrem- ize Sq under appropriate constraints. These constraints are[15]
X^ W
i=
pi = 1 (nor m constr aint) (22)
and
hhi iiq
PW
i=1 p
qi i
PW
i=1 p
q i
= Uq (ener g y constr aint) (23)
where fi g are the eigenvalues of the Hamiltonian of the system. We shall refer to hh:::iiq as the normalized
q-expectation value and to Uq as the generalized inter- nal energy (assumed nite and xed). It is clear that, in the q! 1 limit, these quantities recover the stan- dard mean value and internal energy resp ectively. We immediately verify that, for any observable,
hh:::iiq = h h::: 1 iiq q
The outcome of this optimization pro cedure is given by
pi =
h
1 (1 q ) (i Uq )=
PW
j =1 (pj^ )
q i^1 ^1 q
Z^ q (25)
with
Zq ( )
X^ W
i=
41 (1 q ) (i Uq )=
X^ W
j =
(pj )q
1 ^1 q
It can b e shown that, for the case q < 1, the expression of the equilibrium distribution is supplemented by the auxiliary condition that pi = 0 whenever the argument of the function b ecomes negative (cut-o condition). Also, it can b e shown[15] that
1 =T = @ Sq =@ Uq ; 8 q (T 1 =(k )): (27)
Furthermore, it is imp ortant to notice that, if we add a constant 0 to all fi g, we have (as it can b e self- consistently proved) that Uq b ecomes Uq + 0 , which leaves invariant the di erences fi Uq g, which, in turn, (self-consistently) leaves invariant the set of probabili- ties fpi g, hence al l the thermostatistical quantities. It is also trivial to show that, for the indep endent sys- tems A and B mentioned previously, Uq (A + B ) = Uq (A) + Uq (B ), thus recovering the same form of the standard (q = 1) thermo dynamics. It can b e shown that the following relations hold:
X^ W
i=
(pi )q^ = ( Zq )^1 q^ ; (28)
Fq Uq T Sq = 1 (Zq^ )
1 q (^) 1 1 q (29)
Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 7
The nal equilibrium distribution reads
P (^) i( q^ )= [1^ ^ (1^ ^ q^ )^
(^0) i ] 1 qq
PW
k =1 [1^ ^ (1^ ^ q^ )^0 k^ ]^ 1 qq^ :^ (40)
If the energy sp ectrum fi g is asso ciated with the set of degeneracies fgi g, then the ab ove probability leads to (asso ciated with the level i and not the state i)
P (i ) = gi^ [1^ ^ (1^ ^ q^ )^
(^0) i ] 1 qq
P
all lev els gk^ [1^ ^ (1^ ^ q^ )^0 k^ ]^ 1 qq^ :^ (41)
If the energy sp ectrum fi g is so dense that can prac- tically b e considered as a continuum, then the discrete degeneracies yield the function density of states g (), hence
P () = g () [1^ ^ (1^ ^ q^ )^
(^0) ] 1 qq
R
d^0 g (^0 )[1 (1 q ) 0 ^0 ] 1 qq
The density of states is of course to b e calculated for every sp eci c Hamiltonian (given the b oundary con- ditions). For instance, for a d-dimensional ideal gas of particles or quasiparticles, it is given[30] by g () / d^ ^1 , where is the exp onent characterizing the en- ergy sp ectrum / K ^ where K is the wavevector (e.g., = 1 corresp onds to the harmonic oscillator, = 2 corresp onds to a nonrelativistic particle in an in nitely high square well, etc). In Figs. 2 and 3 we see typical energy distributions for the particular case of a constant density of states. Of course, the q = 1 case repro duces the celebrated Boltzmann factor. Notice the cut-o for q < 1 and the long algebraic tail for q > 1. All the ab ove considerations refer, strictly sp eak- ing, to thermodynamic equilibrium. The word thermo- dynamic makes allusion to \very large" (N! 1 , where N is the numb er of microscopic particles of the phys- ical system). The word equilibrium makes allusion to asymptotically large times (t! 1 limit) (assuming a stationary state is eventually achieved). The ques- tion arises: which of them rst? Indeed, although b oth p ossibilities clearly deserve the denomination \thermo- dynamic equilibrium", nonuniform convergences might b e involved in such a way that l imN !1 l imt!1 could di er from l imt!1 l imN !1. To illustrate this sit- uation, let us imagine a classical Hamiltonian system including two-b o dy interactions decaying at long dis- tances as 1 =r on a d-dimensional space, with 0. If
d the interactions are essentially short-ranged, the
two limits just mentioned are basically interchangeable, and the prescriptions of standard statistical mechanics and thermo dynamics are valid, thus yielding nite val- ues for all the physically relevant quantities. In partic- ular, the Boltzmann factor certainly describ es reality, as very well known. But, if 0 d, nonextensivity is exp ected to emerge, the order of the ab ove limits b e- comes imp ortant b ecause of nonuniform convergences, and the situation is certainly exp ected to b e more sub- tle. More precisely, a crossover (b etween q 6 = 1 and q = 1 b ehaviors) is exp ected to o ccur at t = (N ). If l imN !1 (N ) = 1 , then we would indeed have two (or even more) di erent and equal ly legitimate states of thermodynamic equilibrium, instead of the familiar unique state. The conjecture is illustrated in Fig. 4.
