Constructing a statistical mechanics for Beck-Cohen superstatistics
Constantino Tsallis1,*and Andre M. C. Souza1,2,†1
Centro Brasileiro de Pesquisas Fı´sicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil 2Departamento de Fı´sica, Universidade Federal de Sergipe, 49100-000 Sa˜o Cristo´va˜o-SE, Brazil
共Received 3 July 2002; published 6 February 2003兲
The basic aspects of both Boltzmann-Gibbs 共BG兲 and nonextensive statistical mechanics can be seen through three different stages. First, the proposal of an entropic functional (SBG⫽⫺k兺ipiln pi for the BG
formalism兲 with the appropriate constraints (兺ipi⫽1 and 兺ipiEi⫽U for the BG canonical ensemble兲. Second, through optimization, the equilibrium or stationary-state distribution ( pi⫽e⫺Ei/Z
BG with ZBG⫽兺je⫺Ejfor
BG兲. Third, the connection to thermodynamics 共e.g., FBG⫽⫺(1/)ln ZBGand UBG⫽⫺(/)ln ZBG). Assum-ing temperature fluctuations, Beck and Cohen recently proposed a generalized Boltzmann factor B(E)
⫽兰0⬁d f ()e⫺E. This corresponds to the second stage described above. In this paper, we solve the
corre-sponding first stage, i.e., we present an entropic functional and its associated constraints which lead precisely to B(E). We illustrate with all six admissible examples given by Beck and Cohen.
DOI: 10.1103/PhysRevE.67.026106 PACS number共s兲: 05.70.Ln, 05.20.Gg, 05.70.Ce
The foundation of statistical mechanics and thermody-namics is a fascinating and subtle matter. Its far reaching consequences have attracted deep attention since more than one century ago共see, for instance, Einstein’s remark on the Boltzmann principle关1兴兲. The field remains open to new pro-posals focusing nonequilibrium stationary states, such as me-taequilibrium and others. One of such proposals is nonexten-sive statistical mechanics, advanced in 1988 关2,3兴 共see Ref.
关4兴 for reviews兲. This formalism is based on an entropic
in-dex q 共which recovers usual statistical mechanics for q
⫽1), and has been applied to a variety of systems, covering
certain classes of both共meta兲equilibrium and nonequilibrium phenomena, e.g., turbulence 关5兴, hadronic jets produced by electron-positron annihilation关6兴, cosmic rays 关7兴, motion of
Hydra viridissima 关8兴, quantum chaos 关9兴, and
one-dimensional 关10兴 and two-dimensional 关11兴 maps, among others. In addition to this, it has been noted that it could be appropriate for handling some aspects of long-range interact-ing Hamiltonian systems 共see Ref. 关12兴, and references therein兲.
Let us be more precise. Nonextensive statistical mechan-ics is based on the entropic form共here written for W discrete events兲 Sq⫽k 1⫺
兺
i⫽1 W piq q⫺1冉
i兺
⫽1 W pi⫽1;q苸R冊
, 共1兲 with S1⫽SBG⫽⫺k兺
i⫽1 W piln pi. 共2兲If we focus on the canonical ensemble 共system in contact with a thermostat兲, we must add the following constraint 关3兴:
兺
i⫽1 W piqEi兺
i⫽1 W piq⫽Uq 共U1⫽UBG兲, 共3兲
where兵Ei其is the set of eigenvalues of the Hamiltonian with given boundary conditions. Optimizing Sq, we straightfor-wardly obtain the distribution corresponding to the equilib-rium, metaequilibequilib-rium, or stationary state, namely,
pi⫽ 关1⫺共1⫺q兲q共Ei⫺Uq兲兴1/(1⫺q) Z ¯q , 共4兲 with Z ¯ q⫽
兺
j⫽1 W 关1⫺共1⫺q兲q共Ej⫺Uq兲兴1/(1⫺q), 共5兲 and q⫽ 兺
j⫽1 W pqj , 共6兲 being the Lagrange parameter. We easily verify that, for
q⫽1, we recover the celebrated BG equilibrium distribution
pi⫽ e⫺Ei ZBG , 共7兲 with ZBG⫽
兺
j⫽1 W e⫺Ej. 共8兲*Email address: tsallis@cbpf.br
Very recently, Beck and Cohen have proposed关13兴 a gener-alization of the BG factor. More precisely, assuming that the inverse temperature  might itself be a stochastic variable, they advance
