- Email: [email protected]
Thermostatistical aspects of generalized entropies
Chaos, Solitons and Fractals 20 (2004) 227–233 www.elsevier.com/locate/chaos
Thermostatistical aspects of generalized entropies K.S. Fa *, E.K. Lenzi Departamento de fısica, Universidade Estadual de Maring a, Av. Colombo, 5790, 87020-900 Maring a, PR, Brazil Accepted 15 July 2003
Abstract P We investigate the properties concerning a class of generalized entropies given by Sq;r ¼ kf1 ½ i piq r g=½rðq 1Þ which include TsallisÕ entropy (r ¼ 1), the usual Boltzmann–Gibbs entropy (q ¼ 1), RenyiÕs entropy (r ¼ 0) and normalized TsallisÕ entropy (r ¼ 1). In order to obtain the generalized thermodynamic relations we use the laws of thermodynamics and considering the hypothesis that the joint probability of two independent systems is given by pijA[B ¼ piA pjB . We show that the transmutation which occurs from TsallisÕ entropy to RenyiÕs entropy also occur with Sq;r . In this scenario, we also analyze the generalized variance, covariance and correlation coefficient of a non-interacting system by using extended optimal Lagrange multiplier approach. We show that the correlation coefficient tends to zero in the thermodynamic limit. However, RenyiÕs entropy related to this non-interacting system presents a certain degree of non-extensivity. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction TsallisÕs entropy [1], P 1 i piq ; Sq ¼ k q1
ð1Þ
obeys the pseudo-additivity property SðA þ BÞ ¼ SðAÞ þ SðBÞ þ ð1 qÞSðAÞ SðBÞ=k;
ð2Þ
considering A and B two independent systems and pijA[B ¼ piA pjB . The limit q ! 1 recovers the usual Boltzmann–Gibbs entropy. This entropy maximized with adequate constraints gives power law for probability distribution which generalizes the Boltzmann–Gibbs statistics [1–4]. It has been successfully applied to several situations such as anomalous diffusion [5–7], magnetic systems (manganites) [8,9], to study systems that exhibits long-range interaction [10,11], in dynamic systems [12,13] and electron–phonon interaction [14] (see also [15–17]). However, Eq. (1) represents a particular case of a set of generalized entropies which obeys the pseudo-additivity rule and has a power-law distribution. In fact, the conditions presented in [18] can be modified in a suitable way in order to be incorporated into other entropies, and in particular Sq;r which is defined below. Thus, various properties concerning to Eq. (1) can be extended to a more general context. For instance, the results presented in [19,20] for non-interacting systems and in [21] for the composition of the Tsallis entropy for different q indices can also be obtained for other generalized entropies. In this direction, other points such as the thermodynamic limit, zeroth law of the thermodynamics in the non-extensive scenario may be analyzed in a broad context. P q LetPus start by considering the pseudo-additivity entropy rule and a function which depends on i pi , i.e., q S ¼ S½ i pi . A possible class of generalized entropies which entail the assumptions above may be given by *
Corresponding author.
