Anomalous behavior of q-averages in nonextensive statistical mechanics

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1 Anomalous behavor of -averages n nonextensve statstcal mechancs Sumyosh Abe,2,3 Department of Physcal Engneerng, Me Unversty, Me , Japan 4 2 Insttut Supéreur des Matéraux et Mécanues Avancés, 44 F. A. Barthold, Le Mans, France 3 Inspre Insttute Inc., McLean, Vrgna 220, USA E-mal: suabe@sf6.so-net.ne.jp Abstract. A generalzed defnton of average, termed the -average, s wdely employed n the feld of nonextensve statstcal mechancs. Recently, t has however been ponted out that such an average value may behave unphyscal under specfc deformatons of probablty dstrbutons. Here, the followng three ssues are dscussed and clarfed. Frstly, the deformatons consdered are physcal and may expermentally be realzed. Secondly, n vew of thermostatstcs, the -average s unstable n both fnte and nfnte dscrete systems. Thrdly, a nave generalzaton of the dscusson to contnuous systems msses a pont, and a norm better than the L -norm should be employed for measurng the dstance between two probablty dstrbutons. Conseuently, stablty of the -average s shown not to be establshed n all the cases. 4 Permanent address.

2 . Introducton There exst a number of thermostatstcal systems n nature, whch are exotc from the tradtonal vewpont of Boltzmann-Gbbs statstcal mechancs. They often possess/exhbt broken ergodcty, strong correlatons between elements leadng to nseparablty, nontrval portrats of phase spaces or confguraton spaces and longrange nteractons, for example. In the past decade, nonextensve statstcal mechancs [, 2], whch s a generalzaton of the Boltzmann-Gbbs theory, has been expected to offer a framework for descrbng the propertes of such systems. The prevalng formulaton of nonextensve statstcal mechancs employs a generalzed defnton of average termed the -average [see euaton (5) below]. In ths paper, we rgorously examne f such a generalzaton s possble. Consder measurement of a physcal uantty, Q= { Q } = 2,,...,, of a thermostatstcal system. To obtan relable nformaton on the probablty dstrbuton, the measurement has to be repeated. In realty, two probablty dstrbutons, { p } =, 2,..., and { p' } =, 2,...,, thus obtaned may slghtly be dfferent from each other. Here, stands for the number of accessble mcrostates and s a very large number, typcally beng Such a dfference can be uantfed, for example, by comparng the average values of Q wth respect to those probablty dstrbutons. Snce Q s supposed to behave well, the calculated average values are expected to be close to each other. Ths natural reurement s mathematcally expressed by the followng formal 2

3 predcate [3]: ( > ) ( ) ( ) ( ) ε 0 δ > 0 p p' < δ Q [ p] Q [ p'] < ε () for any par of probablty dstrbutons, { p } =, 2,..., and { p' } =, 2,...,, where p p' = p p' = (2) s the dstance between these two probablty dstrbutons defned n terms of the l - p norm, and Q [ p] ( Q [ p']) stands for the average of Q wth respect to { } =, 2,..., ({ p' } =, 2,..., ). Other knds of norms could also be consdered, but what s relevant to dscrete systems s the present l -norm, whch s able to make p p' ndependent of (see the later dscusson). The average, Q, s sad to be stable or robust, f the condton n euaton () s satsfed. Ths s somewhat analogous to Lesche s stablty condton on entropc functonals [4]-[0] (see also [] for a comment on [8]). Mathematcally, t s concerned wth unform contnuty of a functonal. In the scheme n euaton (), t s mportant to note that the condton on comes after those on ε and δ. Ths mples that the large- lmt, for example, has to be taken at the end of calculaton. The standard defnton of average, referred to as the normal average, 3

4 Q [ p]= Q p (3) = s clearly stable. Ths s smply seen as follows: Q [ p] Q [ p'] = Q ( p p' ) = Q p p' = Q p p', (4) max where Q max Q max,,..., { } =2. Thus, there, n fact, exsts δ such that δ = ε / Q. Ths s actually an obvous result snce Q [ p] s a bounded lnear max functonal of { p } =, 2,...,. Now, n the feld of nonextensve statstcal mechancs [, 2], a possblty s open for generalzng the defnton of average. There, the so-called -average Q [ p] = = Q j = ( p ) ( p ) j ( > 0), (5) s prevalng. Ths uantty s reduced to the normal average n euaton (3) n the lmt. In general, t s a nonlnear functonal of { p } =, 2,...,, and ths fact makes the problem of ts stablty nontrval. 4