B共E兲⫽
冕
0
⬁
d
⬘
f共⬘
兲e⫺⬘E, 共9兲 where the distribution f (⬘
) satisfies冕
⫺⬁ ⬁d
⬘
f共⬘
兲⫽1. 共10兲It is clear that f (
⬘
)⫽␦(⬘
⫺) recovers the usual BG fac-tor. They have also shown that, if f (⬘
) is the ␥ 共or 2) distribution, then the distribution associated with nonexten-sive statistical mechanics is reobtained. They have also illus-trated their proposal with the uniform, bimodal, log-normal, and F distributions. Moreover, they define 共see also Ref.关14兴兲 qBC⫽
具
2典
具
典
2冉
具
•••典
⬅冕
⫺⬁ ⬁ df共兲共•••兲冊
, 共11兲 where we have introduced the notation qBC in order to avoid confusion with the present q. Clearly, if f (⬘
) is the␥ dis-tribution, then qBC⫽q 共see Ref. 关13兴兲. Finally, they argue that whenever兩qBC⫺1兩Ⰶ1, for all admissible f (), we can write the asymptotic expression B(E)⫽具
e⫺E典
⯝e⫺具典E(1⫹2E2/2), where 2⫽
具
2典
⫺具
典
2. This expression coin-cides with the expansion of the power-law function that rep-resents the generalized Boltzmann factor associated with nonextensive statistical mechanics if we use Eq. 共11兲 and identify qBC⫽q. In other words, nonextensive statistical me-chanics would correspond, for this particular mechanism, where nonextensivity is driven by the fluctuations of , to the universal behavior whenever the fluctuations are rela-tively small.This is no doubt a very deep and interesting result, but it does not constitute by itself a statistical mechanics. The rea-son is that the factor B(E) has been introduced through what can, in some sense, be considered as an ad hoc procedure. The basic element which is missing in order to be legitimate to speak of a statistical-mechanical formalism is to be able to
derive the factor B(E) from an entropic functional with
con-crete constraints 共and especially the energetic constraint, which generates the concept of thermostat temperature兲. The purpose of the present paper is to exhibit such entropic form and constraint.
Let us first write a quite generic entropic form共from now on k⫽1 for simplicity兲, namely,
S⫽
兺
i⫽1 W
s共pi兲 关s共x兲⭓0;s共0兲⫽s共1兲⫽0兴. 共12兲 For the BG entropy SBG, we have s(x)⫽⫺x ln x, and for the nonextensive one Sq, we have s(x)⫽(x⫺xq)/(q⫺1). The
function s(x) should generically have a definite concavity
᭙x苸关0,1兴. Conditions 共12兲 imply that S⭓0 and that
cer-tainty corresponds to S⫽0.
Let us now address the constraint associated with the en-ergy. We consider the following form:
兺
i⫽1 W u共pi兲Ei兺
i⫽1 W u共pi兲 ⫽U 关0⭐u共x兲⭐1;u共0兲⫽0;u共1兲⫽1兴. 共13兲For the BG internal energy UBG, we have u(x)⫽x, and for the nonextensive one Uq, we have u(x)⫽xq. The function
u(x) should generically be a monotonically increasing one.
Certainty about Ej implies U⫽Ej. The quantity
u( pi)/兺j⫽1 W u( p
j) constitutes itself a probability distribution 共which generalizes the escort distribution defined in Ref. 关15兴兲. The constraint associated with the energy applies, in
principle, for the共meta兲equilibrium state. However, its valid-ity共either exact or approximate兲 has been verified in various nonequilibrium stationary states related with nonextensive statistical mechanics 共e.g., turbulence 关5兴, granular matter
关18兴兲. For example, we may interpret the energy spectrum
used in the constraint in the same spirit as the temperature used by Beck and Cohen. It represents the mean value of some fluctuation distribution. The general foundational ques-tion and its possible geometrical interpretaques-tion 共in phase space兲 remain, nevertheless, open.
