Nonadditive thermostatistics and thermodynamics

Transcription

1 Journal of Physcs: Conference Seres Nonaddtve thermostatstcs and thermodynamcs To cte ths artcle: P Ván et al 2012 J. Phys.: Conf. Ser Vew the artcle onlne for updates and enhancements. Related content - Thermodynamcs of composton rules T S Bró, K Ürmössy and Z Schram - Temperature n non-equlbrum states J Casas-Vázquez and D Jou - On the generalzed entropy pseudoaddtvty for complex systems Qupng A Wang, Laurent Nvanen, Alan Le Méhauté et al. Recent ctatons - Quas-power laws n multpartcle producton processes Grzegorz Wlk and Zbgnew Wodarczyk - Statstcal Power Law due to Reservor Fluctuatons and the Unversal Thermostat Independence Prncple Tamás Bró et al - Non-extensve statstcs and understandng partcle producton and knetc freeze-out process from pt-spectra at 2.76 TeV Bhaskar De Ths content was downloaded from IP address on 07/07/2018 at 02:54

2 Journal of Physcs: Conference Seres 394 (2012) IOP Publshng do: / /394/1/ Nonaddtve thermostatstcs and thermodynamcs P Ván 1,2, G G Barnaföld 1, T S Bró 1, and K Ürmössy 1 1 Wgner RCP, Hungaran Academy of Scences H-1525 Budapest, P.O.Box 49, Hungary 2 Department of Energy Engneerng, Budapest Unversty of Technology and Economcs, Bertalan Lajos u. 4-6, H-1111, Budapest, Hungary E-mal: van.peter@wgner.mta.hu Abstract. Nonaddtve composton rules for several physcal quanttes are treated n thermodynamcs. It s argued that the zeroth law defnes the exstence of ther addtve forms, the formal logarthms. A further prncple, the unversal thermostat ndependence leads to a partcular formal logarthm, equvalent to Tsalls entropy S q. We connect q wth generalzed susceptbltes of the thermostat. 1. Introducton Nether thermodynamcs nor statstcal physcs ncludes the other. Near-equlbrum statstcal physcs fulflls thermodynamc relatons partcular mcroscopc models can clarfy certan mechansms behnd thermodynamc quanttes. On the other hand the general laws of thermodynamcs are vald also n those cases when the detaled mcroscopc treatment s unavalable or hopelessly complcated. The amng at unversalty n thermodynamcs should be understood from ths pont of vew. Whenever one deduces model relatons from thermodynamc prncples, the results are by constructon ndependent of partculartes of the mcroscopc model. The above relaton s present n attempts of explanng among others power-law tals n the transverse momentum spectra of hgh energy collsons based on statstcal consderatons. These power-law tals can be observed n heavy on collsons at the RHIC and LHC accelerators [1] cf. Fg. 1, n proton-proton collsons at the LHC, RHIC, Fermlab and SPS [2] cf. Fg. 2 and even n electron-postron collsons [3], cf. Fg. 3. One nspects, that n electron-postron collsons, where the number of outcomng partcles s about 10-50, the devaton from Tsalls dstrbuton s salent. The presented data ndcate the unversal nature of the cut power law dstrbutons. These fts perform much better also at low p T than the exponental Boltzmann-Gbbs dstrbuton [4]. Therefore a sngle mcroscopc mechansm, lke perturbatve QCD, cannot explan all observatons. In ths paper we propose a unversal explanaton of the observed Tsalls dstrbutons. Ths s based on general arguments extendng the framework of classcal thermostatcs by abandonng the addtvty property constrant on the extensve quanttes by ntroducng abstract, n a general settng nonaddtve composton rules [5, 6]. By dong so the zeroth law of thermodynamcs reveals that the very concept of separate physcal systems thermodynamc bodes restrct those rules. In the followng we are gong to prove that rules fulfllng the zeroth law are assocatve and can be gven n a specal form wth the help of the so called formal logarthms. Then the thermodynamc temperature can be ntroduced as an ntensve Publshed under lcence by IOP Publshng Ltd 1

