Hidden correlations entailed by q-non additivity render the q-monoatomic gas highly non trivial
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Hidden correlations entailed by q-nonadditivity render the q-monoatomicgas highly non trivial
A. Plastino , M. C. Rocca , La Plata National University and Argentina’s National Research Council(IFLP-CCT-CONICET)-C. C. 727, 1900 La Plata - Argentina
July 21, 2018
Abstract
It ts known that Tsallis’ q-non-additivity entails hidden correlations.It has also been shown that even for a monoatomic gas, both theq-partition function Z and the mean energy < U > diverge and, inparticular, exhibit poles for certain values of the Tsallis non additivityparameter q . This happens because Z and < U > both depend ona Γ -function. This Γ , in turn, depends upon the spatial dimension ν . We encounter three different regimes according to the argument A of the Γ -function. (1) A > 0 , (2)
A < 0 and
Γ > 0 outside thepoles. (3) A displays poles and the physics is obtained via dimensionalregularization. In cases (2) and (3) one discovers gravitational effectsand quartets of particles. Moreover, bound states and gravitationaleffects emerge as a consequence of the hidden q-correlations.Keywords: q-Statistics, divergences, partition function, dimensionalregularization, specific heat. Introduction
Generalized or q-statistical mechanics `a la Tsallis has generated manifoldapplications in the last quarter of a century [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12, 13]. It has been shown (see for instance, [14, 15]) that Tsallis’ q-statisticsis of great importance for dealing with some astrophysical issues involvingself-gravitating systems [16]. Moreover, this statistics has proved its utilityin many different scientific fields, with several thousands of publications andauthors [2], so that studying its structural features is an important issuefor physics, astronomy, biology, neurology, economics, etc. [1]. The successof the q-statistics reaffirms the well grounded notion asserting that there ismuch physics whose origin is of purely statistical nature (not mechanical). Asa spectacular example, me mention the application of q-ideas to high energyexperimental physics, where the q-statistics appears to adequately describethe transverse momentum distributions of different hadrons [17, 18, 19].In this work we show that as yet unexplored gravitational effects characterizethis q-theory on account of divergences that, in some circumstances, emerge,within the q-statistical framework, in both the mean energy < U > and thepartition function Z [20].Divergences constitute an important issue in theoretical physics. Indeed, thestudy and elimination of divergences of a physical theory is perhaps one ofthe most important aspects of theoretical work. The quintessential typicalexample is the thus far failed, attempt to quantify the gravitational field.Some examples of elimination of divergences can be looked at in references[21, 22, 23, 24, 25].We will use here an extremely simplified version, such as that of [26], of theideas of [21, 22, 23, 24, 25] in connection with Tsallis q-statistics [1, 2], withemphasis in its applicability to gravitational issues [14, 15], in particular self-gravitating systems [16]. We will see that the removal of the above mentioneddivergences produces interesting insights.It is to be stressed that for Z < 0 the system can not exist, as no probabilitiescan be introduced. Then, several options are available. < U >> 0 is thenatural state of affairs. We will uncover that, in a Tsallis’ scenario, < U >< 0 is possible for special q-values which entails boundedness. Also negativespecific heats C may emerge, an indication of gravitational effects [16].These interesting results appear after using mathematics known since at least40 years ago and for whose development M. Veltman and G. t’Hooft whereawarded the Nobel prize of physics in 1999. Acquaintance with such mathe-2atics is not really needed to understand this paper. The reader has merelyto accept that their physical significance can not be doubted. In a few words,one needs from the above mathematics analytical extensions and dimensionalregularization [21, 22, 23, 24, 25].In this work we analyze the behavior of Z and < U > in three regions of theargument of the Γ -function contained in them. The nature of the argumentof the Γ -function governs the behavior of Z and < U > . This behavior yieldsthree different regions, for a given spatial dimension ν , Tsallis’ index q andnumber of particles N . The region’s particularities are: ( ) − q − nν2 − ( ) − q − nν2 < 0 Γ (cid:16) − q − nν2 (cid:17) > 0 ( ) − q − nν2 = − p p =
0, 1, 2, 3, 4.....
Normal monoatomic gas behavior is encountered in region (1). Instead, grav-itational effects are discovered in region (2). Finally, in region (3) we haveboth normal behavior and gravitational effects. In particular the N particlesare grouped into quartets that remind one of alpha particles. It shouldbe noted than in case (3) we are making a regularization of thecorresponding theory and NOT a renormalization.