Figure 2. Generalizatio n (Eq. (42)) of the Boltzmann factor (recovered for q = 1) as function of the energy E at a given renormalized temp erature T 0 , assuming a con- stant density of states. From top to b ottom at low ener- gies: q = 0 ; 1 = 4 ; 1 = 2 ; 2 = 3 ; 1 ; 3 ; 1 (the vertical line at E =T 0 = 1 b elongs to the limiting q = 0 distribution; the q! 1 distribution collapses on the ordinate). All q > 1 curves have a (T 0 =E )q^ =(q^ 1)^ tail; all q < 1 curves have a cut-o at E =T 0 = 1 =(1 q ).
A wealth of works has shown that the ab ove de- scrib ed nonextensive statistical mechanics retains much of the formal structure of the standard theory. Indeed, many imp ortant prop erties have b een shown to b e q - invariant. Among them, it is mandatory to mention (i) the Legendre transformations structure of thermo- dynamics [14, 15 ]; (ii) the H-theorem (macroscopic time irreversibility),
8 Constantino Tsallis
more precisely, that, in the presence of some irre- versible physical evolution, dSq =dt 0 ; = 0 and 0 if q > 0 ; = 0 and < 0, resp ectively, the equalities holding for equilibrium[3 1 , 32]; (iii) the Ehrenfest theorem (corresp ondence principle b etween classical and quantum mechanics)[33]; (iv) the Onsager recipro city theorem (microscopic time reversibility)[34, 35]; (v) the Kramers and Wannier relations (causality)[35]; (vi) the factorization of the likeliho o d function (Ein- stein' 1910 reversal of Boltzmann's formula)[19]; (vii) the Bogolyub ov inequality[36]; (viii) thermo dynamic stability (i.e., a de nite sign for the sp eci c heat)[37]; (ix) the Pesin equality[38].
Figure 3. Log-log plot of some cases like those of Fig. 2 (T 0 = 1 ; 5 for each value of q ).
In contrast with the ab ove quantities and prop er- ties, which are q -invariant, some others do dep end on q , such as (i) the sp eci c heat [39]; (ii) the magnetic susceptibility [40]; (iii) the uctuation-dissipation theorem (of which the two previous prop erties can b e considered as particular cases) [40]; (iv) the Chapman-Enskog expansion, the Navier-Stokes equations and related transp ort co ecients[41]; (v) the Vlasov equation[42, 43 ]; (vi) the Langevin, Fokker-Planck and Lindblad equations[44, 45, 46, 47, 48]; (vii) sto chastic resonance[49 ]; (viii) the mutual information or Kullback-Leibler
entropy[32, 50]. A remark is necessary with regard to b oth sets just mentioned. Indeed, these prop erties have in fact b een studied, whenever applicable, within unnormal- ized q -exp ectation values for the constraints, rather than within the normalized ones that we are using herein. Nevertheless, they still hold b ecause they have b een established for xed , which, through Eq. (33), implies xed 0.
Figure 4. Central conjecture of the present work, assuming a Hamiltonian system which includes two-b o dy (attractive) interactions which, at long distances, decay as r . The crossover at t = is exp ected to b e slower than indicated in the gure (for space reasons).