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00369-2
228
K.S. Fa, E.K. Lenzi / Chaos, Solitons and Fractals 20 (2004) 227–233
P
Sq;r ¼ k
q r
1 i pi rðq 1Þ
;
ð3Þ
where q; r 2 R. For simplicity, we consider k ¼ 1 hereafter. It is interesting to note that Sq;r includes several entropies presented in the literature. The last expression recovers TsallisÕ entropy (r ¼ 1Þ, the usual Boltzmann–Gibbs entropy (q ¼ 1), RenyiÕs entropy (r ¼ 0) and normalized TsallisÕ entropy (r ¼ 1) [22–24]. A similar family of entropy functions has been introduced in [25]. These generalized entropies have been explored in different contexts. Our aim is to study the thermostatistical aspects of these generalized entropies in an unified way by employing the optimal Lagrange multiplier (OLM) approach [4]. In order to obtain the generalized thermodynamic relations we use the laws of thermodynamics and considering, implicitly, the hypothesis that the joint probability of two independent systems pijA[B ¼ piA pjB is valid. We verify that the transmutation which occurs from TsallisÕ entropy to RenyiÕs entropy [26] also occurs with Sq;r . In this case, it is interesting to analyze a non-interacting system and verify the separability of the joint probability of RenyiÕs entropy which was implicitly assumed. We calculate the quantities like generalized variance, covariance and correlation coefficient of this system. We show that the generalized correlation coefficient is vanished in the thermodynamic limit. On the other hand, we calculate, in particular, RenyiÕs entropy of two equal parts of the non-interacting system. We show that RenyiÕs entropy presents a certain degree of non-extensivity by calculating the sum of RenyiÕs entropy of two equal parts of the system which is not equal to RenyiÕs entropy of the whole system. This result is very important to the discussion of implementation of the zeroth law of thermodynamics in the nonextensive statistical mechanics [19].
2. Thermodynamic relations The basic prescription for obtaining the generalized thermodynamic relations are based on the laws of thermodynamics and thermodynamic Legendre transform structure. In order to apply the zeroth law of thermodynamics we consider a system to be composed of two independent systems A and B and the hypothesis of the joint probability of the total system to be pijA[B ¼ piA pjB . Thus, the entropy Sq;r ðA; BÞ can be written as Sq;r ðA þ BÞ ¼ Sq;r ðAÞ þ Sq;r ðBÞ þ rð1 qÞSq;r ðAÞ Sq;r ðBÞ:
ð4Þ
In thermal equilibrium, the entropy Sq;r is maximized, i.e., dSq;r ðA; BÞ ¼ 0, and it yields dSq;r ðAÞ dSq;r ðBÞ ¼ : 1 þ rð1 qÞSq;r ðAÞ 1 þ rð1 qÞSq;r ðBÞ
ð5Þ
For a closed system the total internal energy Uq;r ðA þ BÞ ¼ Uq;r ðAÞ þ Uq;r ðBÞ is fixed and dUq;r ðA þ BÞ ¼ 0. Using Eq. (5) we obtain the following quantity bðAÞ bðBÞ ¼ ; 1 þ rð1 qÞSq;r ðAÞ 1 þ rð1 qÞSq;r ðBÞ
ð6Þ
where b ¼ oSq;r =oUq;r . We see that, from Eq. (6), the temperature must be defined as 1 T ¼ ½1 þ rð1 qÞSq;r : b
ð7Þ
Generalized pressure can be obtained by taking a fixed total volume V ðA þ BÞ ¼ V ðAÞ þ V ðBÞ and using Eq. (5) we obtain Pq;r ¼
T oSq;r : 1 þ rð1 qÞSq;r oV
ð8Þ
This last relation suggests that generalized free energy must be given by Fq;r ¼ Uq;r T
ln½1 þ rð1 qÞSq;r : rð1 qÞ
ð9Þ
We see that the generalized thermodynamic relations advanced by Abe et al. [27] are maintained for Sq;r too. Moreover, the transmutation that occurs from TsallisÕ entropy to RenyiÕs entropy also occurs with Sq;r .