3 Journal of Physcs: Conference Seres 394 (2012) IOP Publshng do: / /394/1/ Fgure 1. Tsalls fts to transverse momentum dstrbutons of dentfed partcles stemmng from AuAu collsons at s = 200 AGeV collson energy. The expanson of the quark-gluon plasma s taken nto account by the blast wave model. X = 1 a ln(1+aγ(m T vp T )) and β = 1/T. (Fg. 1b n Ref. [1]). Fgure 2. Tsalls fts on transverse momentum spectra of charged hadrons n proton-proton collson on a logarthmc plot (Fg. 1a n Ref. [2]). 2

4 Journal of Physcs: Conference Seres 394 (2012) IOP Publshng do: / /394/1/ Fgure 3. Fragmentaton functons from electron-postron collsons measured at varous collson energes and ftted one-dmensonal Tsalls dstrbutons. The energy fracton x = ɛ/e s the energy of the produced hadron scaled by the maxmal acqurable energy n a 2-jet event E = s/2 (Fg. 5 n Ref. [3]). quantty characterzng the thermodynamc bodes separately. The extenson of the maxmum entropy prncple for nonaddtve composton rules becomes straghtforward. Nonaddtvely extended thermodynamcs ensures that thermostatstcal relatons represented by dfferent statstcal entropy formulas can be nterpreted n a thermodynamc framework [7]. Ths framework can be further refned when dealng wth the choce among several dfferent generalzatons of thermostatstcal entropes (e.g. [8, 9, 10, 11, 12, 13, 14]. One would welcome a reasonable crtera formulated n general thermodynamcs [15]. We show that consderng a canoncal, maxmum entropy based approach to an open statstcal physcal system. The requrement of vanshng frst order correctons n the mcrocanoncal treatment of a system ncludng the thermostat, leads to a partcular formal logarthm, that transforms the Boltzmann Gbbs Shannon entropy densty to a Tsalls entropy densty. Ths leads to a physcal nterpretaton of the q parameter n the entropy formula, related to the heat capacty of the reservor. In ths way the Tsalls entropy densty and the Rény total entropy are connected by the selected formal logarthm. 2. The zeroth law of thermodynamcs and nonaddtve composton rules 2.1. Separablty The basc varables of classcal thermodynamcs are the so called extensve physcal quanttes, lke the energy (E), partcle number of the th chemcal component (N ) and entropy (S). The 3

5 Journal of Physcs: Conference Seres 394 (2012) IOP Publshng do: / /394/1/ adjectve extensve s often assocated to a dfferent characterstcs of these fundamental physcal quanttes. It s customary to assume that the the extensve quanttes are addtve f we merge two thermodynamc bodes: X 12 = X 1 + X 2, (1) where X denotes one of the above mentoned quanttes of the th body. Ths specal property of composton s addtvty. On the other hand tradtonally one assumes that these quanttes characterze the systems up to the largest meanngful scale operatng wth fnte denstes of the extensve quanttes [16]: 1 ρ X = lm N N N X <. (2) Here one dvdes the body to N dfferent parts and X belongs to the th of them. Ths property s the extensvty. These two propertes are related, but far from beng equvalent. If a quantty s addtve, then t s necessarly extensve, but there are extensve and yet non-addtve quanttes [6, 9]. In our treatment the classcally extensve quanttes are n general nonaddtvely composed, therefore n the followngs we refer to them as basc quanttes. Another mportant class of thermodynamc quanttes are the ntensves. Ther defnton s related to thermodynamc equlbrum and to the zeroth and the second laws of thermodynamcs. The zeroth law s formulated as the requrement of the transtvty of the thermodynamc equlbrum state [17, 18]. Somewhat mplct, but mportant n ths respect, that the thermodynamc bodes are assumed to be separable of each other: n thermodynamc equlbrum ther respectve equaton of states do not depend on the propertes of the partner bodes or of the nteracton. Ths condton of physcal separablty s requred also n case of nonaddtve composton of the extensves. We shall ntroduce the concept of ntensves accordngly. In the followng we generalze our reasonng wth a nonaddtve entropy and nonaddtve energy as basc quanttes developed n [7] to nonaddtve entropy and addtonal n nonaddtve quanttes. Let us consder two thermodynamc bodes, characterzed by n basc quanttes denoted by X1 and X 2, = 1,..., n for the two bodes respectvely and X0 1 = S 1, X2 0 = S 2 are the entropes. The entropy plays a dstngushed role, t defnes the equaton of state of the respectve bodes, S(X 1,..., X n ). In our treatment the basc quanttes are not addtve, but composable, therefore ther value n the two-body system s expressed by the composton rules =1 X 12(X 1, X 2) X 1 + X 2, = 0,..., n. (3) Ths functonal relatonshp s n general nonaddtve. We assume that our thermodynamc system of two nteractng bodes s solated, therefore the basc quanttes, X12 are fxed. Ther conservaton s expressed wth the help of dfferentals: dx 12 = X 12 X 1 d 1 + X 12 X2 dx2 = 0, = 1,..., n. (4) The requrement of maxmum entropy n solated systems n equlbrum reads as ds 12 (S 1, S 2 ) = n =1 ( S12 S 1 S 1 X1 dx1 + S ) 12 S 2 S 2 X2 dx2 = 0. (5) The unque condton of the equatons (4) and (5) havng arbtrary dx1 and dx 2 changes n the basc quanttes s gven by X 12 X 2 S 12 S 1 S 1 X 1 = X 12 X 1 S 12 S 2 S 2 X2, = 1,..., n. (6) 4