Finally, let us mention various imp ortant theoreti- cal to ols which enable the thermostatistical discussion of complex nonextensive systems, and which are now available (within the unnormalized and/or normalized versions for the q -exp ectation values) for arbitrary q. We refer to (i) Linear resp onse theory[35]; (ii) Perturbation expansion[51]; (iii) Variational metho d (based on the Bogoliub ov inequality)[51]; (iv) Many-b o dy Green functions[52];
10 Constantino Tsallis
can check that hhx^2 ii 1 = hx^2 i 1 = R^ 1^1 dx x^2 pq (x) is
nite if q < 5 = 3 and diverges if 5 = 3 q 3 (the norm constraint cannot b e satis ed if q 3). Finally, let us mention that the Gaussian (q = 1) solution is recovered
in b oth limits q! 1 + 0 and q! 1 0 by using the q > 1 and the q < 1 solutions resp ectively. This family of solutions is illustrated in Fig. 5.
Figure 5. The one-jump distributions pq (x) for typical values of q. The q! 1 distribution is the uniform one in the interval [ 1 ; 1]; q = 1 and q = 2 resp ectively corresp ond to Gaussian and Lorentzian distributions; the q! 3 is completely at. For q < 1 there is a cut-o at jxj= = [(3 q )=(1 q )]^1 =^2.
We fo cus now the N -jump distribution pq (x; N ) = pq (x) pq (x) ::: pq (x) (N -folded convolution pro duct). If q < 5 =3, the standard central limit theorem applies, hence, in the limit N! 1 , we have
pq (x; N ) (^) ^1
h 5 3 q
2 (3 q )N
i 1 = 2
exp
(^) 2(3^5 ^3 qq )Nx
2 2
i.e., the attractor in the distribution space is a Gaus- sian, consequently we have normal di usion. If, how- ever, q > 5 =3, then what applies is the Levy-Gnedenko central limit theorem, hence, in the limit N! 1 , we have
pq (N ; x) L (x=N 1 =^ ) (49)
where L is the Levy distribution with index < 2 given by
= (^3) q^ ^ q 1 (5= 3 < q < 3) (50)
Through the Fourier transforms of b oth Eq. (48) and (49), we can characterize the width q (dimensionless
di usion co eecient) of pq (x; N ). We obtain
q (^53) ^ 3 qq (q < 5 =3) (51)
and
q = (^) 12 = 2
h q 1
3 q
i 2(^3 q q1)
h 3 q 5
2(q 1)
i
; (5= 3 < q < 3) : (52) These results are depicted in Fig. 6. This result should b e measurable in sp eci cally devised exp eri- ments. More details can b e found in [80] and refer- ences therein. What we wish to retain in this short review is that the present formalism is capable of (thermo)statistically founding, in an uni ed and simple manner, b oth Gaussian and Levy b ehaviors, very ubiq- uitous in Nature (resp ectively asso ciated with normal di usion and a certain typ e of anomalous sup erdi u- sion).
Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 11
Figure 6. The q -dep endence of the dimensionl ess di u- sion co ecient q (width of the prop erly scaled distribution pq (x; N ) in the limit N! 1 ). In the limits q! 5 = 3 0 and q! 5 = 3 + 0 we resp ectively have q [4=9]=[(5=3) q ] and q [4=(9 1 =^2 ]=[q (5=3)]; also, lim (^) q! 3 q = 2 = 1 =^2.