We stress that here we use normal (linear in the probability) expectationvalues, and not the weighted ones usually attached to Tsallis’ theory [1]. Thisis done for simplicity. Other ways of evaluating mean values pose difficultiesin this context that will be tackled in the future [28].Remark that in this case, restricting ourselves to the interval [ ≤ ] ,the so-called Tsallis cut-off [1] does not apply.The q-partition function of a monoatomic gas is given by: Z = V n ∞ Z − ∞ (cid:20) + β ( − q ) ( p + p + · · · p ) (cid:21) − d ν p d ν p · · · d ν p n , (2.1)The mean energy is defined by: < U > = V n Z ∞ Z − ∞ (cid:20) + β ( − q ) ( p + p + · · · p ) (cid:21) − + p + · · · p
2m d ν p d ν p · · · d ν p n . (2.2)Both integrals are in general divergent for many values of q . This canbe proved by a simple powers count. We appeal to techniques developed in[21, 22, 23, 24, 25] so as to deal with them.In ref.[26] we have computed both Z and < U > obtaining: Z = V n (cid:20) ( − q ) (cid:21) νn2 Γ (cid:16) − q − νn2 (cid:17) Γ (cid:16) − q (cid:17) . (2.3) < U > = V n Z νn2β ( − q ) (cid:20) ( − q ) (cid:21) νn2 Γ (cid:16) − q − νn2 − (cid:17) Γ (cid:16) − q (cid:17) . (2.4) < U > = νnβ [ − νn ( − q )] . (2.5)The derivative with respect to T yields for the specific heat at constantvolume C V = νnk2q − νn ( − q ) . (2.6) We briefly review now results obtained in Ref. [26]. Some related work byLivadiotis, McComas, and Obregon, should be cited [12, 13, 27]. We pass firstto analyze the Gamma functions involved in computing Z and < U > , forthe region [ ≤ ] . From (2.3) we have, for positive Gamma-argument − q − νn2 > 0. (3.1)In the same vein we have from (2.4) − q − νn2 − (3.2)We reach then two conditions that pose severe limitations on the particle-number n , i.e., ≤ n < 2qν ( − q ) (3.3)4here is a maximum permissible n . For example, if q = − − , ν = , onehas ≤ n < 666. (3.4)We can not have more than 665 particles. Keeping the dimensionality equalto three, for q = just one particle is permitted and for q = , noparticles can be present exist. Roughly, for a number of particles of theorder of n , q has to be of the order of − − n . The exposition of our present results starts here. We pass first to consider-ing negative Gamma arguments in (2.3), which will require analytical exten-sion/dimensional regularization in integrals (2.1) and (2.2). One has − q − νn2 < 0, (4.1)together with Γ (cid:18) − q − νn2 (cid:19) > 0, (4.2)We use now Γ ( z ) Γ ( − z ) = π sin ( πz ) , (4.3)to find Γ (cid:18) − q − νn2 (cid:19) = − π sin π (cid:16) νn2 − − q (cid:17) Γ (cid:16) νn2 − − q (cid:17) > 0. (4.4)This is true if sin π (cid:18) νn2 − − q (cid:19) < 0, (4.5)so that + − − q < 2 ( p + ) (4.6)where p =
0, 1, 2, 3, 4, 5..... , or equivalently5 p + + ( − q ) < n < 4 ( p + ) ν + ( − q ) . (4.7)Now, from (2.3), (2.5), and (2.6) we obtain (a) Z > 0 , (b) < U >< 0 , (c)
C < 0 , which entails bound states, on account of (b) and self-gravitationaccording to (c) [16].