K.S. Fa, E.K. Lenzi / Chaos, Solitons and Fractals 20 (2004) 227–233
229
3. Entropy maximization Now, let us to use the following constraints to obtain probability distribution: X pi ¼ 1;
ð10Þ
i
and X
!r1 pjq
j
X
piq ðei Uq;r Þ ¼ 0;
ð11Þ
i
where P q p ei Uq;r ¼ Pi i q ; i pi
ð12Þ
and ei are the eigenvalues of the Hamiltonian of the system. We note that the constraint (11) is a generalization of the constraint used in the OLM approach [4]. Our choice is due to the fact that the constraint used in OLM approach presents several advantages with relation to other constraints introduced in the literature [1–3], for instance, the Hessian matrix assumes diagonal form, probability distribution is not explicitly self-referential and temperature retains its ordinary meaning, i.e., temperature is proportional to the inverse of the Lagrange multiplier. The use of our constraint, Eq. (11), can reproduce these properties of OLM approach except the measure Sq;r does give the diagonal form for Hessian matrix only for r ¼ 1 (TsallisÕ measure). Probability distribution is obtained by extremizing Sq;r subject to the constraints (10) and (11), and we obtain pi ¼
expq fbðei Uq;r Þg ; Zq;r
ð13Þ
where b is a Lagrange multiplier, Zq;r is given by X expq fbðei Uq;r Þg Zq;r ¼
ð14Þ
i 1
and expðxÞ ½1 þ ð1 qÞx1q . Moreover, we have X q pi ¼ ðZq;r Þ1q :
ð15Þ
i
It is interesting to note that the probability and partition function do not depend, explicitly, on the parameter r and they are equal to those obtained in [4]. Following the same reasoning given in [28,29], we can conclude that temperature also retains its ordinary meaning in this extended OLM approach, i.e. b ¼ 1=T .
4. Ideal gas system P Non-interacting system is characterized by the Hamiltonian H ¼ i Pi2 =ð2mÞ. We consider this system in the threedimensional space. The motivation for this study is that the ideal gas system can be analytically worked and it serves to study some important statistical properties of non-extensive statistics. In particular, we are interested in the quantities like generalized variance, covariance and correlation coefficient in order to analyze the separability of a system. Here, we limit our analysis in the range 0 < q < 1 [20,27]. The generalized partition function and the normalized internal energy of a classical ideal gas [27] are given by Zq;r ¼
Uq;r
VN N!h3N
Z Y N
( d3 Pi 1 þ ðq 1Þb
i¼1
VN ¼ 3N cN !h ðZq;r Þq
X Pj2 Uq;r 2m j
1 !)1q
;
q ( !)1q 2 2 X X P P j i 1 þ ðq 1Þb Uq;r d3 Pl ; 2m 2m j i l¼1
Z Y N
ð16Þ
ð17Þ
230
K.S. Fa, E.K. Lenzi / Chaos, Solitons and Fractals 20 (2004) 227–233
and VN c¼ 3N N !h ðZq;r Þq
Z Y N
( 3
d Pi
i¼1
X Pj2 1 þ ðq 1Þb Uq;r 2m j
q !)1q
;
ð18Þ
where h is the linear dimension of the elementary cell in phase space. Eqs. (16)–(18) can be explicitly evaluated by using the following integral representation of the gamma function [30] 8 < 1 b 1z Z 1 ibt eb e for b > 0; dt ð19Þ CðzÞ z ¼ : 2p ð1 þ itÞ 1 0 for b < 0: The results are
N
Zq;r ¼
V 2pm N !h3N ð1 qÞb VN
Uq;r ¼
h
3N 2b
i
cN !h3N ðZq;r Þq
3N2 ½1 þ ð1 qÞbUq
2pm ð1 qÞb
1 3N 1q þ 2
3N2
C 2q 1q ; 2q C 1q þ 3N 2 1 3N 1q þ 2
½1 þ ð1 qÞbUq;r
C 2q 1q ; 2q C 1q þ 3N 2
ð20Þ
ð21Þ
and N
c¼
V 2pm N !h3N ðZq;r Þq ð1 qÞb
3N2 ½1 þ ð1 qÞbUq;r
q 3N 1q þ 2
1 C 1q : 1 þ 3N C 1q 2
ð22Þ