5 The purpose of ths paper s to dscuss and clarfy that the -average s unstable unless. To explctly do so, we consder specfc deformatons of probablty dstrbutons. Then, we present the followng three results. Frstly, the deformatons consdered here are physcal and may be realzed n a laboratory. Secondly, n vew of thermostatstcs, stablty of the -average cannot be verfed n both fnte and nfnte dscrete systems. Then, thrdly, a nave generalzaton of the l -norm n euaton (2) to the- L norm for contnuous systems msses an mportant physcal pont. Thus, we conclude that stablty of the -average cannot be establshed n all the cases, and accordngly the formulaton of nonextensve statstcal mechancs has to be amended. 2. Specfc deformatons of probablty dstrbutons and ther physcalty To evaluate the uantty Q [ p] Q [ p'], let us examne the followng specfc deformatons of the probablty dstrbutons: ) for 0< < ; p δ δ = δ, p' = p +, (6) 2 2 ) for > ; 5

6 p = δ δ ( δ ), p' = p + δ. (7) 2 2 These have been consdered n the context of stabltes of generalzed entropes n [4, 5]. In both ) and ), holds p p = δ, whch s n fact ndependent of, makng ' stablty analyss possble freely from the system sze. Here, we wsh to menton that the deformatons from p to p' n euatons (6) and (7) are ndeed physcal and may even be realzed n uantum-mechancal experments. Ths s because as follows. Recall that the uantum -average reads ˆ [ ˆ] ( ˆ Q ρ = Tr Q ˆ ρ )/ Tr (ˆ ρ ), where ˆρ and ˆQ are a densty matrx and an observable, respectvely. The deformed densty matrx, ˆ' ρ, correspondng to p' s n euatons (6) and (7) s gven by the convex combnaton of the completely random state, I ˆ / (wth Î beng the dentty matrx), and the normalzed pure egenstate, u (correspondng to the frst egenvalue Q of ˆQ) n dmensons: u Iˆ ˆ' ρ = λ + ( λ) u u, (8) where 0 λ. λ s gven n terms of the fdelty, F, wth respect to the reference state, u u, as λ = ( F ) /( ), provded that / F. In case ), λ = ( δ / 2) /( ), whereas λ = ( δ / 2) /( ) n case ). ˆ' ρ n euaton 6

7 (8) s referred to as the erner state [2] when u s maxmally entangled. It s known [3, 4] that such a state can be generated for a bpartte system usng spontaneous parametrc down-converson. For a large number of uanta, there s a techncal dffculty n realzng a maxmally entangled state. However, u does not have to be maxmally entangled: any pure egenstate s suffcent n our dscusson, avodng the man techncal dffculty. In addton, we pont out the fact [5] that the state n euaton (8) s actually the thermal state ˆ' ρ = e Z ( β) β Hˆ ˆ β H ( Z( β) = Tr e ) (9) wth the projector Hamltonan, Ĥ = g u u, where g s a postve constant havng the dmenson of temperature, β. λ s related to the nverse temperature β and g as β λ = /( + e g ). 3. Dscrete systems Frst, let us evaluate Q [ p] Q [ p'] for the deformatons n euatons (6) and (7). A straghtforward calculaton shows that, n case ), 7

8 Q [ p] Q [ p'] = Q ( ) ( [ ]) ( δ / 2) Q + δ / 2( ) Q Q ( δ / 2) + ( δ / 2) ( ), (0) and, n case ), Q [ p] Q [ p'] = Q Q ( ) ( ) + [ δ / 2 ( δ / 2)/( ) ] Q Q Q ( δ / 2) + ( δ / 2) ( ), () = where Q = ( / ) Q s the arthmetc mean of Q. In the lmt, Q [ p] Q [ p'] converges to Q Q n both cases ) and ). Therefore, the condton n euaton () s volated, and the -average s unstable n such a lmt. It s worth mentonng that the lmts and do not commute, snce the normal average s stable as shown n euaton (4). Next, let us evaluate euatons (0) and () wth that s fnte but very large, 23 0 typcally beng ~2 n thermostatstcal systems. In case ), n order for Q [ p] Q [ p'] to be small, t s necessary that ( δ / 2) << ( δ / 2) holds. Ths condton leads to 8

9 δ << (2) for = / 2. Smlarly, n case ), should hold ( δ / 2) << ( δ / 2) leads to, whch δ << / 3 (3) for = 3/ 2. These values of δ are extremely small, and t s unlkely that realzaton of such overwhelmngly hgh precson s physcally possble n measurements of probablty dstrbutons. Bascally, t s reasonable to beleve that f the result s dfferent between 23 0 ~2 and, then the model tself must be pathologcal. Conseuently, n vew of thermostatstcs, the -averages are unstable n both fnte and nfnte dscrete systems. 4. Contnuous systems In a recent paper [6], t has been clamed that the -averages are stable n contnuous systems. The dscusson gven there s based on a nave generalzaton of the l -norm n dscrete systems to the L -norm, wth whch the dstance between two normalzed probablty denstes, f ( x) and f'( x), defned n the range 0 x s gven by 9

