- Email: [email protected]
The thermodynamic limit in the non-extensive thermostatistics
Physica A 344 (2004) 403 – 408
www.elsevier.com/locate/physa
The thermodynamic limit in the non-extensive thermostatistics R. Boteta , M. P loszajczakb;∗ , K.K. Gudimab; c , A.S. Parvanc; d , V.D. Toneevb; d a Laboratoire
de Physique des Solides, Centre d’ Orsay CNRS/UMR8502, Universit!e Paris-Sud, Batiment 510, F-91405 Orsay Cedex, France b Grand Acc! el!erateur National d’Ions Lourds, CEA/DSM -CNRS/IN2P3, BP5027, F-14076 Caen Cedex 05, France c Institute of Applied Physics, Moldova Academy of Sciences, MD-2028 Kishineu, Moldova d Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
Available online 2 July 2004
Abstract Thermodynamic limit is more than just: ‘the limit of the system when size tends to in4nity’. This rough de4nition can be su7cient in the simplest cases, but precise de4nition of the thermodynamic limit involves subtle constraints in order to compare systems of various sizes N but in the similar thermodynamic state. We discuss in this paper how these constraints must be handled in the non-extensive thermostatistics, within the canonical ensemble. Role of the entropy index q on the proper de4nition of the thermodynamic limit, is emphasized. In particular, the only choice to get non-trivial thermodynamic limit is: N (q − 1) = constant. It can lead to a steady state di
∗
Corresponding author. Tel.: +33-169-1565-925; fax: +33-169-1560-86. E-mail addresses: [email protected], [email protected] (M. Ploszajczak).
c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.06.007
404
R. Botet et al. / Physica A 344 (2004) 403 – 408
1. Context Thermodynamic limit is a fundamental concept in physics as it describes the route between microscopic interactions and the macroscopic system. It is directly related to 4nite-size scaling, as one has to de4ne and compare systems with di
R. Botet et al. / Physica A 344 (2004) 403 – 408
405
is no dynamical information in the pi : since ergodicity is realized, pi is e
i
i
with and the Lagrange multipliers. This condition yields the relation. 1 pi = exp(−Ei =k) : Z The factor Z = i e−Ei =k depends only of the microstate energy distribution. The macroscopic entropy at the BG equilibrium is then: S = kB ln Z + U . Di
from which one deduces that is the inverse of the system temperature T , because of the thermodynamic relation T dS = dU + p dV . The free energy F and the partition function Z are related through: F = U − TS = −kB T ln Z. Now, let us discuss the non-extensive case throughout the same lines. Tsallis introduced the generalized microscopic entropy: Si =−k lnq pi , with the q-logarithm function (q a real parameter) lnq x =
x1−q − 1 : 1−q
The constant k, in the de4nition of the Si , relates to an entropy scale. Moreover, the average values are now de4ned through the formula Oi q ≡ thermodynamic q i pi O i = i pi , similar to the Boltzmann case, except for the probability weight pi which has been changed into piq , at the numerator only. Since these averages are no more normalized, one has to consider the macroscopic observable average as Oq ≡ Oi q =1q (also called escort averages). One should interpret the change pi → piq , as the result of dynamics. Suppose the system be always prepared into some preferred initial state, then the e
i
i
406
R. Botet et al. / Physica A 344 (2004) 403 – 408
which is the generalization of (1). Performing the di
(2)
with the q-exponential function, which is the inverse function of lnq : expq (x) = (1 + (1 − q)x)1=(1−q) : The normalization factor 1q , needed in the observable averages, is simply q pi = 1 + (1 − q)Si q =k = Zq1−q + (1 − q) Ei q : k i
(3) (4)
One obtains then a generalized formula for the macroscopic entropy: Si q = k lnq Zq + Ei q . De4nition of the free energy: Fi q =Ei q − Si q , with the state parameter , yields ≡ 1= , and Fi q = −k lnq Zq . This is just a formal change of Zq and , into Fq and , and this relation is unable to de4ne properly the thermometric temperature of the system, since holds for the unnormalized quantities (Si q ; Ei q ; Fi q , etc.). Instead, one should ask what is the observable temperature Tq , which veri4es both the usual thermodynamics relations for the normalized quantities, and the 0th law. To characterize Tq , one can notethat the probability normalization ( i pi = 1), with (2) and de4nition (3), leads to: i (1 − (1 − q)Ei =k ) dpiq = 0. Now because of (4), one obtains the general di
(7)
R. Botet et al. / Physica A 344 (2004) 403 – 408
407
