onuniueness of canonical enseble theory arising fro icrocanonical basis arxiv:uant-ph/99097 v2 25 Oct 2000 Suiyoshi Abe and A. K. Rajagopal 2 College of Science and Technology, ihon University, Funabashi, Chiba 274-850, Japan 2 aval Research Laboratory, Washington, D.C. 20375-5320, USA Given physical systes, counting rule for their statistical echanical descriptions need not be uniue, in general. It is shown that this nonuniueness leads to the existence of various canonical enseble theories, which eually arise fro the definite icrocanonical basis. Thus, the Gibbs theore for canonical enseble theory is not universal, and axiu entropy principle is to be appropriately odified for each physical context. PACS nubers: 05.20.-y, 05.20.Gg, 05.45.Df, 05.90.+
It is generally believed that canonical enseble theory arising fro the icrocanonical basis is uniue. This conviction sees to have its origins in the deonstration given by Gibbs []. In this Letter, we show that it is actually not true. A crucial point regarding this rearkable fact is that, given a syste, rule for counting its physical configurations is not uniue and is context-dependent, in general. A counting rule is an algorith, which establishes a connection between entropy, a acroscopic uantity, and the nuber of icroscopic configurations of the syste under consideration. The ost celebrated rule is that of Boltzann, who eployed the logarithic function for counting the icroscopic configurations. A wide class of physical systes is covered by this logarithic counting rule. However, strictly speaking, there is no a priori reason to assue such a special counting rule to be copletely universal. In fact, there ay yet exist a class of systes which sees to prefer alternate kinds of counting rules. Specifically, nonextensive systes, such as systes with longrange interactions, long-range eories or (ulti)fractal structure, belong to this class. In what follows, we first suarize the derivation of ordinary canonical enseble theory fro icrocanonical basis with the principle of eual a priori probability based on the logarithic counting rule. Then, we present a new counting rule, which leads to a non-gibbsian canonical enseble, showing the nonuniueness of canonical enseble theory. { } =2 Consider a syste s and take replicas s, s 2, L, s of it. S = s α α,, L, is referred to as a supersyste. Let A α be a physical uantity associated with s α and is assued to be bounded fro below. It can be thought of as the syste energy, for exaple. Its value is denoted by a ( α ), where α labels the allowed configurations of s α. A uantity of interest is given by its average over the supersyste: ( / ) A α. Microcanonical enseble theory states that the probabilities of finding S in the configurations where the values of the acroscopic uantity lies around a certain value a, α 2
i.e. a a < ε () α α = are all eual. Here, ε goes to zero with. The ordinary central liit theore and law of large nubers indicate that ε ~ O( / ). However, if the Lévy-type power-law canonical distribution is in ind, relevant atheatics is the Lévy-Gnedenko generalized central liit theore, and accordingly the assuption on the order of -dependence of ε should appropriately be odified. In Ref. [2], a way of deriving ordinary canonical enseble theory is presented based on the logarithic counting rule. The probability p of finding the syste s s in its th configuration is given by the ratio of two nubers of eually-probable configurations {,, L, }, i.e. p W W 2 =. Here, W is the total nuber of configurations satisfying e. (), whereas W is deterined by the following two conditions: a) s is found in =, and b) a in the condition in e. () is the arithetic ean over the configurations of S. To calculate it explicitly, we rewrite e. () as follows: a a [ ]+ a a α α = 2 < ε. (2) The nuber of configurations Y satisfying a a α α = 2 < ε (3) 3
is counted in the large- liit as follows: ln Y ln a Y a S a [ ], (4) where Sa is soe function of a and will be identified with the entropy. This logarithic counting rule iposes the additional extensivity assuption on the entropy. Fro es. (2)-(4), we have ln W ln Y ln a a a Y a a a S = [ ] [ ] a. (5) Therefore, defining as Sa = a (6) and putting Z ( )= li W Y a, (7) we obtain the canonical distribution p = exp [ a a] Z { }, (8) 4
= with the ordinary canonical partition function Z where Z Z exp a [ ] exp a. = In axiu entropy principle [3], the distribution in e. (8) is derived fro the Boltzann-Shannon entropy S = p ln p (9) under the constraints on the noralization of p and the expectation value pa = a. (0) Then, is the Lagrange ultiplier associated with this constraint. ote that euation (0) is euivalent to the condition Z a = 0. () On the other hand, the entropy in e. (9) is calculated to be S = ln Z ( ). (2) Therefore, euation () is actually the axiu entropy condition. Consistency of e. (2) with e. (4) identifies 5
[ ] = Z li Y a. (3) Also, fro e. (7), W is found to be + [ ] W li Y a =. (4) In the above derivation, the logarithic counting in e. (4) plays an essential role. ow, let us exaine another kind of counting rule. Specifically, we exaine to replace e. (4) by ln Y ln a Y a [ ] S a. (5) Here, ln x is the -logarith defined by ln x = x ( ), (6) which is the inverse function of the -exponential e [ + ( x ) ] + ( x ) > 0 ( x) = 0 ( otherwise), (7) where is a positive paraeter. In the liit, ln x and e ( x) converge to the 6