6 Journal of Physcs: Conference Seres 394 (2012) IOP Publshng do: / /394/1/ The most general form of the partal dervatves now may be wrtten by a multplcatve separaton of the varables as Then (6) requres, that S 12 (S 1, S 2 ) S 1 = F 1 (S 1 ) G 2 (S 2 ) H 1 (S 1, S 2 ), S 12 (S 1, S 2 ) S 2 = F 2 (S 2 ) G 1 (S 1 ) H 2 (S 1, S 2 ), X12 X1 (X1, X2) = A 1(X1) B2(X 2) C1(X 1, X2), X12 X2 (X1, X2) = A 2(X2) B1(X 1) C2(X 1, X2). (7) A 2B 1C 2 F 1 G 2 H 1 S 1 X 1 = A 1B2C 1 S 2 F 2 G 1 H 2 X2. (8) Ths equaton factorzes to (X 1, S 1) and (X 2, S 2) dependent terms only f C2 (X 1, X 2 ) C1 (X 1, X 2 ) = H 2(S 1, S 2 ) H 1 (S 1, S 2 ) for = 1,..., n (9) The physcal separablty of the thermodynamc bodes has to bendependent of any partcular form of the equaton of state, S(X 1,..., X n ). Therefore the above rato can only be constant. Its value can easly be absorbed nto one of the factorzng component functons of the entropy F or G. As an mmedate consequence one obtans C 1(X 1, X 2) = C 2(X 1, X 2), H 1 (S 1, S 2 ) = H 2 (S 1, S 2 ). (10) These equaltes are the bass for consderng formal logarthms for the basc quanttes separately. The factorzed form of eq. (6) B1 F 1 S 1 A 1 G 1 X1 = B 2 F 2 S 2 A 2 G 2 X2, (11) defnes the followng generalzed ntensve quanttes Y Fnally, usng the defntons Y = B (X )F (S) A (X )G(S) S X (X1,...X n ). (12) F (S) ˆL(S) := G(S) ds, A L (X (X ) ) := B (X ) dx, (13) we arrve at Y = ˆL(S) L (X ). (14) 5