If the Gamma’s argument is such that − q − νn2 = − p for p =
0, 1, 2, 3, ......, (5.1) Z displays a single pole.For ν = we have − q − n2 = − p for p =
0, 1, 2, 3, ....... (5.2)Since ≤ q < 1 , the concomitant q values are q =
12 , 23 , 34 , 45 , ......, (5.3) n even, n ≥ and q =
13 , 35 , 57 , 79 , ......, (5.4) n odd, n ≥ .For ν = − q − n = − p for p =
0, 1, 2, 3, ......, (5.5)Again, since ≤ q < 1 , q =
12 , 23 , 34 , 45 , ......, (5.6) n ≥ .For ν = − q − = − p for p =
0, 1, 2, 3, ......, (5.7)6nd because ≤ q < 1 , q =
12 , 23 , 34 , 45 , ......, (5.8) n even, n ≥ , and q =
13 , 35 , 57 , 79 , ......, (5.9) n odd, n ≥ .The location of these poles will change if escort mean values where used.We discuss now poles in < U > , given by − q − νn2 − = − p for p =
0, 1, 2, 3, ......, (5.10)For ν = − q − n2 − = − p for p =
0, 1, 2, 3, ......, (5.11)On account on the condition (G): ≤ q < 1 we have q =
12 , 23 , 34 , 45 , ......, (5.12)for n even, n ≥ and q =
13 , 35 , 57 , 79 , ......, (5.13)for n odd , n ≥ .For ν = − q − n − = − p for p =
0, 1, 2, 3, ......, (5.14)minding (G) we have q =
12 , 23 , 34 , 45 , ......, (5.15) n ≥ .For ν = − q − − = − p for p =
0, 1, 2, 3, ......, (5.16)and, from (G), q =
12 , 23 , 34 , 45 , ......, (5.17)7 even, n ≥ and q =
13 , 35 , 57 , 79 , ......, (5.18) n odd, n ≥ . As an example of dimensional regularization [21, 22, 23, 24, 25] we will gointo some detail concerning the poles at q = and q = . q = pole In this case n is even, n ≥ . We have Z = V n (cid:18) (cid:19) νn2 Γ (cid:16) − νn2 (cid:17) . (6.1)Using Γ (cid:16) − νn2 (cid:17) Γ (cid:16) νn2 − (cid:17) = − π sin (cid:0) πνn2 (cid:1) (6.2)or, equivalenly Γ (cid:16) − νn2 (cid:17) Γ (cid:16) νn2 − (cid:17) = − (− ) + π sin (cid:2) πn2 ( ν − ) (cid:3) , (6.3)so that Z = V n (cid:18) (cid:19) νn2 (− ) + π sin [ πn2 ( ν − )] Γ (cid:0) νn2 − (cid:1) . (6.4)Sincesin [ πn2 ( ν − )] = πn2 ( ν − ) (cid:14) + ∞ X m = (− ) m ( + )! h πn2 ( ν − ) i (cid:15) = (6.5) = πn2 ( ν − ) X, (6.6)with X = (cid:14) + ∞ X m = (− ) m ( + )! h πn2 ( ν − ) i (cid:15) , (6.7)8e get Z = V n (cid:18) (cid:19) (− ) + Γ (cid:0) νn2 − (cid:1) X n2 ( ν − ) (cid:20) + n2 ( ν − ) ln (cid:18) (cid:19) + · · · (cid:21) (6.8)The term independent of ν − is, following dimensional regularization pre-scriptions [21, 22, 23, 24, 25] Z = V n (cid:18) (cid:19) (− ) + Γ (cid:0) − (cid:1) ln (cid:18) (cid:19) (6.9)This is then the physical Z-value at the pole [21, 22, 23, 24, 25]. For themean energy we have Z < U > = V n nνβ (cid:18) (cid:19) νn2 Γ (cid:16) − νn2 (cid:17) . (6.10)Using Γ (cid:16) − νn2 (cid:17) Γ (cid:16) νn2 (cid:17) = π sin (cid:0) πνn2 (cid:1) (6.11)or, equivalently Γ (cid:16) − νn2 (cid:17) Γ (cid:16) νn2 (cid:17) = − (− ) π sin (cid:2) πn2 ( ν − ) (cid:3) (6.12)one finds for < U >Z < U > = V n nνβ (cid:18) (cid:19) νn2 (− ) π sin [ πn2 ( ν − )] Γ (cid:0) νn2 (cid:1) . (6.13) < U > can be recast as Z < U > = V n n ( ν − ) β (cid:18) (cid:19) νn2 (− ) π sin [ πn2 ( ν − )] Γ (cid:0) νn2 (cid:1) + V n (cid:18) (cid:19) νn2 (− ) π sin [ πn2 ( ν − )] Γ (cid:0) νn2 (cid:1) . (6.14)Repeating the Z-treatment yields for < U > :9 < U > = V n (cid:18) (cid:19) (− ) Γ (cid:0) (cid:1) + V n (cid:18) (cid:19) (− ) Γ (cid:0) (cid:1) ln (cid:18) (cid:19) (6.15)or, equivalently Z < U > = V n (cid:18) (cid:19) (− ) Γ (cid:0) (cid:1) (cid:20) − ln (cid:18) (cid:19)(cid:21) . (6.16)Appealing here to (6.9) for the physical Z we finally obtain < U > = ( − ) (cid:20) ln β − ln − (cid:21) . (6.17)We discuss first (− ) + = − and then n =