ordinary ln x and e x, respectively. As the ordinary logarith, ln x is also a onotonically increasing function of x. S a in e. (5) is a certain uantity dependent on a and will be shown to be the Tsallis nonextensive entropy [4]. Instead of e. (5), now our algorith is given as follows: ln W = ln Y a [ a a ] ln Y ( a) ln a a a [ ] Y ( a), (8) provided that ay be large but finite. Defining S( a) = a (9) and using e. (5), we have = [ ] Y a Y a. (20) a Therefore, we find ln Y( a)= Y( a) a [ ] ( ). (2) Substituting e. (2) into e. (8), we have 7
ln = [ ] [ ] W ln Y a Y a a a. (22) Taking advantage of the relation x ln = y ln x ln y y, (23) we obtain W Y a ( [ ]) e * a a, (24) where * c (25) with c Y a S a. (26) [ ] = + Fro e. (24) and the total nuber of configurations W = W, we construct * Y a Z W ( [ ]) e a a. (27) 8
Conseuently, we find the probability p W W = to be p = * e( [ a a] ). (28) Z The property corresponding to e. () here translates to Z a = 0, (29) leading to the result a =< A> P a, (30) where P is the escort distribution [5] P [ p ] = p [ ]. (3) Euation (30) shows that the arithetic ean is identical with the noralized - expectation value introduced in Ref. [6]. ow, we show that consistency of the whole discussion is achieved if the generalized entropy is given by S = ln Z ( ), (32) 9
which should be copared with e. (2). Accordingly, euation (29) becoes the axiu generalized entropy condition. Consistency between es. (32) and (5) leads to the identification Z Y a [ ], (33) as in e. (3). Furtherore, fro e. (27), W is seen to be + [ ] W Y a, (34) which is forally siilar to e. (4). ow, note that the identical relation [ ] [ p ] = Z ( ) (35) holds for the distribution in e. (28) with e. (30). Cobining this with e. (33), we have [ Y ( a) ] = p [ ]. (36) Therefore, c in e. (26) is found to be c = p. (37) [ ] 0
Finally, using es. (33), (36) and (37), we ascertain that both es. (26) and (32) consistently lead to S [ ] = p, (38) which is precisely the Tsallis entropy [4]. In the liit, es. (28), (30) and (38) converge to es. (8), (0) and (9), respectively. We end our discussion with ephasizing the following points. p ) in e. (28) is ( the generalized canonical distribution in nonextensive statistical echanics [6], which is derived via the axiu entropy principle based on the Tsallis entropy in e. (38) subject to the constraints on the noralization of p and the noralized -expectation value of A given in e. (30) and is known to lead to the consistent therodynaic foralis. in e. (9) is the Lagrange ultiplier associated with e. (30). Another iportant point is that if the liit is strictly taken, for exaple, in e. (27), then the difference between Z ( ) in e. (7) and Z ( ) in e. (27) ay disappear. In other words, sees to go to unity in the liit. This suggests that nonuniueness of counting rules ay be intiately related to the fact that physically is large but actually finite. Such an observation is consistent with the discussion developed in Refs. [7,8]. This leads to a fundaental uestion regarding a connection between the concepts of extensivity and therodynaic liit. These considerations, in turn, indicate that nonextensive statistical echanics would also be iportant for understanding therodynaic properties of sall systes. In conclusion, we have found that canonical enseble theory arising fro the
conventional icrocanonical basis with the principle of eual a priori probability is not uniue. We have shown the nonuniueness by eploying a non-logarithic counting rule, the -logarithic counting, as an exaple. We have found that, in this case, the resulting canonical enseble theory is Tsallis nonextensive foralis. We stress that the concept of the arithetic ean is the sae throughout the discussion: the nonuniueness is brought about through counting rule and concoitant definition of the statistical expectation value. We wish to ention that the sae result is obtained by using the ethod of steepest descents [9] and the Lévy-Gnedenko generalized central liit theore [0]. The authors would like to thank Professor Roger Balian for drawing their attention to Ref. [2], where discussions about uniueness of the Gibbs canonical enseble theory are developed. The present work arose fro a careful reexaination of Ref. [2]. S. A. was supported in part by the GAKUJUTSU-SHO Progra of College of Science and Technology, ihon University. A. K. R. acknowledges the support of the U.S. Office of aval Research. References [] J. W. Gibbs, Eleenrary Principles in Statistical Mechanics (Yale University Press, ew Haven, 902). [2] R. Balian,. L. Balazs, Ann. Phys. (Y) 79 (987) 97. [3] E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics, edited by R. D. Rosenkrantz (D. Reidel, Boston, 983). 2
[4] C. Tsallis, J. Stat. Phys. 52 (988) 479. [5] C. Beck, F. Schlögl, Therodynaics of chaotic systes: An introduction (Cabridge University Press, Cabridge, 993). [6] C. Tsallis, R. S. Mendes, A. R. Plastino, Physica A 26 (998) 534. For applications of nonextensive statistical echanics, see, for exaple, Braz. J. Phys. 29 Special Issue (999), which can be obtained fro http://sbf.if.usp.br/www_pages/journals/bjp/vol29/u/index.ht A coprehensive list of references is currently available at http://tsallis.cat.cbpf.br/biblio.ht [7] A. R. Plastino and A. Plastino, Phys. Lett. A 93 (994) 40. [8] D. H. E. Gross, Micro-canonical statistical echanics of soe non-extensive systes e-print (cond-at/0004268). [9] S. Abe and A. K. Rajagopal, Microcanonical foundation for systes with power-law distributions, to appear in J. Phys. A. [0] S. Abe and A. K. Rajagopal, Justification of power-law canonical distributions based on generalized central liit theore, to appear in Europhys. Lett.. 3