10 f f ' = dx f( x) f '( x) 0. (4) It s shown [6] that the -average, Q [ f] dxq( x)[ f( x)] / dx' [ f( x')] = 0 0, satsfes Q [ f] Q [ f'] < cδ α, where δ = f f ', and c and α are postve constants. However, the above dscusson msses a pont. The dstance n euaton (4) poorly descrbes closeness between f ( x) and f'( x). To see t, let us look at the followng smple deformaton: 2 2δ f ( x)=, f'( x)= 2 3 2δ 2 2δ 0 x 2. (5) 2 2δ < x 2 The dstance between them defned by the L -norm s calculated to be f f ' = δ, although they are actually ute dstnct from each other. In mathematcs, better norms are known. One such example s [7] f f ' = sup f( x) f ' ( x). (6) 0 x 0

11 Clearly, the followng neualty holds: f f' f f', (7) mplyng that the dstance n euaton (6) ntroduces, n the space of probablty dstrbutons on the unt nterval, metrc topology fner than that defned by the L - norm. In fact, euaton (6) for f ( x) and f'( x) n euaton (5) yelds f f = δ ' + 2δ, (8) δ whch s larger than /δ f δ < 2. Thus, we see that stablty of the -averages cannot be establshed also n contnuous systems. 5. Concludng remarks e have dscussed and clarfed that stablty of the -averages cannot be establshed n both dscrete and contnuous systems. In addton, we have also ponted out that the specfc deformatons of the probablty dstrbutons consdered n the dscusson are ute physcal and may be realzed n an optcal-physcs laboratory. In a recent work [8], t has been shown that the normal-average formalsm of

12 nonextensve statstcal mechancs (and the references uoted theren) s consstent wth the generalzed H-theorem, whereas the -average formalsm s not. There are also dscussons about a number of conceptual dffcultes wth the -averages [9]. In addton, t s suggested n [20] that based on these observatons the homogeneous entropy, nstead of Tsalls entropy, should be used for descrbng asymptotcally power-law dstrbutons. The problem of the defnton of averages s one of the cruxes n nonextensve statstcal mechancs. The present results combned wth those presented n [3, 8] show that what to be employed n nonextensve statstcal mechancs s the normal averages, and not the -averages. Acknowledgment Ths work was supported n part by a Grant-n-Ad for Scentfc Research from the Japan Socety for the Promoton of Scence. References [] Abe S and Okamoto Y (eds.), 200 Nonextensve Statstcal Mechancs and Its 2

13 Applcatons (Hedelberg: Sprnger-Verlag,) [2] Tsalls C, 2009 Introducton to Nonextensve Statstcal Mechancs: Approachng a Complex orld (New York: Sprnger Scence+Busness Meda) [3] Abe S, 2008 Europhys. Lett [4] Lesche B, 982 J. Stat. Phys Lesche B, 2004 Phys. Rev. E [5] Abe S, 2002 Phys. Rev. E Abe S, 2004 Physca D Abe S, 2004 Contn. Mech. Thermodyn [6] Naudts J, 2004 Rev. Math. Phys [7] Abe S, Kanadaks G and Scarfone A M, 2004 J. Phys. A [8] Curado E M F and Nobre F D, 2004 Physca A [9] Abe S, Lesche B and Mund J, 2007 J. Stat. Phys [0] Ubraco M R, 2009 Preprnt [] El Kaabouch A, ang Q A, Ou C J, Chen J C, Su G Z and Le Méhauté A, 2009 Preprnt [2] erner R F, 989 Phys. Rev. A [3] Zhang Y-S, Huang Y-F, L C-F and Guo G-C, 2002 Phys. Rev. A [4] Barber M, De Martn F, D Nep G, Matalon P, D Arano G M and Macchavello C, 2003 Phys. Rev. Lett

14 [5] Abe S, Usha Dev A R and Rajagopal A K, 2008 Preprnt [6] Hanel R, Thurner S and Tsalls C, 2009 Europhys. Lett [7] Yosda K, 97 Functonal Analyss 3rd edton (Berln: Sprnger-Verlag) [8] Abe S, 2009 Phys. Rev. E [9] Abe S, Conceptual dffcultes wth the -averages n nonextensve statstcal mechancs, to appear n Proceedngs of Statstcal Mechancs and Mathematcs of Complex Systems. [20] Lutsko J F, Boon J P and Grosfls P, 2009 Preprnt , to appear n Europhys. Lett. 4