I I I.2 Correlated-typ e anomalous di usion
There are some phenomena exhibiting anomalous (sup er and sub) di usion of a typ e which di ers from the one discussed in the previous subsection. We refer to the so called correlated-typ e of di usion. We consider here a quite large class of them, namely those asso ci- ated with the following generalized, Fokker-Planck-like equation:
@ @ t [p(x;^ t)]
@ x fF^ (x)[p(x;^ t)]
g + D @^2 x^2 [p(x;^ t)]
(53) where (; ) 2 R^2 , D is a dimensionless di usion- like constant, F (x) dV =dx is a dimensionless ex- ternal force (drift) asso ciated with a p otential V (x), and (x; t) is a dimensionless 1 + 1 space-time. If = 1, we can interpret p(x; t) as a probability dis- tribution since
R
dx p(x; t) = 1 ; 8 t can b e satis- ed. If 6 = 1, then p(x; t) must b e seen as a density function. The word \correlated" is frequently used in this context due to the fact that D (@ 2 =@ x^2 )[p(x; t)]^ = (@ =@ x)fD [p(x; t)]^ ^1 (@ =@ x) p(x; t)g, i.e., an e ective di usion emerges, for 6 = 1, which dep ends on p(x; t) itself, a feature which is natural in the presence of cor- relations. The = 1 particular case of this nonlinear equation is commonly denominated \Porous medium equation", and corresp onds to a variety of physical sit- uations (see [46] and references therein for several ex- amples). The rst connection of Eq. (53) with the present nonextensive statistical mechanics was established in 1995 by Plastino and Plastino[45]. They considered a
particular case, namely = 1 and F (x) = k 2 x with k 2 > 0 (so called Uhlenb eck-Ornstein pro cesses), and found an exact solution which has the form of Eq. (43- 45). Their work was generalized in [46] where arbitrary and F (x) = k 1 k 2 x were considered. The explicit exact solution of Eq. (53), for al l values of (x; t), was once again found by prop osing an Ansatz of the form of Eqs. (45-47), i.e., the form which optimizes Sq with the asso ciated simple constraints. This form eventu- ally turns out to b e the Barenblatt one, useful in re- lated problems. Here, let us restrict ourselves to just repro duce the exact solution of Eq. (53) assuming that p(x; 0) = (x), this is to say, a Dirac delta distribution. We obtain[46]
pq (x; t) = f^1 ^ (1^ ^ q^ )^ (t)[x^ ^ xM^ (t)]
(^2) g 1 =(1 q ) Zq (t) (54) where q = 1 + (55) and (t) (0) =
h Zq (0)
Zq (t)
i 2
with
Zq (t) = Zq (0)
h
1 K^1
2
e t= ^ + (^) K^1 2
i 1 =(+ )
K 2 (^2) D (0)[kZ^2 q (0)] ^
and (^) k 2 (^ +^ ^ )^
Summarizing, by using the form which optimizes Sq , it has b een p ossible to nd the physically relevant solution of a nonlinear equation in partial derivatives with integer derivatives. It can b e shown[81] that the problem that was solved in the previous subsection cor- resp onds to a linear equation in partial derivatives but with fractional derivatives. We b elieve that we are al- lowed to say that an unusual mathematical versatility has b een observed, within the present nonextensive for- malism, in this couple of nontrivial examples of anoma- lous di usion.
I I I.3 Stellar p olytrop es and other self- gravitating systems
The present formalism has b een applied to a vari- ety of astrophysical[42, 43] and cosmological[82] self- gravitating systems. In some sense, this is something
Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 13
c
hhg ainiitakq e^ the^ money= 85 ; 000 (65)
and hhg ainiiplayq the^ g^ ame= 100 ;^000 ^0 :^85
q (^) + 0 0 : 15 q 0 : 85 q^ + 0 : 15 q^ =^
100 ; 000 0 : 85 q 0 : 85 q^ + 0 : 15 q^ (66) Since most p eople would prefer the money, this means that most p eople have q < 1 for this particular decision problem. For the loss prob em we have: hhg ainiitakq e^ the^ money= 85 ; 000 (67)
and hhg ainiiplayq the^ g^ ame= ^100 ;^000 ^0 :^85
q (^) + 0 0 : 15 q 0 : 85 q^ + 0 : 15 q^ =^
100 ; 000 0 : 85 q 0 : 85 q^ + 0 : 15 q^ (68)
d Since in this case most p eople would prefer to play, this means that, consistently with the previous result, most p eople have q < 1 for the particular decision problem we are considering now. In some sense, we have some epistemological progress. Indeed, the state- ment \most p eople have (for this typ e of amount of money) q < 1", uni es the previous two separate state- ments concerning exp ectation to gain and exp ectation to lo ose.