Let us consider now the functional
⌽⬅S⫺␣
兺
i⫽1 W pi⫺兺
i⫽1 W u共pi兲Ei兺
i⫽1 W u共pi兲 , 共14兲where ␣ and  are Lagrange parameters. The condition
⌽/pj⫽0 implies s
⬘
共pj兲⫺␣⫺ 兺
i⫽1 W u共pi兲 u⬘
共pj兲共Ej⫺U兲⫽0. 共15兲Let us now heuristically assume
u
⬘
共x兲⫽⫹s⬘
共x兲, 共16兲u共x兲⫽x⫹s共x兲⫹. 共17兲
But the condition s(0)⫽u(0)⫽0 implies⫽0, and the con-ditions s(1)⫽0 and u(1)⫽1 imply that ⫽1, hence,
u共x兲⫽x⫹s共x兲 共18兲
and
Observe that if ⫽0, we have u(x)⫽x and 兺Wj⫽1u( pj)
⫽兺j⫽1 W
pj⫽1.
The replacement of Eq.共19兲 into Eq. 共15兲 yields
s
⬘
共pi兲⫽ ␣⫹ Ei⫺U兺
j⫽1 W u共pj兲 1⫺ Ei⫺U兺
j⫽1 W u共pj兲 , 共20兲 hence pi⫽共s⬘
兲⫺1冉
␣⫹ Ei⫺U兺
j⫽1 W u共pj兲 1⫺ Ei⫺U兺
j⫽1 W u共pj兲冊
, 共21兲where (s
⬘
)⫺1(•••) is the inverse function of s⬘
(•••). The condition 兺iW⫽1pi⫽1 enables the 共analytical or numerical兲 elimination of the Lagrange parameter ␣. The function共21兲 is to be identified with B(E)/兰0⬁dE⬘
B(E⬘
) from Ref.关13兴. In other words, if E(y ) is the inverse function ofB(E)/兰0⬁dE
⬘
B(E⬘
), we have thats共x兲⫽
冕
0 x d y ␣⫹E共y兲⫺U兺
j⫽1 W u共pj兲 1⫺ E共y兲⫺U兺
j⫽1 W u共pj兲 , 共22兲which, together with Eq.共18兲, completely solves the problem once is determined. Summarizing, given an admissible function B(x), we have uniquely determined the functions
s(x) and u(x), which replaced into Eqs.共12兲 and 共13兲
con-cludes the formulation of the statistical mechanics associated with the Beck-Cohen superstatistics. An important remark remains to be made, namely, any monotonic function of S given by Eq. 共12兲 also is a solution at this stage. Which of those is to be retained for a possible connection with ther-modynamics is a different matter, and remains an open issue at the present stage. For example, for nonextensive statistical mechanics, in what concerns the stationary distribution, Sq and the Renyi entropy SqR⫽ln关1⫹(1⫺q)Sq兴/(1⫺q) are equivalent. In other words, at this level, we could indistinc-tively use Sq or Sq
R
. There is, however, a variety of physical arguments which are out of scope of the present work but which nevertheless point, in that particular case, Sq as being the correct physical quantity to be used for thermodynamic
and dynamic purposes. A strong argument along this line concerns the stability of the entropy under arbitrarily small deformation of the statistical state probabilities 关16兴. For in-stance, Abe关17兴 and Lesche 关16兴, respectively, showed that
Sq is stable and Sq
R is unstable.
Let us now compare the B(E) factor obtained by Beck-Cohen formalism with the distribution presented here in Eq.
