Brazilian Journal of Physics, vol. 29, no. 1, March, Ensemble and their Parameter Dierentiation. A. K. Rajagopal. Naval Research Laboratory,

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1 Brazilian Journal of Pysics, vol. 29, no. 1, Marc, Fractional Powers of Operators of sallis Ensemble and teir Parameter Dierentiation A. K. Rajagopal Naval Researc Laboratory, Wasington D. C , USA Received 7 December, 1998 We develop four identities concerning parameter dierentiation of fractional powers of operators appearing in te sallis ensembles of quantum statistical mecanics of nonextensive systems. In te appropriate limit tese reduce to te corresponding dierentiation identities of exponential operators of te Gibbs ensembles of extensive systems derived by Wilcox. I Introduction Wilcox [1] in 1967 publised a seminal paper entitled \Exponential Operators and Parameter Dierentiation in Quantum Pysics". is paper was centered around te following remarkable identity: ten If ^H() is an operator depending on a parameter, c ^Q(; ) =, du ^Q(; ) ^Q (; u) ^H() ^Q(; u) = exp,u ^H() : ^Q(; u) ; (1) d From tis e went on to obtain several oter identities in elegant ways wic are all central in te development of quantum time evolution, Gibbsian ensembles in equilibrium quantum statistical mecanics, perturbation expansions, inequalities concerning correlation functions etc., all of wic depend on te appearance of te exponential operator of te form introduced in Eq.(1). For a compreensive account of te ramications of tis identity, see Appendix A in Vol. I of Grandy's extbook on Statistical Mecanics [2]. In Appendix D of te Vol.II of is text, e uses tis identity to establis inequalities of various covariance functions wic are just te quantum mecanical variances and covariances of quantities of interest. It is useful to recall tat te exponential form in te Gibbsian ensemble arises from te principle of maximum von Neumann entropy, S 1 =,r^ ln ^; wit r^ = 1, subject to te given mean energy of te system, E = r^ ^H, asexplained in [2], for example. In 1988, sallis [3] introduced a dierent ensemble for describing a large variety of nonextensive systems for wic te exponential form of te operator is replaced by a monomial fractional power of te form i q=(1,q) ^Q (; ) = 1, (1, q) ^H() ; (2) q species te system nonextensivity. Nonextensivity comes about in systems wit long-range interactions between constituent particles of te system. is goes over to te Gibbsian form given in Eq.(1) wen q is set equal to unity, wic is appropriate wenever

2 62 A.K. Rajagopal te system consists of particles wic are eiter noninteracting or interacting wit sort-range forces. e operator in Eq.(2) replaces te exponential operator in Eq.(1) wen statistical expectation values of quantities of interest are to be calculated in te sallis formalism. Here te sallis entropy S q =,r(^, ^ q )=(1, q), is maximized subject to te given q-expectation value (a dierent form of te q-expectation value to be de- ned later is sometimes preferred) of te Hamiltonian, E q = r^ q ^H, besides te usual normalization, r^ =1. It is in tis form (as well as te dierent form alluded to above) tat te formal structure of Statistical Mecanics is preserved namely - Legendre transform nature of te free energy. is ensemble as now been used to develop [4] a Green function teory of manyparticle nonextensive systems in muc te same way as te Gibbsian ensemble is used in te Green function teory of te corresponding extensive systems. Wit tis development, te program of statistical mecanics of nonextensive systems is acieved in a manner tat is formally complete in parallel to te conventional statistical mecanics of extensive systems. It may be important to give te reader some compelling reasons for setting up anoter ensemble different from te traditional Boltzmann-Gibbs ensemble for dealing wit nonextensive systems. We rst recall tat among te statistical distributions, te exponential - class played very important role in te analysis of many penomena. (see E.. Jaynes [5] for a discussion of tese aspects). ese can all be derived from a maximum entropy principle subject to some constraints in wic te entropy functional is cosen to be te Gibbs-von Neumann form, S 1 =,r^ ln ^. ere are many oter probability distributions possessing long tails suc as Pareto, Levy, etc. wic are of te monomial - class, not related to te exponential class. ese are not derivable from te maximum entropy principle wit te Gibbs-von Neumann form for te entropy functional. ese cover many penomena wic do not come under te rubric of \extensive" class of systems wic were traditionally treated in pysical and oter sciences. It is tis important gap tat te sallis ensembles cover. As far as we are aware, tere is no matematical or pysical argument to rule out te applicability of te sallis ensemble nor do we know any demonstration tat te exponential-class covers every conceivable situation in pysical and oter sciences so tat te universality of te Boltzmann{Gibbs ensemble may be considered as te only one paramount form. In discussing sensitivity to initial conditions of nonlinear dynamical systems, similar \exponential"and \power" law sensitivities ave been discussed recently [6,7] in te context of te use of von Neumann (Kolmogorov and Sinai) and sallis entropies in teir quantication. e sallis ensemble wit q 6= 1 deals wit Hamiltonians of systems wit long-range interactions wic may exibit nontrivial anomalies in teir ergodicity and mixing properties. For systems wic are noninteracting or interacting systems wit sort-range forces, one certainly as q = 1 (Boltzman-Gibbs class). II Four teorems In tis paper, we rst obtain te counterpart of te Wilcox teorem, Eq.(1), for te sallis form of te operator, and ten deduce tree oters. It is not out of place ere to mention tat some of tese ave been recently stated witout derivation [8] in developing te dynamic linear response of a nonextensive system based on sallis framework. It is wort pointing out tat every one of tese teorems as its counterpart in te exponential version, derived by Kubo, Karplus and Scwinger in dierent pysical situations of te respective autors' interests. Wilcox's teorem unies all of tese in a single, elegant form, from wic all oters follow. c HEOREM I: ^Q (; ) =,q du ^Q (; ) ^Q (; u) ^~ H(; u) ^Q (; u) ;