Let us address now the following question: how can we measure the value of q asso ciated with a particular individual? We illustrate this interesting p oint with the example of the gain. The p erson is asked to cho ose b etween having V dollars or playing a game in which, if the p erson wins, the prize will b e 100 ; 000 dollars and, if the p erson lo oses, he (she) will receive nothing. As b efore, the p erson is informed that his (her) probability of winning is 0.85 (hence, the probability of lo osing is 0.15). Then we keep gradually changing the value V and asking what is the preference. At a certain critical value, noted Vc , the p erson will change his (her) mind. Then, the value of q to b e asso ciated with that p erson, for that problem, is given by the following equality
100 ; 000 0 : 85 q 0 : 85 q^ + 0 : 15 q^ =^ Vc^ (69)
(See Fig. 7). The ideally rational op erator corresp onds to q = 1. For this gain problem, the risk-averse op era- tors corresp ond to q < 1, and the risk-seeking ones to q > 1.
Figure 7. The index q to b e asso ciated with a p erson whose critical value corresp onding to Eq. (69) is Vc. People with q < 1 (q > 1) tend to avoid (seek) risks for that particular game. The case q = 1 corresp onds to an ideally rational agent.
In order to b etter understand the unifying p ower of the present prop osal, let us analyze another classical problem, namely that of the two b oxes. A p erson is prop osed to cho ose b etween two games. He (she) is in- formed that in b ox A there are (exactly) 100 balls, 50 of them b eing red, the other 50 b eing white. The p er- son declares a color, and randomly takes o a ball. If it has the chosen color, the p erson will earn 100 dollars. If it has the other color, the p erson will earn nothing.
14 Constantino Tsallis
The p erson is also informed that in b ox B there are also (exactly) 100 balls, some are red, some are white but we do not know how many of each, though we do know that no other colors are in the b ox. As b efore, the p erson is asked to declare a color, and then randomly take o a ball. If it has the chosen color, the p erson will earn 100 dollars. If it has the other color, the p er- son will earn nothing. These are the two games. The p erson is now asked to cho ose the b ox to play. The exp erimental outcome is that most of the p eople cho ose b ox A (p ossibly b ecause their anxiety is smaller with regard to that particular b ox, b ecause they have some supplementary information ab out it...even though this information is completely useless !). Let us write down the asso ciated normalized q -exp ectation values: It is clear that mo dels for sto ck exchange can b e formulated by using these remarks. Such an e ort is presently in
progress[87].
I I I.6 Physiology of vision
Physiological p erceptions such as the visual p ercep- tion are since long known to fo cus up on rare events (e.g., a red sp ot on a white wall). Barlow[88], among others, has recurrently stressed our attention on the fact that, at the action decision level, the various p ossibili- ties should enter with a weight prop ortional to ln pi , and not prop ortional to pi , pi b eing the a priori prob- ability of o ccurrence of that particular event; indeed, ln pi diverges when pi! 0. He even argues that evo- lutionary arguments hold very well together with such hyp othesis. To privilege rare events is precisely what happ ens, in the present formalism, whenever q < 1. Let us b e more sp eci c: if we consider the 0 < q << 1 limit, we obtain[89]
c
Sq =k = 1 ^
PW
i=1 p
q i q 1 ^ W^ ^1 +^ q^ [W^ ^1
X^ W
i=
( ln pi )]; (70)
hO iq
X^ W
i=
pqi Oi
X^ W
i=
Oi q
X^ W
i=
( ln pi ) Oi (71)
and
hhO iiq
PW
Pi=1^ pqi^ Oi
W i=1 p
q i
PW
i=1 Oi W
n
1 + q
h PWi=1 ( ln pi )
W
PW
i=1 P ( ^ ln^ pi^ )^ Oi
W i=1 Oi
io
d
where O is an arbitrary observable. Leaving aside sev- eral constant quantities that app ear ab ove, we imme- diately observe the prominent role which ln pi plays in these expressions. Consistently, the q! 0 limit of the present formalism could well b e of some utility in the theoretical analysis of the physiological phenomena fo cused here.