7 Journal of Physcs: Conference Seres 394 (2012) IOP Publshng do: / /394/1/ The zeroth law requres that ths common value s to be ntroduced as the generalzed entropc ntensve quantty. These functons of the orgnal thermodynamcal varables, ˆL(S) and L (X ) defned n eq.(13) can be used to map the orgnal composton rules to a smple addton. Namely consderng the condtons (10) one obtans B1 A 1 X 12 X 1 = B 2 A 2 X12 X2, G 1 F 1 S 12 S 1 = G 2 F 2 S 12 S 2. (15) Utlzng now the defntons (13) for X 1, X 2, = 1,..., n and and S 1, S 2 separately, the partal dervatves smplfy: X12 L 1 = X 12 L, 2 S 12 ˆL 1 = S 12 ˆL 2. (16) The general soluton of such partal dfferental equatons s an arbtrary functon of the sum of varables: X 12 = Φ (L 1 + L 2), for all = 1,...n. S 12 = Ψ(ˆL 1 + ˆL 2 ). (17) If the Φ and Ψ functons are nvertble we can ndex ther nverse arrvng at a more symmetrc notaton: L 12(X12) = L 1(X1) + L 2(X2), ˆL 12 (S 12 ) = ˆL 1 (S 1 ) + ˆL 2 (S 2 ). (18) Snce the L and ˆL functons map a non-addtve composton rule to an addtve one, they are formal logarthms. One recovers the classcal addtve composton when the formal logarthms are the respectve dentty functons. The role of the zeroth law n abstract composton rules was nvestgated by several authors wth dfferent generalty and purpose [19, 20, 5, 21, 22, 23] and wth dvergng conclusons. Composton rules, that can be transformed nto the above form wth a formal logarthm, are assocatve. Assocatve composton rules are also results of an nfnte repettve applcaton of the same rule onto nfntesmal peces of the same materal, as t was proved n [6]. However, n ths specal case the formal logarthms n (18) are related to the same materal and therefore are dentcal. In the general, heterogeneous case these functons L 1 and L 2 may dffer, and the mappng related to the nteracton L 12 can be a further one Transtvty Up to now we requred the separablty of the thermodynamc bodes n equlbrum and dd not nvestgate the transtvty aspect of the zeroth law applyng nonaddtve composton rules. Ths formulaton s not connected to the second law drectly. Transtvty unversally requres 6

8 Journal of Physcs: Conference Seres 394 (2012) IOP Publshng do: / /394/1/ that f bodes 1 and 2 are n thermal equlbrum and ndependently the bodes 2 and 3, then also the bodes 1 and 3 are n equlbrum 1. So far we have establshed addtvty of composte functons of the basc quanttes of the respectve subsystems, L I (E I ), I = 1, 2. Therefore t s natural to assume that these functons are characterstc to the bodes and only the double-ndexed formal logarthms, L IJ (X IJ ), I, J = 1, 2 are characterstc to the nteracton between bodes n equlbrum. Ths s vald for the entropy composton, as well. By ths constructon all subsystems develop the same ndvdual formal logarthm, rrespectve to whch other system they equlbrate wth. Assumng namely the opposte,.e. a partner-dependent ndvdual formal logarthm, the transtvty would be volated. Let us regard three possble parngs of three subsystems. The composte basc quanttes satsfy X 12 = Φ 12(L 1(X 1) + L 2(X 2)), X 23 = Φ 23( L 2(X 2) + L 3(X 3)), X 13 = Φ 13( L 1(X 1) + L 3(X 3)). (19) If L L, then the equlbrum condton s not automatcally transtve. The same s vald for the nonaddtve entropy, too: S 12 = Ψ 12 (ˆL1 (S 1 ) + ˆL ) 2 (S 2 ), S 23 = Ψ 23 (Ľ2 (S 2 ) + ˆL ) 3 (S 3 ), Equaton (6) requres, that S 13 = Ψ 13 (Ľ1 (S 1 ) + Ľ3(S 3 ) ). (20) S 12 S 1 X12 S 1 X1 X2 S 23 S 2 X23 S 2 X2 X3 S 13 S 1 X13 S 1 X1 X3 = S 12 S 2 X12 S 2 X2 X1 = S 23 S 3 X23 S 3 X3 X2 = S 13 S 3 X13 S 3 X3 X1 (21) From here the followng separaton s obtaned, also consderng (18): S 12 S 1 S 1 X1 S 12 S 2 S 2 X2 = X 12 X 1 X 12 X 2 = ˆL 1 S 1 ˆL 2 S 2 = L 1 L 2, S 23 S 2 S 2 X2 S 23 S 3 S 3 X3 = X 23 X 2 X 23 X 3 = Ľ 2 S 2 ˆL 3 S 3 = L 2 L 3, S 13 S 1 S 1 X1 S 13 S 3 S 3 X3 = X 13 X 1 X 13 X 3 = Ľ 1 S 1 Ľ 3 S 3 = L 1 L 3. (22) 1 J. C. Maxwell was the frst who formulated the zeroth law n ths form [17]. A representatve summary of the hstory s gven here: 7