From Eqs. (21) and (22) we verify that Uq;r ¼
3N : 2b
ð23Þ
Now, using the fact that b¼
1 ; T
ð24Þ
one has Uq;r ¼
3N T: 2
ð25Þ
Therefore, one has obtained the classical result of an ideal gas which is independent of the parameters q and r. The specific heat reads Cq;r ¼
oUq;r : oT
ð26Þ
From Eq. (25) we immediately obtain that Cq;r ¼
3N : 2
ð27Þ
Other quantities like generalized variance, covariance and correlation coefficient are also important to analyze the statistical properties of an ideal gas. Following the definitions given in [19], we obtain ðMq;r Hi Þ2 ¼ hHi2 iq;r hHi i2q;r ¼
3 2 þ 3ð1 qÞðN 1Þ ; 2b2 4 2q þ ð1 qÞ3N
Cq;r ðHi ; Hj Þ ¼ hHi Hj iq;r hHi iq;r hHj iq;r ¼
9ð1 qÞ ; 2b ½4 2q þ ð1 qÞ3N 2
ð28Þ ð29Þ
K.S. Fa, E.K. Lenzi / Chaos, Solitons and Fractals 20 (2004) 227–233
Cq;r ðHi ; Hj Þ 3ð1 qÞ : qq;r ðHi ; Hj Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 þ 3ð1 qÞðN 1Þ 2 2 ðDq;r Hi Þ ðDq;r Hj Þ
231
ð30Þ
We note that these quantities above are independent of the parameter r. In the thermodynamic limit (N , V ! 1) yield 3 ; 2b2
ð31Þ
Cq;r ðHi ; Hj Þ ! 0;
ð32Þ
qq;r ðHi ; Hj Þ ! 0:
ð33Þ
ðDq;r Hi Þ2 !
We see that the generalized entropies Sq;r , with the use of the constraint (11), are also able to reproduce the ordinary results of a classical ideal gas. Finally, Sq;r of ideal gas is given by rð1qÞ h i3N2 1 3N Cð2qÞ N 2pm 1 NV!h3N ð1qÞb ½1 þ ð1 qÞbUq;r 1q þ 2 C 2q1q ð1qþ3N2 Þ Sq;r ¼ : ð34Þ rðq 1Þ In particular, we take r ! 0 (RenyiÕs entropy) and then take N large. For r ! 0, we have 8 1 þ 3N 1 9 1q 1 2 < C 1q = 1 þ ð1 qÞ 3N 2 3 2pmT Sq;0 ¼ N ln V þ ln ln N ! þ ln : 3N 2 : ; 2 h C 1 þ 3N ð1 qÞ 2 1q
ð35Þ
2
1 large too. Using StirlingÕs formula CðnÞ ffi þ 3N Now, take N large and consider the argument of C 1q 2 p ffiffiffiffiffiffi n we 2pn 1=2 expðnÞ we obtain 8 9 1 1 1 > 1 1q 2 1q > < = ð1 qÞ C e 1q V 3 2pmT 5 ffi : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ln Sq;0 ¼ N ln þ ln þ ð36Þ 2 > N 2 h 2 : ; 2p 1 þ ð1 qÞ 3N > 2
This is the generalization of the standard result. The difference between them is the last term which is absent in the standard expression [31]. As we can see that, the last term of Eq. (36) may represent a problem to the extensivity of RenyiÕs entropy and to establish the zeroth law of thermodynamics in the non-extensive scenario. However, the last term of Eq. (36) is not linear in N and it grows less than the linear term. Therefore, the dominant term is linear and it is an extensive quantity as can be viewed below. To see that, we suppose that a container of gas is divided into two equal 1 2 and Sq;0 the entropies of the two parts, thus parts by a partition. We consider Sq;0 8 9 1 1 1 > 1 1q2 1q > < = C e ð1 qÞ 1q V1 3 2pmT 5 1 2 ffi ; þ ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sq;0 þ ¼ Sq;0 ¼ N1 ln þ ln ð37Þ 2 > h 2 N1 2 : 2p 1 þ ð1 qÞ 3N1 > ; 2
and Sq;0 is the entropy of the whole gas without partition given by 8 9 1 1 1 1 < C 1q ð1 qÞ1q2 e1q = V1 3 2pmT 5 þ ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Sq;0 ¼ 2N1 ln þ ln þ : 2p½1 þ ð1 qÞ3N1 ; h2 2 N1 2 1 yields The difference Sq;0 2Sq;0 8 9 pffiffiffiffiffiffi 1 < = 2p 1 þ ð1 qÞ 3N2 1 e1q 1 Sq;0 2Sq;0 ¼ ln ; 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : C 1 ð1 qÞ1q 2 1 þ ð1 qÞ3N ; 1 1q
ð38Þ
ð39Þ