IV Exp erimental evidences and
connections
IV.1 D = 2 turbulence in pure-electron plasma
A few years ago, in 1994, Huang and Driscoll[6] ex- hibited some quite interesting nonneutral plasma exp er-
imental results obtained in pure-electron plasma (con- ned in a 20 cm long and 6 cm wide metallic cylindric Penning trap with a 10 ^10 torr vacuum in its interior) in the presence of an external axial magnetic eld ( Gauss). In the interval 2-100 ms after every single elec- tric shot (generating the electron plasma), it was ob- served a turbulent axisymmetric metaequilibrium state, the electronic density radial distribution of which was measured. Its average (over typically 100 shots) mono- tonically decreased with the radial distance, disapp ear- ing at some radius sensibly smal ler than the radius of the container (i.e., a cut-o was observed). The ex- p eriment was recently redone[90] under slightly mo di- ed exp erimental conditions (a slow external rotation was imp osed in such a way as to comp ensate the small
16 Constantino Tsallis
c
Sq [g ] (^) q ^1
Z
(g g q^ )d^2 r; (73)
Z
g d^2 r = 1 (mass conser v ation)
R
Rr 2 g^ q^ d^2 r
g q^ d^2 r =^ Lq^ ^ L^ (ang^ ul^ ar^ momentum^ conser^ v^ ation) (^12)
R
R ?^ g^ q^ d^2 r
g q^ d^2 r =^ Uq^ ^ U^ (ener^ g^ y^ conser^ v^ ation);^ (74)
where g (r ) is the probability distribution. Moreover, the scaled electrostatic p otential
(r ) ?^
R
g q^ (r R 0 ) G(r; r^0 )d^2 r^0
g q^ d^2 r with^ r
(^2) G(r; r (^0) ) = 4 (r r (^0) ); (75)
satis es r^2 ? = 4 g^
q
R
g q^ d^2 r :^ (76)
The constrained optimization of Sq [g ] ( (Sq
R
g d^2 r Lq Uq )) now yields 1 q g q q^ ^1 q 1 ^ ^
N q^ r^
(^2) g q q (^1) + N q^
q ?^ g^
q q 1 + q (L N ^2 U^ N )g^
q q 1 = 0 (77)
(where
R
g q^ d^2 r N ) or g (^) q^1 q q q 1 ^ q^
1 q (^) N q^ r^
N q^
q ?^ +^ q^ (L^
N ^2 U^ N )^ =^0 (78)
or, taking the Laplacian of b oth sides,
[1 + (1 q )]
r^2 g (^1) q q q 1 ^4
N q^ +^4 ^ N 2 q^ g^
qq = 0 (79)
which can b e rewritten as
g (^00) q q
(g (^0) q )^2 gq^ +^
g (^) q^0 r =^ g^
qq (B y (^) g (^) qq Ay (^) ) (80)
where Ay^ 4 q (^) N =[1 + (1 q )] and B y^ 4 q (^) N 2 =[1 + (1 q )].
Alternatively, identifying q g (^) qq =N , we have
^00 q 2 q^ q ^1
(^0 q )^2 q^ +^
^0 q r =^ q^
2 q q 1 q (B q A); (81)
d
with A Ay^ N
q q 1 and B B y^ N
2 q q 1
. This equa- tion precisely is the one app earing in [91], which, for q = 1 =2, recovers that of [43]. For any chosen q , the values of the parameters (A; B ) are obtained by imp os-
ing the exp erimental values of total angular momentum and energy. This phenomenological theory has, there- fore, only one tting parameter (q ). As said b efore, q = 1 = 2 exactly repro duces the Huang and Driscoll's
Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 17
Restricted Minimum Enstrophy pro le. The b est over- all tting is, however, obtained for a value of q slightly ab ove 1 =2.