3 Brazilian Journal of Pysics, vol. 29, no. 1, Marc, ^~ H(; u) ^H() = 1, u(1, q) ^H() 1, u(1, q) ^H() : (3) (4) Prof: Let ^F (; ) = ^Q (; ) : (5) en dierentiating wit respect to, intercanging te order of dierentiation, using te denition (4), and using te denition (3) given above, we obtain te dierential equation ^F (; ) + q =,q ^~ H(; ) 1, (1, q) ^H()i ^H() ^F (; ) ^Q (; ) : Using te explicit form given in Eq.(2), and dening te inverse operator, ^Q (; ), in te usual way, we recast Eq.(6) in te form i ^Q (; ) ^Q (; ) ^F (; ) =,q ^~ H(; ) ^Q (; ) : (7) (6) From tis expression te teorem I is establised upon integration of bot sides of Eq.(7), because ^Q (; =)=1 and ^F (; =)=. e above eorem reduces to te Wilcox teorem [1] wen q is set equal to unity. is derivation is more direct tan te one used in Ref.[1,2] for te Gibbsian ensemble. is is our central teorem. From tis, we deduce te following teorem by applying eorem I using a similarity transformation to generate te -dependence of ^H(). HEOREM II: If ^A is an arbitrary operator, ten te commutator ^A; ^Q ()i ^Q () = ^A(u) = = q ^Q () 1, (1, q) ^H ^A; ^Q (; )i is given by du ^Q(u) ^H; A(u) i ^ ^Q (u) i q=(1,q) ^Hi 1, u(1, q) ^A ; 1, u(1, q) ^Hi : (8) Proof: We deduce tis teorem by taking ^H() in eorem I as a similarity transformation of te following form en it follows tat ^H() = (exp ^A) ^H (exp, ^A) : (9) ^Q(; ) = (exp ^A) ^Q ()(exp, ^A) ; (1) we used te denition in Eq.(8). en, we obtain te following expressions tat appear in Eq.(3) i ^A ^Q () = exp ^A; ^Q () exp, ^A ; and (11) H() ^~ = exp ^A^ A(u); ^H exp, ^A i