that is not equal to zero, except for q ! 1. 5. Summary and conclusion P We have considered a generalized entropy function given by Sq;r ¼ ½1 ð i piq Þr =½rðq 1Þ which includes several entropies presented in the literature. We have studied their thermostatistical properties by using the extended OLM
232
K.S. Fa, E.K. Lenzi / Chaos, Solitons and Fractals 20 (2004) 227–233
approach. In this approach we have used the constraints (10) and (11) in order to maintain several properties investigated in [4]. We have shown that the transmutation which occurs from TsallisÕ entropy to RenyiÕs entropy is also verified for Sq;r . Moreover, P it can be shown this transmutation to RenyiÕs entropy also occurs for an arbitrary entropy function given by S ¼ S½ i piq . We have also studied a non-interacting system due to the fact that this system permits us to analyze the separability of the joint probability which was implicitly assumed in order to obtain the generalized thermodynamic relations. In fact, we have two important quantities which are related to the separability of joint probability. First, we have shown that the correlation coefficient of RenyiÕs entropy is suppressed in the thermodynamic limit. However, this result does not guarantee that the probabilities are independent [32,33]. Second, our result (36) presents a certain degree of non-extensivity due to the non-linear term in N . However, this non-linear term grows less than the linear term and, the dominant term is an extensive quantity. Therefore, the equilibrium entropy can be considered as an extensive quantity. This last result seems to be valid for other generalized entropies Sq;r .
Acknowledgement E.K. Lenzi thanks CNPq for financial support.
References [1] Tsallis C. Possible generalization of Boltzmann–Gibbs statistics. J Stat Phys 1988;52:479–87. [2] Curado EMF, Tsallis C. Generalized statistical mechanics––connection with thermodynamics. J Phys A 1991;24:L69–72, Corrigenda: 1991;24:3187 and 1992;25:1019. [3] Tsallis C, Mendes RS, Plastino AR. The role of constraints within generalized nonextensive statistics. Physica A 1998;261:534–54. [4] Martınez S, Nicol as F, Pennini F, Plastino A. TsallisÕ entropy maximization procedure revisited. Physica A 2000;286:489–502. [5] Plastino AR, Plastino A. Non-extensive statistical mechanics and generalized Fokker–Planck equation. Physica A 1995;222: 347–54. [6] Tsallis C, Bukman DJ. Anomalous diffusion in the presence of external forces: exact time-dependent solutions and their thermostatistical basis. Phys Rev E 1996;54:R2197–200. [7] Malacarne LC, Mendes RS, Pedron IT, Lenzi EK. Nonlinear equation for anomalous diffusion: unified power-law and stretched exponential exact solution. Phys Rev E 2001;63:30101R-1-4. [8] Reis MS, Araujo JP, Amaral VS, Lenzi EK, Oliveira IS. Magnetic behavior of a nonextensive S-spin system: possible connections to manganites. Phys Rev B 2002;66:134417-1-6. [9] Reis MS, Freitas JCC, Orlando MTD, Lenzi EK, Oliveira IS. Evidences for Tsallis non-extensivity on CMR manganites. Europhys Lett 2002;58:42–8. [10] Salazar R, Toral R. Scaling laws for a system with long-range interactions within Tsallis statistics. Phys Rev Lett 1999;83:4233–6; Vollmayr-Lee BP, Luijten E. Comment on Scaling laws for a system with long-range interactions within Tsallis statistics. Phys Rev Lett 2000;85:470; Toral R, Salazar R. Comment on Scaling laws for a system with long-range interactions within Tsallis statistics––Toral and Salazar reply. Phys Rev Lett 2000;85:471. [11] Anteneodo C, Tsallis C. Breakdown of exponential sensitivity to initial conditions: role of the range of interactions. Phys Rev Lett 1998;80:5313–6. [12] Baldovin F, Robledo A. Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics. Phys Rev E 2002;66:045104-1-4. [13] Baldovin F, Robledo A. Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: rigorous nonextensive solutions. Europhys Lett 2002;60:518–24. [14] Koponen I. Thermalization of an electron–phonon system in a nonequilibrium state characterized by fractal distribution of phonon excitations. Phys Rev E 1997;55:7759–62. [15] Salinas SRA, Tsallis C. Nonextensive statistical mechanics and thermodynamics. Braz J Phys 1999;29:1–214. [16] Abe S, Okamoto Y. Nonextensive statistical mechanics and its applications. Heidelberg: Springer-Verlag; 2001. [17] Grigolini P, Tsallis C, West BJ. Classical and quantum complexity and nonextensive thermodynamics. Chaos, Solitons & Fractals 2002;13:367–616. [18] Santos RJV. Generalization of ShannonÕs theorem for Tsallis entropy. J Math Phys 1997;38:4104–7. [19] Abe S. Correlation induced by TsallisÕ nonextensivity. Physica A 1999;269:403–9. [20] Abe S, Martinez S, Pennini F, Plastino A. Classical gas in nonextensive optimal Lagrange multipliers formalism. Phys Lett A 2001;278:249–54. [21] Sasaki K, Hotta M. Generalized entropy composition with different q indices: an attempt. Chaos Solitons & Fractals 2002;13: 513–9. [22] Rajagopal AK, Abe S. Implications of form invariance to the structure of nonextensive entropies. Phys Rev Lett 1999;83:1711–4.
K.S. Fa, E.K. Lenzi / Chaos, Solitons and Fractals 20 (2004) 227–233
233
[23] Lenzi EK, Mendes RS, Silva LR. Normalized Tsallis entropy and its implications for the nonextensive thermostatistics. Physica A 2001;295:230–3. [24] Suyari H. Nonextensive entropies derived from form invariance of pseudoadditivity. Phys Rev E 2002;65:066118-1-7. [25] Czachor M, Naudts J. Thermostatistics based on Kolmogorov–Nagumo averages: unifying framework for extensive and nonextensive generalizations. Phys Lett A 2002;298:369–74. [26] Abe S. Heat and entropy in nonextensive thermodynamics: transmutation from Tsallis theory to Renyi-entropy-based theory. Physica A 2001;300:417–23. [27] Abe S, Martinez S, Pennini F, Plastino A. Nonextensive thermodynamic relations. Phys Lett A 2001;281:126–30. [28] Martınez S, Pennini F, Plastino A. The concept of temperature in a nonextensive scenario. Physica A 2001;295:246–9. [29] Rama SK. Tsallis statistics: averages and a physical interpretation of the Lagrange multiplier beta. Phys Lett A 2000;276:103–8. [30] Gradshteyn IS, Ryzhik IM. Table of integrals, series and products. New York: Academic Press; 1980. [31] Reif F. Fundamentals of statistical and thermal physics. New York: McGraw-Hill; 1985. [32] Feller W. In: An introduction to probability theory and its applications, 2nd ed, vol. 1. New York: John Wiley and Sons; 1965. [33] Risken H. The Fokker–Planck equation. Berlin: Springer-Verlag; 1984.