IV.2 Solar neutrino problem
As easily conceivable, the core of the Sun is a very complex and turbulent plasma, within which an enor- mous amount of nuclear reactions take place. Many of them constitute chains of nuclear reactions in which neutrinos are pro duced. For instance, the p-p chain is describ ed in [93]. Through a quite complete analy- sis of the pro duction of neutrinos within the so called Solar Standard Model (SSM), it is p ossible to predict the neutrino ux onto the Earth. However, the actual ux measured in a variety of underground lab oratories (Gallex, Sage, Kamiokande, Sup er-Kamiokande, Home- stake) roughly amounts to only half of the predicted value. This problem is currently referred to as the "so- lar neutrino problem". Two nonexclusive sources of ex- planation of this enigmatic discrepancy are: (i) the p os- sible neutrino oscillations, which would make that only part of the predicted value would b e detectable on the Earth; (ii) the current use of the SSM might b e incor- rect b ecause it uses BG thermal statistics, which could b e inappropriate for the solar plasma. Clayton[10] was the rst to address the second p ossibility, as far as 25 years ago. Indeed, he assumed an hyp othetic distribu- tion of energies essentially given by
p(E ) / e ^ E^ e ^ (^ E^ )^2 (82)
The particular value = 0 obviously recovers BG statistics. Clayton showed that a small value of ( ' 0 :01) was enough to make the theory consistent with the exp erimental data that were available at that time. Quarati and co-workers remarked (preliminar- ily in 1996[94], and in more re ned calculations since then[95]) that, since the needed is very small, the ansatz distribution could as well b e the p ower-law one which app ears in the present formalism. By identifying the rst corrections (to BG) of b oth distributions, they obtained = 1 2 q (83)
Consequently, values of q quite close to unity are enough to t the solar neutrino discrepancy. Once again, we
verify the extreme eciency that mo di cations of the statistics can have.
IV.3 Peculiar velo cities in Sc galaxies
From the data obtained by the Cosmic Background Explorer (COBE), it has b een p ossible to infer the dis- tribution of p eculiar velo cities of certain groups of spi- ral (Sc) galaxies (we recall that by peculiar velo city we mean the residual velo city after the global universe ex- pansion velo city has b een substracted). Bahcall and Oh[11] develop ed four theoretical attempts (namely Cold Dark Matter with = 0 : 3 and with = 1 :0, Hot Dark Matter with = 1 : 0 and Primeval Barionic Isotropic with = 0 :3). All the attempts were done within BG statistics. The less unsatisfactory tting was obtained for the CDM mo del with = 0 :3. In fact, all the attempts exhibit a long tail towards high velo cities, whereas the exp erimental data show a pronounced cut- o at ab out 500 K m s ^1. It is relevant to mention that all the mo dels that were used had several tting parameters, and nevertheless could not get rid of the tail. A tting was then advanced[96] using the present formalism with only two free parameters, one of them b eing q and the other one a characteristic velo city. The function that was used was the q -generalized Maxwell distribution, essenti ally corresp onding to an ideal clas- sical gas. The quality of the tting is quite remarkable, far b etter than those corresp onding to the already men- tioned four attempts. Once again, one sees that mo di- cations of the statistics can b e sensibly more ecient than mo di cations of the mo del. A famous example along this line is provided by the completely di erent physics asso ciated with a gas of free fermions or of free b osons, i.e., a Fermi-Dirac ideal gas or a Bose-Einstein ideal gas (same mo del but di erent statistics).
IV.4 Nonlinear inverse bremsstrahlung absorp- tion in low pressure argon plasma
Liu et al[12] provided in 1994 strong evidence of the existence of non-Maxwellian velo city distributions in a sp eci c plasma exp eriment, where low pressure argon is exp osed to pulsed discharges. During the afterglow, measurements of the inverse bremsstrahlung of intense
Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 19
measurements done at the Mount Pamir lead cham- b ers). This distribution was recently tted by Wilk and Wlo darcsyc[105] with the q = 1 : 3 function which emerges within the present formalism.
Figure 8. Distribution of the transverse momenta pT ob- tained in electron-p ositron frontal collisi ons of energy W varying from 14 to 161 Gev. The dotted line corresp onds to q = 1 (i.e., a Hagedorn typ e of tting as given by Eq. (87)) for all values of W. The solid lines corresp ond to q 6 = 1 ttings.
Figure 9. The values of q and T 0 used in the ttings of Fig. 8. When W approaches zero, q approaches unity, i.e., Hagedorn's theory; T 0 is essentially insensitive to W , as physically desirable.