9 Journal of Physcs: Conference Seres 394 (2012) IOP Publshng do: / /394/1/ Ths condton can be reduced easly to obtan ˆL 1Ľ 2Ľ 3 = L 1 L L 2 L 3 ˆL ˆL 2 3Ľ 1 2 L 3 for = 1,..., n. The left hand sde of the n equaltes depends on the body entropes S 1, S 2 and S 3 and the rght hand sde depends on the th basc quanttes of the bodes X1, X 2 and X 3, respectvely. Moreover t s vald for any permuted arrangement of the lowercase body ndces 1, 2 and 3. Therefore one concludes that the transtvty of thermal equlbrum can be satsfed only f the ˆL (S) Ľ (S) = const. and L (X ) = const. (24) L (X ) Ths s a necessary and suffcent condton. Physcally sensble composton rules must satsfy two addtonal smple requrements. The trvalty condton says, that a composton wth zero does not change the value, therefore L(0) = 0 s also requred. Accordng to the compatblty condton for small values of the basc quanttes the nonaddtve effects are reduced and addton emerges, therefore L (0) = 1. Due to these two condtons, from eq. (24) t follows that the L and L functons of the basc quanttes are dentcal, as well as the ˆL and Ľ functons of the entropy. One aspect of the zeroth law the separablty enforces that composton rules are expressed by formal logarthms. Another aspect of the zeroth law the transtvty ensures that ˆL I and L I are characterstcs of the thermodynamc body I and only L IJ and ˆL IJ depend on the nteractons. It s straghtforward to see, that zeroth law compatble composton rules are assocatve f they are homogeneous, that s L IJ = L I = L J Frst order compostons A partcular non-addtve entropy formula, advanced by Tsalls, underles the followng composton rule [9, 8] S 12 (S 1, S 2 ) = S 1 + S 2 + as 1 S 2. (25) We have ntroduced the shorthand notaton â = 1 q. Addtve entropy systems realze a = 0, non-addtve systems a non-zero value of ths parameter. One can get the above formula as a Taylor seres expanson of an arbtrary S 12 (S 1, S 2 ) up to frst order n S 1 and S 2 and requerng the homogenety of the composton. Accordng to our prevous result a thermodynamc framework requres formal logarthms. The formal logarthm for the above rule s easy to derve from L 1 (23) 1 + as 12 = 1 + as 1 + as 2 + a 2 S 1 S 2 = (1 + as 1 ) (1 + as 2 ). (26) The product s related to the addton by the logarthm and scaled down to satsfy ˆL (0) = 1: ˆL(S) = 1 ln(1 + as). (27) a From the vewpont of zeroth law compatblty t s strongly advsed to consder the addtve formal logarthm. In case of the Tsalls entropy, ( ) S T = 1 p 1 â p, (28) a wth â = 1 q and the normalzaton p = 1, ts formal logarthm s gven by the well known Rény entropy [24]: S R = ˆL(S T ) = 1 â ln(1 + âs T ) = 1 1 q ln p q. (29) 8