Thermostatistical aspects of generalized entropies K.S. Fa *, E.K. Lenzi Departamento de fısica, Universidade Estadual de Maring a, Av. Colombo, 5790, 87020-900 Maring a, PR, Brazil Accepted 15 July 2003
Abstract P We investigate the properties concerning a class of generalized entropies given by Sq;r ¼ kf1 ½ i piq r g=½rðq 1Þ which include TsallisÕ entropy (r ¼ 1), the usual Boltzmann–Gibbs entropy (q ¼ 1), RenyiÕs entropy (r ¼ 0) and normalized TsallisÕ entropy (r ¼ 1). In order to obtain the generalized thermodynamic relations we use the laws of thermodynamics and considering the hypothesis that the joint probability of two independent systems is given by pijA[B ¼ piA pjB . We show that the transmutation which occurs from TsallisÕ entropy to RenyiÕs entropy also occur with Sq;r . In this scenario, we also analyze the generalized variance, covariance and correlation coefficient of a non-interacting system by using extended optimal Lagrange multiplier approach. We show that the correlation coefficient tends to zero in the thermodynamic limit. However, RenyiÕs entropy related to this non-interacting system presents a certain degree of non-extensivity. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction TsallisÕs entropy [1], P 1 i piq ; Sq ¼ k q1
ð1Þ
obeys the pseudo-additivity property SðA þ BÞ ¼ SðAÞ þ SðBÞ þ ð1 qÞSðAÞ SðBÞ=k;
ð2Þ
considering A and B two independent systems and pijA[B ¼ piA pjB . The limit q ! 1 recovers the usual Boltzmann–Gibbs entropy. This entropy maximized with adequate constraints gives power law for probability distribution which generalizes the Boltzmann–Gibbs statistics [1–4]. It has been successfully applied to several situations such as anomalous diffusion [5–7], magnetic systems (manganites) [8,9], to study systems that exhibits long-range interaction [10,11], in dynamic systems [12,13] and electron–phonon interaction [14] (see also [15–17]). However, Eq. (1) represents a particular case of a set of generalized entropies which obeys the pseudo-additivity rule and has a power-law distribution. In fact, the conditions presented in [18] can be modified in a suitable way in order to be incorporated into other entropies, and in particular Sq;r which is defined below. Thus, various properties concerning to Eq. (1) can be extended to a more general context. For instance, the results presented in [19,20] for non-interacting systems and in [21] for the composition of the Tsallis entropy for different q indices can also be obtained for other generalized entropies. In this direction, other points such as the thermodynamic limit, zeroth law of the thermodynamics in the non-extensive scenario may be analyzed in a broad context. P q LetPus start by considering the pseudo-additivity entropy rule and a function which depends on i pi , i.e., q S ¼ S½ i pi . A possible class of generalized entropies which entail the assumptions above may be given by *
Corresponding author.
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00369-2
228
K.S. Fa, E.K. Lenzi / Chaos, Solitons and Fractals 20 (2004) 227–233
P
Sq;r ¼ k
q r
1 i pi rðq 1Þ
;
ð3Þ
where q; r 2 R. For simplicity, we consider k ¼ 1 hereafter. It is interesting to note that Sq;r includes several entropies presented in the literature. The last expression recovers TsallisÕ entropy (r ¼ 1Þ, the usual Boltzmann–Gibbs entropy (q ¼ 1), RenyiÕs entropy (r ¼ 0) and normalized TsallisÕ entropy (r ¼ 1) [22–24]. A similar family of entropy functions has been introduced in [25]. These generalized entropies have been explored in different contexts. Our aim is to study the thermostatistical aspects of these generalized entropies in an unified way by employing the optimal Lagrange multiplier (OLM) approach [4]. In order to obtain the generalized thermodynamic relations we use the laws of thermodynamics and considering, implicitly, the hypothesis that the joint probability of two independent systems pijA[B ¼ piA pjB is valid. We verify that the transmutation which occurs from TsallisÕ entropy to RenyiÕs entropy [26] also occurs with Sq;r . In this case, it is interesting to analyze a non-interacting system and verify the separability of the joint probability of RenyiÕs entropy which was implicitly assumed. We calculate the quantities like generalized variance, covariance and correlation coefficient of this system. We show that the generalized correlation coefficient is vanished in the thermodynamic limit. On the other hand, we calculate, in particular, RenyiÕs entropy of two equal parts of the non-interacting system. We show that RenyiÕs entropy presents a certain degree of non-extensivity by calculating the sum of RenyiÕs entropy of two equal parts of the system which is not equal to RenyiÕs entropy of the whole system. This result is very important to the discussion of implementation of the zeroth law of thermodynamics in the nonextensive statistical mechanics [19].