with the density of energy uq = Uq =N , independent on the system size since energy is extensive, and k as used in (2), that is 1 pi ( ) = expq (−Ei =k ) : (8) Zq One can notice from formula (7) that if the value of q is 4xed while the system size N goes to in4nity, the de4nition of the observable temperature Tq is in trouble. It appears here clearly that, for the thermodynamic limit to be de4ned, one has to vary the value of the entropy index with the system size, in such a way that # = N (q − 1) stays constant. The method proposed above to determine the correct thermometric temperature (the one which can be measured by thermal contact with a thermometer at equilibrium), is probably the most direct. But it is essentially equivalent to the method of the escort-averages, as introduced by Tsallis et al. [2]. In the latter method, one has to maximize the entropy q q pi Ei i d =0 ; pi (−k lnq pi ) − pi − 1 − q − U q i pi i i where the (normalized) internal energy Uq is directly used. Now, (8) holds too, with the thermal parameter = i piq = + (1 − q)Uq =k. This shift in temperature was discussed in Ref. [3], but the present method indicates directly that the temperature satisfying the 0th law is Tq , or equivalently i piq = in the Tsallis approach (both temperatures are identical because of (6) and (4)). Note also that in the extensive systems, ergodicity yields equivalence of the two limits: limN →∞ limt→∞ and limt→∞ limN →∞ . This is no more the case for non-extensive systems, as they correspond to di
and
PV = NkTq ;
which show that things are not so di
408
R. Botet et al. / Physica A 344 (2004) 403 – 408
when all the couples of spins interact with the same strength. In this particular case, the system exhibits a second-order critical behavior for the BG thermodynamics at the thermodynamic limit. The same derivation for the NE thermostatistics has been discussed in Ref. [6], without taking into account for the observable temperature (i.e., with the thermal variable ). Expressing with the real temperature Tq , this leads to a behavior similar to the BG case, with a simple shift of the critical temperature. 5. Conclusion In this short paper, we discussed the proper characterization of the thermodynamic limit in the framework of the non-extensive thermostatistics at equilibrium. Two points are delicate: the de4nition of the observable temperature Tq , and the role of the entropy index q. Concerning the latter parameter, the thermodynamic limit is de4ned provided the long-range correlations, hidden in the parameter q, are renormalized with the system size as: q = 1 + #=N , with the parameter # constant. This new parameter should govern the relative strength of the correlations due to non-extensive entropy, in comparison with the natural system-size correlations (those coming from the extensive theory). When taking the correct values for the state variables (Tq ; Pq ; q ), one 4nds that the limit state of the system does not depend crucially of the NE hypothesis. Only quantitative, but not qualitative, di
C. Tsallis, J. Stat. Phys. 52 (1988) 479. C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A 261 (1998) 534. A.R. Lima, J.S. SSa Martins, T.J.P. Penna, Physica A 268 (1999) 553. V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64 (2001) 056134. S. Abe, S. Martinez, F. Pennini, A. Plastino, Phys. Lett. A 281 (2001) 126. R. Botet, M. Ploszajczak, J.A. GonzSalez, Phys. Rev. E 65 (2002) 015103.
www.elsevier.com/locate/physa
The thermodynamic limit in the non-extensive thermostatistics R. Boteta , M. P loszajczakb;∗ , K.K. Gudimab; c , A.S. Parvanc; d , V.D. Toneevb; d a Laboratoire
de Physique des Solides, Centre d’ Orsay CNRS/UMR8502, Universit!e Paris-Sud, Batiment 510, F-91405 Orsay Cedex, France b Grand Acc! el!erateur National d’Ions Lourds, CEA/DSM -CNRS/IN2P3, BP5027, F-14076 Caen Cedex 05, France c Institute of Applied Physics, Moldova Academy of Sciences, MD-2028 Kishineu, Moldova d Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
Available online 2 July 2004
Abstract Thermodynamic limit is more than just: ‘the limit of the system when size tends to in4nity’. This rough de4nition can be su7cient in the simplest cases, but precise de4nition of the thermodynamic limit involves subtle constraints in order to compare systems of various sizes N but in the similar thermodynamic state. We discuss in this paper how these constraints must be handled in the non-extensive thermostatistics, within the canonical ensemble. Role of the entropy index q on the proper de4nition of the thermodynamic limit, is emphasized. In particular, the only choice to get non-trivial thermodynamic limit is: N (q − 1) = constant. It can lead to a steady state di
∗
Corresponding author. Tel.: +33-169-1565-925; fax: +33-169-1560-86. E-mail addresses: [email protected], [email protected] (M. Ploszajczak).