10 Journal of Physcs: Conference Seres 394 (2012) IOP Publshng do: / /394/1/ Unversal Thermostat Independence The analyss of the prevous secton shows the condtons of a thermodynamc treatment and also enlghtens why the nonaddtve composton rule eq. (25) s leadng order n a mathematcal sense. However, nvestgatng the most researched partcular case of nonaddtve entropes one can fnd dozens of dfferent formulas that fulfll ether the same, or dfferent but suffcently smple nonaddtve composton rules [12, 25]. They frequently lead to the same equlbrum dstrbutons n a sutable maxmum entropy framework. Moreover, acceptng one of the suggested formulas one should ask about the orgn of the ntroduced parameters. E.g. how could we determne the a = 1 q parameter of the Tsalls dstrbuton from mcroscopc models beyond fttng? Gven a partcular mcroscopc system, such as an deal gas, a mcrocanoncal treatment assumes that the partcles are correlated because the total basc quanttes are fxed. A canoncal treatment assumes ndependent dstrbutons wth gven average basc quanttes. The fxed averages are physcally provded by sutable reservors. A canoncal treatment s useful also when the reservor concept seems purely hypothetc, snce ts smplcty compared to mcrocanoncal approaches s a great advantage. The Tsalls dstrbuton ntroduces a partcular correlaton. It s known that some smple mcrocanoncal dstrbutons are of Tsalls type [26]. If the orgn of Tsalls dstrbutons s unversal, what s the dstnctve property of these correlatons? We seek for a generalzed entropy formula frst examnng the two-body thermodynamcs of a subsystem and a reservor, slghtly generalzng the results of [15]. We consder fnte reservors and the conservaton of the basc quanttes: X 0 = X 1 + X 2 = const., = 1,..., n. We am at a monotonc functon of the Boltzmann Gbbs entropy K(S) that s maxmal n the body-reservor system and absorbs fnte sze correlaton effects n the Taylor-expanson of the maxmum entropy prncple. The respectve entropy contrbutons are n general nonaddtve and satsfy a composton rule formulated n terms of formal logarthms. We consder homogeneous rules K(S 12 ) = K(S 1 ) + K(S 2 ). (30) The maxmum entropy prncple together wth the conservaton of the basc quanttes reads: K(S(X 1 1,..., X n 1 )) + K(S(X 1 0 X 1 1,..., X n 0 X n 1 )) = max. (31) The necessary condton of the maxmum n case of twce dfferentable functons requres vanshng dervatves by the subsystem basc quanttes X 1 : Y1 = K (S(X1, 1..., X1 n )) S X1 (X1, 1..., X1 n ) = = K (S(X0 1 X1, 1..., X0 n X1 n )) S X1 (X0 1 X1, 1..., X0 n X1 n ) = Y2. (32) Subsystem 1 s much smaller than the reservor, X0 X 1. Ths equalty n a tradtonal canoncal approach declare the equalty of the reservor and subsystem ntensves n the X1 0 lmt. Now, consderng effects hgher order n X1 /X 0 we request that ther leadng term vanshes on the rght hand sde of eq. (32). The Taylor seres expanson up to the frst order reads as Y 1 = K (S 0 ) S 0 X ( K (S 0 ) S 0 X S 0 X j + K (S 0 ) 2 S 0 X X j ) dx j +... (33) where S 0 = S(X0 1,..., Xn 0 ) denotes the functon values at the constant basc quanttes of the system. It s not obvous that a sngle functon K(S 0 ) could annul the bracket coeffcent of dx j 9

11 Journal of Physcs: Conference Seres 394 (2012) IOP Publshng do: / /394/1/ n general. One of the reasons s that S 0 S 0 s not nvertble. Therefore we restrct ourselves X X j to one addtonal basc quantty that we denote smply by X and the dervatve of the entropy by X s denoted b S. Then the prevous equalty reduces to Y = K (S 0 )S 0 ( K (S 0 )S K (S 0 )S 0 ) dx +... (34) In ths case the vanshng lnear term requres for a general K(S) that: K (S) K (S) = S (X) S (X) 2. (35) Here the rght hand sde s a functon of X, therefore t s a constant solvng the above equaton for an S dependent K. Therefore K (S) K (S) = a, (36) and the soluton of the above equaton wth the compatblty and trvalty condtons K (0) = 1 and K(0) = 0 gves K(S) = eas 1. (37) a The dervatves of the S(X) equaton of state do have a physcal meanng: a = 1/χ(X 0 ), (38) where χ characterzes the change of the reservor entropy per unt change of the ntensve Y = S. χ s the generalzed susceptblty. It s remarkable that functon (37) s dentcal to the nverse of the formal logarthm of the Tsalls addtvty composton (27). Now we generalze the result of the two body analyss to the classcal entropy formula S = p ln p and recover the Tsalls entropy formula as follows: S T salls = p K( ln p ) = 1 1 q (p q p ), (39) where we have ntroduced the usual q = 1 a notaton. Ths can be consdered as a modfcaton of the entropy densty n the total average, the entropy densty representng the lmt of the mnmal subsystem n an extended state space. Accordng to the prevous secton the zeroth law requres, that the thermodynamc total entropy of the system must be the Rény entropy S Rény = K 1 (S Tsalls ) = 1 1 q ln p q. (40) The correspondng maxmum entropy prncple reads as 1 1 q ln p q β p E α p = max.. (41) Ths s the bass of calculatng the equlbrum dstrbuton n a canoncal form. 10