4, 8, 12, 16...... , so that Z = V n Γ (cid:0) − (cid:1) (cid:18) (cid:19) ln (cid:18) β4mπ (cid:19) (6.18)If (− ) + = , then n =
2, 6, 10, 14...... and Z = V n Γ (cid:0) − (cid:1) (cid:18) (cid:19) ln (cid:18) (cid:19) . (6.19)From (6.17) - (6.18) and requiring Z > 0 and < U >> 0 one deduces < T < 14mπk . (6.20)From (6.17) - (6.19) and demanding
Z > 0 y < U >< 0 one finds ≤ T < 14mπke (6.21)The specific heat derives from (6.17) for < U > . One has C = − (cid:20) ln β − ln + ( ln β − ln ) − (cid:21) . (6.22)10 .2 The q = pole Here Z is Z = V n (cid:18) (cid:19) νn2 Γ (cid:0) − νn2 (cid:1) Γ (cid:0) (cid:1) . (6.23)Using again Γ (cid:18) − νn2 (cid:19) Γ (cid:18) νn2 − (cid:19) = − π cos (cid:0) πνn2 (cid:1) , (6.24)or, equivalently Γ (cid:18) − νn2 (cid:19) Γ (cid:18) νn2 − (cid:19) = − (− ) − π sin (cid:2) πn2 ( ν − ) (cid:3) , (6.25)so that Z = V n Γ (cid:0) (cid:1) (cid:18) (cid:19) νn2 (− ) − π sin [ π2 ( ν − )] Γ (cid:0) νn2 − (cid:1) . (6.26)We can dimensionally regularize Z - < U > as above, to find Z = V n Γ (cid:0) (cid:1) (cid:18) (cid:19) (− ) − Γ (cid:0) − (cid:1) ln (cid:18) (cid:19) , (6.27) < U > = ( − ) (cid:20) ln β − ln − (cid:21) . (6.28)We tackle first (− ) − = − and then n =
1, 5, 9, 13......Z = V n Γ (cid:0) (cid:1) Γ (cid:0) − (cid:1) (cid:18) (cid:19) ln (cid:18) β3mπ (cid:19) . (6.29)For (− ) − = , then n =
3, 7, 11, 15......Z = V n Γ (cid:0) (cid:1) Γ (cid:0) − (cid:1) (cid:18) (cid:19) ln (cid:18) (cid:19) . (6.30)From (6.28) - (6.29) and demanding e Z > 0 - < U >> 0 we find < T < 13mπk . (6.31)11rom (6.28)-y (6.30) and requiring Z > 0 - < U >< 0 we get ≤ T < 13mπke . (6.32)Finally, for C one has C = − (cid:20) ln β − ln + ( ln β − ln ) − (cid:21) . (6.33) We have used an elementary regularization procedure to investigate the polesin both Z and < U > for specific, discrete q-values, in Tsallis’ q-scenario.We analyzed the thermal behavior at the poles and encountered suggestivefeatures. The study was undertaken in one, two, three, and N dimensions.Amongst the ensuing pole-features, rather unexpected, but nonetheless true,we focus on: • There is an upper bound to the temperature at the poles, re-confirmingthe discoveries of Ref. [29]. • In some instances, Tsallis’ entropies are positive just for a restrictedtemperature-range. • Negative specific heats, characteristic feature of self-gravitating systems[16], are found. See the illuminating by Silva-Alcaniz [30]. We will, in afuture work, try to connect our methodology with that of this reference.Our physical results are deduced only from statistics and not from me-chanical properties. This fact brings to mind of a similar feature that emergesin the case of the entropic force conjectured by Verlinde [31].12 eferences [1] M. Gell-Mann and C. Tsallis, Eds.
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