4 64 A.K. Rajagopal wic in eorem I lead to te result in Eq.(8). In te limit wen q = 1, tis goes to te well-known Kubo identity [2] wic was of muc use in is teory of irreversible processes. We now employ eorem I to deduce anoter important result concerning te parametric derivative of an sallisexpectation value of an arbitrary operator, often useful in computing te linear response function of a sallis-mean value of a quantity of interest. We must remark ere tat tis denition of te mean value diers from te denition rst proposed in [3] and as te advantage of aving te property tat te mean value of a scalar constant, C, is C itself, wic was not te case in its original formulation. Wile tis entails some canges in te formalism it does not cange te basic Legendre structure of te statistical mecanical principles. For a discussion of te implications and ramications of tese aspects, one may refer to [9]. For q = 1 tis goes over to te result for te usual termodynamic average [2]. HEOREM III: en Dene te sallis-expectation value of an arbitrary operator ^B [9] as. ^Bi = r^b ^Q (; ) r^q (; ) (12) E D ^B =,q =,q ^A = ^A, D ^A E D E dund ^A(; u) ^B, ^A(; u) E o nd ^A(; y) du E ^B ; A(; u) = ^Q (; u) ^~ H(; u) D ^BE o ; (13) ^Q (; u) : Proof: is follows upon dierentiation of te dening expression in Eq.(12) and subsequently using eorem I directly. Quite often, one uses tis expression wen ^H() = ^H + ^V, and evaluates te expression in Eq.(13) for = to obtain te correlation function of interest. Finally,we develop a Karplus-Scwinger perturbation type teory for te sallis ensemble in te next eorem. is teorem is not in te form of eorem I but its proof involves steps similar to te one used in eorem I and ence its inclusion ere. HEOREM IV: If ^H = ^H + ^H 1, ten ^Q () R (; ^H) 1, u(1, q) ^Q ; ^H + ^H1 = ^Q() (; ^H ), q ^Q() (; ^H ) ^H + ^H1 1, u(1, q)( ^H ) H1 ^~ (u) ^Q (u; ^H + ^H1 ) ; du ^~H 1 (u) =1, u(1, q) ^H ^H1 1, u(1, q) ^H : (14) Here ^Q (; ^H + ^H1 ) = 1, (1, q)( ^H + ^H1 ) q=(1,q) ; (15) ^Q () (; ^H ) = q=(1,q) 1, (1, q) ^H :

5 Brazilian Journal of Pysics, vol. 29, no. 1, Marc, Proof: Consider n o ^Q (; ^H ) ^Q (; ^H + ^H 1 ) n = ^Q (; ^H )o ^Q (; ^H + ^H 1 )+ n ^Q (; ^H + ^H 1 )o ^Q (; ^H ) : (16) Performing te indicated dierentiations, and after some algebra we nd o nq (; ^H ) ^Q (; ^H + ^H 1 ) =,q ^Q (; ^H ) 1, (1, q)( ^H + ^H) ^H1 1, (1, q) ^H ^Q (; ^H + ^H 1 ) (17) d Introducing te notation given in Eq.(14), upon integration of bot sides of Eq.(17) wit respect to, from to, and using ^Q ( =; ^H) = 1, we obtain te result stated in eorem IV above. III Summary In summary, weave ere deduced a set of four teorems on parametric dierentiation of te operator dening te sallis ensemble, wic we ope are of use in te same way as te Wilcox teorems were for te operator dening te Gibbsian ensemble. Several versions of tis paper were read by Professor sallis. anks are due to im for making valuable suggestions to improve te presentation. is work is supported in part by te Oce of Naval Researc. References [1] R. M. Wilcox, J. Mat. Pys. 8, 962 (1967). [2] W..Grandy, Jr. Foundations of Statistical Mecanics, Vol. I: Equilibrium eory, and Vol. II:Nonequilibrium Penomena, D. Reidel Publising Co., Boston (1988). [3] C. sallis, J. Stat. Pys. 52, 479 (1988). Since tis paper rst appeared, a large number of papers based on tis work ave been and continues to be publised in a wide variety of topics. A list of tis vast literature wic is continually being updated and enlarged may be obtained from ttp://tsallis.cat.cbpf.br/biblio.tm [4] A. K. Rajagopal, R. S. Mendes, and E. K. Lenzi, Pys. Rev. Lett. 8, 397 (1998). See also an expanded version of tis work by E. K. Lenzi, R. S. Mendes, and A. K. Rajagopal, Pys. Rev. E 59, 1398 (1999). [5] E.. Jaynes: Papers on Probabilty, Statistics, and Statistical Pysics, edited by R. D. Rosenkrantz, D. Reidel Publising Co., Boston (1983). [6] M. L. Lyra and C. sallis, Pys. Rev. Lett. 8, 53 (1998). [7] C. Anteneodo and C. sallis, Pys. Rev. Lett. 8, 5313 (1998). [8] A. K. Rajagopal, Pys. Rev. Lett. 76, 3469 (1996). [9] C. sallis, R. S. Mendes, and A. R. Plastino, Pysica A 261, 534 (1998).