c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.06.007
404
R. Botet et al. / Physica A 344 (2004) 403 – 408
1. Context Thermodynamic limit is a fundamental concept in physics as it describes the route between microscopic interactions and the macroscopic system. It is directly related to 4nite-size scaling, as one has to de4ne and compare systems with di
R. Botet et al. / Physica A 344 (2004) 403 – 408
405
is no dynamical information in the pi : since ergodicity is realized, pi is e
i
i
with and the Lagrange multipliers. This condition yields the relation. 1 pi = exp(−Ei =k) : Z The factor Z = i e−Ei =k depends only of the microstate energy distribution. The macroscopic entropy at the BG equilibrium is then: S = kB ln Z + U . Di
from which one deduces that is the inverse of the system temperature T , because of the thermodynamic relation T dS = dU + p dV . The free energy F and the partition function Z are related through: F = U − TS = −kB T ln Z. Now, let us discuss the non-extensive case throughout the same lines. Tsallis introduced the generalized microscopic entropy: Si =−k lnq pi , with the q-logarithm function (q a real parameter) lnq x =
x1−q − 1 : 1−q
The constant k, in the de4nition of the Si , relates to an entropy scale. Moreover, the average values are now de4ned through the formula Oi q ≡ thermodynamic q i pi O i = i pi , similar to the Boltzmann case, except for the probability weight pi which has been changed into piq , at the numerator only. Since these averages are no more normalized, one has to consider the macroscopic observable average as Oq ≡ Oi q =1q (also called escort averages). One should interpret the change pi → piq , as the result of dynamics. Suppose the system be always prepared into some preferred initial state, then the e
i
i
406
R. Botet et al. / Physica A 344 (2004) 403 – 408
which is the generalization of (1). Performing the di
(2)
with the q-exponential function, which is the inverse function of lnq : expq (x) = (1 + (1 − q)x)1=(1−q) : The normalization factor 1q , needed in the observable averages, is simply q pi = 1 + (1 − q)Si q =k = Zq1−q + (1 − q) Ei q : k i
(3) (4)
One obtains then a generalized formula for the macroscopic entropy: Si q = k lnq Zq + Ei q . De4nition of the free energy: Fi q =Ei q − Si q , with the state parameter , yields ≡ 1= , and Fi q = −k lnq Zq . This is just a formal change of Zq and , into Fq and , and this relation is unable to de4ne properly the thermometric temperature of the system, since holds for the unnormalized quantities (Si q ; Ei q ; Fi q , etc.). Instead, one should ask what is the observable temperature Tq , which veri4es both the usual thermodynamics relations for the normalized quantities, and the 0th law. To characterize Tq , one can notethat the probability normalization ( i pi = 1), with (2) and de4nition (3), leads to: i (1 − (1 − q)Ei =k ) dpiq = 0. Now because of (4), one obtains the general di
(7)
R. Botet et al. / Physica A 344 (2004) 403 – 408
407
with the density of energy uq = Uq =N , independent on the system size since energy is extensive, and k as used in (2), that is 1 pi ( ) = expq (−Ei =k ) : (8) Zq One can notice from formula (7) that if the value of q is 4xed while the system size N goes to in4nity, the de4nition of the observable temperature Tq is in trouble. It appears here clearly that, for the thermodynamic limit to be de4ned, one has to vary the value of the entropy index with the system size, in such a way that # = N (q − 1) stays constant. The method proposed above to determine the correct thermometric temperature (the one which can be measured by thermal contact with a thermometer at equilibrium), is probably the most direct. But it is essentially equivalent to the method of the escort-averages, as introduced by Tsallis et al. [2]. In the latter method, one has to maximize the entropy q q pi Ei i d =0 ; pi (−k lnq pi ) − pi − 1 − q − U q i pi i i where the (normalized) internal energy Uq is directly used. Now, (8) holds too, with the thermal parameter = i piq = + (1 − q)Uq =k. This shift in temperature was discussed in Ref. [3], but the present method indicates directly that the temperature satisfying the 0th law is Tq , or equivalently i piq = in the Tsallis approach (both temperatures are identical because of (6) and (4)). Note also that in the extensive systems, ergodicity yields equivalence of the two limits: limN →∞ limt→∞ and limt→∞ limN →∞ . This is no more the case for non-extensive systems, as they correspond to di
and
PV = NkTq ;
which show that things are not so di
408
R. Botet et al. / Physica A 344 (2004) 403 – 408
when all the couples of spins interact with the same strength. In this particular case, the system exhibits a second-order critical behavior for the BG thermodynamics at the thermodynamic limit. The same derivation for the NE thermostatistics has been discussed in Ref. [6], without taking into account for the observable temperature (i.e., with the thermal variable ). Expressing with the real temperature Tq , this leads to a behavior similar to the BG case, with a simple shift of the critical temperature. 5. Conclusion In this short paper, we discussed the proper characterization of the thermodynamic limit in the framework of the non-extensive thermostatistics at equilibrium. Two points are delicate: the de4nition of the observable temperature Tq , and the role of the entropy index q. Concerning the latter parameter, the thermodynamic limit is de4ned provided the long-range correlations, hidden in the parameter q, are renormalized with the system size as: q = 1 + #=N , with the parameter # constant. This new parameter should govern the relative strength of the correlations due to non-extensive entropy, in comparison with the natural system-size correlations (those coming from the extensive theory). When taking the correct values for the state variables (Tq ; Pq ; q ), one 4nds that the limit state of the system does not depend crucially of the NE hypothesis. Only quantitative, but not qualitative, di
C. Tsallis, J. Stat. Phys. 52 (1988) 479. C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A 261 (1998) 534. A.R. Lima, J.S. SSa Martins, T.J.P. Penna, Physica A 268 (1999) 553. V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64 (2001) 056134. S. Abe, S. Martinez, F. Pennini, A. Plastino, Phys. Lett. A 281 (2001) 126. R. Botet, M. Ploszajczak, J.A. GonzSalez, Phys. Rev. E 65 (2002) 015103.