12 Journal of Physcs: Conference Seres 394 (2012) IOP Publshng do: / /394/1/ Summary We have derved the requrements of the zeroth law when the basc thermodynamc quanttes are nonaddtve. We have obtaned that then the concept of separate thermodynamc bodes enforces the exstence of formal logarthms. The transtvty condton ensure that the formal logarthms represent materal propertes. Then we were lookng for a unversal orgn of nonaddtve compostons and derved the Tsalls entropy formula. We requred that the nonaddtve composton cancels lnear correctons due to a fnte X 0 basc quantty of the reservor. That requrement s the prncple of Unversal Thermostat Independence [15]. Ths dervaton explans the partcular functonal form of the Tsalls and Rény formulas as generalzed entropy expressons. Acknowledgments The work was supported by the Hungaran Natonal Scence Research Fund OTKA NK778816,NK106119,H07/C 74164,K68108, K81161, K104260, NIH TET and ZA- 15/2009. GGB also thanks the János Bolya Research Scholarshp of the HAS. [1] Bró T S, Ürmössy K and Schram Z 2010 Journal of Physcs G [2] Barnaföld G G, Ürmössy K and Bró T S 2011 Journal of Physcs: Conference Seres [3] Ürmössy K, Barnaföld G and Bró T S 2011 Physcs Letters B [4] CMS collaboraton 2010 Journal of Hgh Energy Physcs [5] Abe S 2002 Physcal Revew E [6] Bro T S 2008 EPL [7] Bró T S and Ván P 2011 Physcal Revew E [8] Tsalls C 1988 Journal of Statstcal Physcs [9] Tsalls C 2009 Introducton to Nonextensve Statstcal Mechancs (Sprnger) [10] Nauenberg M 2003 Physcal Revew E [11] Arndt C 2001 Informaton Measures (Informaton and ts descrpton n scence and engneerng) (Berln- Hedelberg-New York: Sprnger) [12] Taneja I Generalzed Informaton Measures and Ther Applcatons (SprngerOnlne book, taneja/book/book.html) [13] Kanadaks G 2001 Physca A [14] Landsberg P T and Vedral V 1998 Physcs Letters A [15] Bró T S, Barnaföld G G and Ván P 2012 Preprnt ArXv: [16] Mackey M C 1992 Tme s Arrow: The Orgns of Thermodynamc Behavour (Sprnger) [17] Maxwell J C 1902 Theory of Heat 10th ed (London: Longmans, Green and Co.) [18] Fowler R and Guggenhem E 1939 Statstcal Thermodynamcs: A Verson of Statstcal Mechancs for Students of Physcs and Chemstry (Cambrdge Unversty Press (prnted by W. Lews)) [19] Abe S 2001 Physca A [20] Abe S 2001 Physcal Revew E [21] Johal R S 2003 Physcs Letters A [22] Johal R S 2004 Physcs Letters A [23] Scarfone A 2010 Physcs Letters A [24] Rény A 1961 On measures of entropy and nformaton Proceedngs of the 4th Berkeley Symposum on Mathematcal Statstcs and Probablty, 1960 vol I ed Neyman J (Berkeley: Unversty of Calforna Press) pp [25] Zarpov R 2010 Prncples of nonextensve statstcal mechancs and geometry of measures of the dsorder and the order (Kazan: Kazan A. N. Tupolev State Techncal Unversty Press) (n Russan) [26] Naudts J 2011 Generalzed Thermostatstcs (Sprnger) 11