共21兲. In addition to the fact that the former does not include
the normalization constant whereas the latter does, we notice that the B(E) factor has parameters such as0⬅
具
典
, instead of the parameters ␣, U, , and /兺Wj u( pj) 关generalization of Eq. 共6兲兴 appearing in Eq. 共21兲. This problem will be handled as follows. Since our aim is to determine thefunc-tional forms of s(x) and u(x), it is enough to work with only
one variable. So, we take 0⫽/兺j W
u( pj)⫽1 and U⫽0. Let us now determine. Using Eqs.共15兲 and 共19兲, and inte-grating, we obtain
u共x兲⫽共1⫹␣兲
冕
0
x dy
1⫺E共y兲. 共23兲
It is physically reasonable to assume that u(x) monotonically increases with x, hence du/dx⭓0, and (1⫹␣)/(1⫺E)
⭓0. We shall verify later that 1⫹␣⬎0, hence it must be
⭐1/E. If we note E*, the lowest admissible value of E, we are allowed to consider ⫽1/E*. In particular, if E*→
⫺⬁, then it must be⫽0. An example where E*is finite is nonextensive statistical mechanics with q⬎1. In this case,
E*⫽1/(1⫺q), hence⫽1⫺q. We can trivially verify that this value for , together with s(x)⫽(x⫺xq)/(q⫺1) and
u(x)⫽xq precisely satisfy Eq.共18兲.
Summarizing, the final form of u(x) is given by
u共x兲⫽共1⫹␣/E*兲
冕
0 x dy 1⫺E共y兲/E*, 共24兲 and, therefore, s共x兲⫽冕
0 x dy ␣⫹E共y兲 1⫺E共y兲/E*. 共25兲 In what follows, we shall illustrate the above procedure by addressing all the admissible examples appearing in Ref.关13兴. The cases associated with the Dirac␦ and the␥ distri-butions for f ()共respectively corresponding to BG and non-extensive statistical mechanics兲 can be handled analytically. The other four cases 关uniform, bimodal, log-normal, and F distributions for f ()] have been treated numerically as fol-lows. We first choose f (), then calculate B(E), and from this calculate 兰0⬁dE
⬘
B(E⬘
). By inverting the axes of the variables, we find the inverse E(y ) of Beck-Cohen supersta-tistics. From this, we obtain E*. Two cases are possible. The first one corresponds to E*→⫺⬁, hence ⫽0, u(x)⫽x, and s(x)⫽␣x⫹兰0xd y E( y ). The condition s(1)⫽0deter-mines␣, which is therefore given by␣⫽⫺兰01d y E( y ). The
second case corresponds to a finite and known value of E*, which determines ⫽1/E*. From this, we calculate
兰0
Eq. 共24兲, it is easy to see that 1⫹␣⫽1⫹␣/E*
⫽1/兰0
1d y /关1⫺E(y)/E*兴⬎0 and it is direct to determine␣, which in turn enables the calculation of u(x) using once again Eq.共24兲. Finally, from Eq. 共18兲, we calculate s(x), and the problem is solved.
In Figs. 1共a兲 and 1共b兲, typical examples of s(x) and u(x) are presented for all the cases addressed in Ref.关13兴. In Fig. 2, we show the entropies associated with all these examples assuming W⫽2.
Let us conclude by saying that it has been possible to find expressions for the entropy and for the energetic constraint that lead to a generic Beck-Cohen superstatistics. Of course,
similar considerations are valid for other constraints if we were focusing on say the grandcanonical ensembles. The step we have discussed is necessary for having the statistical mechanics generating these superstatistics through a varia-tional principle. What remains to be done is the possible connection with thermodynamics. This is not a trivial task because unless we are dealing with a nonlinear power law for
u(x) 共which precisely is nonextensive statistical mechanics兲,
the Lagrange parameter ␣ is not factorizable in Eq. 共21兲, hence no partition function can be defined in the usual sense, i.e., a partition function which depends on  共and other analogous parameters兲, but does not depend on␣. Summa-rizing, nonextensive statistical mechanics not only paradig-matically represents, as shown in Ref. 关13兴, the universal behavior of all Beck-Cohen superstatistics in the limit qBC ⯝q⯝1, but it is the only one for which an ␣-independent partition function can be defined.