IV.8 Reasso ciation of heme-ligands in folded proteins
In the folded conformational state, proteins might exhibit fractal e ects. One such case might b e the
time evolution of the re-asso ciation of molecules that have b een taken away from their natural p ositions. For instance, if O 2 molecules are disso ciated, through light ashes, from their natural F e p ositions in a heme protein and reach p ositions outside the heme p o cket, they tend to start rebinding, and, for so doing, they might have to follow a fractal path, or b e under the dynamical in uence of fractal excitations (e.g., frac- tons). Anyhow, this re-asso ciation phenomenon has b een lengthily studied by Frauenfelder et al[106]. If we de ne N (t)=N (0) where N (t) is the numb er of molecules that have not yet re-asso ciated at time t, the (t) monotonically vanishes with t. The results ob- tained by photo-disso ciating C O molecules from Sigma Typ e 2 sp erm whale Myoglobin (Mb) dissolved in a glycerol-water solution are shown in Fig. 10. For times not to o long, the exp erimental data have b een tted by Frauenfelder et al[106] with
= (1 + t=t 0 ) n^ (88)
where t 0 and n smo othly dep end on the temp erature T. Bemski, Mendes and myself[107] have argued that, within the generalized formalism, the following equa- tion naturally app ears: d dt =^ q^ ^
q (^) (q 0; q 1) (89)
Its solution is given by
= 1 [1 + (q 1)q t] q ^11 (90)
This expression recovers, for q = 1, the usual ex- p onential relaxation, and repro duces the Frauenfelder form through the identi cations 1 =(q 1) n and 1 =[(q 1)q ] t 0. Besides reobtaining the Frauenfelder empiric law, the present scheme allows for a b etter ap- proximation if a crossover is admitted. More precisely, the ab ove di erential equation can b e generalized as follows: d dt =^ r^ ^
r (^) (q r ) (r q ) (91)
The general solution involves[107] a hyp ergeometric function. The tting is shown in Figs. 10 and 11. A detailed mo del which would justify the ab ove phe- nomenological di erential equation would b e welcome.
20 Constantino Tsallis
Figure 10. Time evolution of N (t)=N (0) asso ciated with M bC O in glycerol-water. Dots: exp erimental data. Dashed lines: ttings with Frauenfelder's empiric law (Eq. (88) or Eq. (90)). Solid lines: ttings with the solutions of Eq. (91) (see Fig. 11).
Figure 11. Temp erature dep endences of the parameters used to t the exp erimental data of Fig. 10.
IV.9 Di usion of Hydra Vulgaris
Upadhyaya et al[108] are presently p erforming in- teresting exp eriments on Hydra Vulgaris (a cylindrical b o dy column with inner and outer cells, resp ectively re- ferred to as endo dermal and ecto dermal resp ectively) in
physiological solution. The endo dermal cells are more adhesive than the ecto dermal ones. The authors have measured the velo city distribution P (jVy j) of the \ver- tical" comp onent of the velo city during the di usion of endo dermal Hydra cells in an ecto dermal aggregate. The results are presented in Fig. 12, where the velo city unit is 10 ^6 m=hour and the probability is represented by the histogram of the numb er of counts. These results were tted with
P (jVy j) = (^) (1 + bajV y j^2 )c^
with the values of (a; b; c) indicated in the gure. Through the identi cation
a = P (0); b = (q 1)=V 02 ; c = (^) q q 1 (93)
we precisely have the law which emerges within the present formalism, namely
P (jVy j) = (^) [1 + (q 1)(PV^ (0) y =V 0 )^2 ]q^ =(q^ 1)^
with q = 1 :53. The next desirable step of course is to formulate a sp eci c mo del for Hydra which would lead to this law, but this remains to b e done.
IV.10 Citation of scienti c pap ers
An interesting study was recently done by Redner[109], in which the statistics of citations of scien- ti c pap ers is fo cused. He exhibited the numb er N (x) of pap ers which have b een cited x times for two long se- ries, namely one (6 716 198 citations of 783 339 pap ers) from the Institute of Scienti c Information (ISI) and another one (351 872 citations of 24 296 pap ers) from the Physical Review D (PRD). As exp ected, in b oth examples, N (x) monotonically decreases with x. Red- ner tted the (relatively) low-x data with a stretched exp onential of the form
N (x) = N (0) e (x=x^0 )^ (95)
with = 0 : 44 and 0 : 39 for the series ISI and PRD resp ectively. Also, he remarked that the large-x data exhibit a p ower law, namely close to / 1 =x^3. He argues that this di erent functional b ehavior for low and large values of x must re ect di erent phenomenologies in