Last but not least, let us emphasize that the present results strengthen the idea that the statistical-mechanical methods can be in principle used out of equilibrium as well. To be more specific, we can think of using them共i兲 in equilibrium
共e.g., in the t→⬁ limit of noninteracting or short-range
in-teracting Hamiltonians, as well as in the lim
N→⬁limt→⬁ of long-range interacting many-body Hamiltonian systems; this is essentially BG statistical mechanics兲, 共ii兲 in
metaequilib-rium 共e.g., in the lim
t→⬁limN→⬁ of long-range interacting many-body Hamiltonian systems; see, for instance, Ref.
关12兴兲, and 共iii兲 for appropriate classes of stationary states 共see, for instance, Refs. 关5,18兴兲. Further foundational work
would be welcome for case共iii兲.
Useful remarks from F. Baldovin are gratefully acknowl-edged. Partial support from PCI/MCT, CNPq, PRONEX, and FAPERJ 共Brazilian agencies兲 is also acknowledged.
FIG. 1. Functions s(x) 共a兲 and u(x) 共b兲 for illustrative ex-amples of the cases focused on in Ref. 关13兴. From top to bottom:
共i兲 F distribution 关 f ()⫽24/(2⫹)4兴; 共ii兲 bimodal 关 f ()
⫽0.5␦(⫺1/2)⫹0.5␦(⫺3/2)兴; 共iii兲 log normal 关 f ()
⫽(1/
冑
2)e⫺[(0.5⫹ln)2/2]兴; 共iv兲 uniform 关 f ()⫽1 on the interval关1/2,3/2兴, and zero otherwise兴; 共v兲Dirac ␦ 关 f ()⫽␦(⫺1)兴; 共vi兲
␥ 关 f ()⫽关()0.25
/(0.8)1.25⌫(1.25)兴e⫺1.25兴, which corresponds to
q⫽1.8.
FIG. 2. W⫽2 entropies associated with the examples presented in Fig. 1.
关1兴 A. Einstein, Ann. Phys. 共Leipzig兲 33, 1275 共1910兲; 关‘‘Usually W is put equal to the number of complexions. In order to
calculate W, one needs a complete 共molecular-mechanical兲 theory of the system under consideration. Therefore it is dubi-ous whether the Boltzmann principle has any meaning without a complete molecular-mechanical theory or some other theory which describes the elementary processes. S⫽(R/N)log W
⫹const seems without content, from a phenomenological point
of view, without giving in addition such an
Elementartheo-rie’’.兴
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http://tsallis.cat.cbpf.br/biblio.htm
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共2002兲; Nonextensive Entropy—Interdisciplinary Applications,
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关6兴 I. Bediaga, E.M.F. Curado, and J. Miranda, Physica A 286, 156 共2000兲; C. Beck, ibid. 286, 164 共2000兲.
关7兴 C. Tsallis, J.C. Anjos, and E.P. Borges, e-print
astro-ph/0203258.
关8兴 A. Upadhyaya, J.-P. Rieu, J.A. Glazier, and Y. Sawada, Physica
A 293, 549共2001兲.
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214101共2002兲.
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Fractals 8, 885 共1997兲; U.M.S. Costa, M.L. Lyra, A.R. Plas-tino, and C. Tsallis, Phys. Rev. E 56, 245共1997兲; M.L. Lyra and C. Tsallis, Phys. Rev. Lett. 80, 53共1998兲; U. Tirnakli, C. Tsallis, and M.L. Lyra, Eur. Phys. J. B 10, 309共1999兲; J. Yang and P. Grigolini, Phys. Lett. A 263, 323 共1999兲; F.A.B.F. de Moura, U. Tirnakli, and M.L. Lyra, Phys. Rev. E 62, 6361
共2000兲; V. Latora, M. Baranger, A. Rapisarda, and C. Tsallis,
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共2002兲.
关11兴 F. Baldovin, C. Tsallis, and B. Schulze, Physica A 共to be
pub-lished兲, e-print cond-mat/0203595.
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056134共2001兲.
关13兴 C. Beck and E.G.D. Cohen, Physica A 共to be published兲,
e-print cond-mat/0205097.
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