2. Thermodynamic relations The basic prescription for obtaining the generalized thermodynamic relations are based on the laws of thermodynamics and thermodynamic Legendre transform structure. In order to apply the zeroth law of thermodynamics we consider a system to be composed of two independent systems A and B and the hypothesis of the joint probability of the total system to be pijA[B ¼ piA pjB . Thus, the entropy Sq;r ðA; BÞ can be written as Sq;r ðA þ BÞ ¼ Sq;r ðAÞ þ Sq;r ðBÞ þ rð1 qÞSq;r ðAÞ Sq;r ðBÞ:
ð4Þ
In thermal equilibrium, the entropy Sq;r is maximized, i.e., dSq;r ðA; BÞ ¼ 0, and it yields dSq;r ðAÞ dSq;r ðBÞ ¼ : 1 þ rð1 qÞSq;r ðAÞ 1 þ rð1 qÞSq;r ðBÞ
ð5Þ
For a closed system the total internal energy Uq;r ðA þ BÞ ¼ Uq;r ðAÞ þ Uq;r ðBÞ is fixed and dUq;r ðA þ BÞ ¼ 0. Using Eq. (5) we obtain the following quantity bðAÞ bðBÞ ¼ ; 1 þ rð1 qÞSq;r ðAÞ 1 þ rð1 qÞSq;r ðBÞ
ð6Þ
where b ¼ oSq;r =oUq;r . We see that, from Eq. (6), the temperature must be defined as 1 T ¼ ½1 þ rð1 qÞSq;r : b
ð7Þ
Generalized pressure can be obtained by taking a fixed total volume V ðA þ BÞ ¼ V ðAÞ þ V ðBÞ and using Eq. (5) we obtain Pq;r ¼
T oSq;r : 1 þ rð1 qÞSq;r oV
ð8Þ
This last relation suggests that generalized free energy must be given by Fq;r ¼ Uq;r T
ln½1 þ rð1 qÞSq;r : rð1 qÞ
ð9Þ
We see that the generalized thermodynamic relations advanced by Abe et al. [27] are maintained for Sq;r too. Moreover, the transmutation that occurs from TsallisÕ entropy to RenyiÕs entropy also occurs with Sq;r .
K.S. Fa, E.K. Lenzi / Chaos, Solitons and Fractals 20 (2004) 227–233
229
3. Entropy maximization Now, let us to use the following constraints to obtain probability distribution: X pi ¼ 1;
ð10Þ
i
and X
!r1 pjq
j
X
piq ðei Uq;r Þ ¼ 0;
ð11Þ
i
where P q p ei Uq;r ¼ Pi i q ; i pi
ð12Þ
and ei are the eigenvalues of the Hamiltonian of the system. We note that the constraint (11) is a generalization of the constraint used in the OLM approach [4]. Our choice is due to the fact that the constraint used in OLM approach presents several advantages with relation to other constraints introduced in the literature [1–3], for instance, the Hessian matrix assumes diagonal form, probability distribution is not explicitly self-referential and temperature retains its ordinary meaning, i.e., temperature is proportional to the inverse of the Lagrange multiplier. The use of our constraint, Eq. (11), can reproduce these properties of OLM approach except the measure Sq;r does give the diagonal form for Hessian matrix only for r ¼ 1 (TsallisÕ measure). Probability distribution is obtained by extremizing Sq;r subject to the constraints (10) and (11), and we obtain pi ¼
expq fbðei Uq;r Þg ; Zq;r
ð13Þ
where b is a Lagrange multiplier, Zq;r is given by X expq fbðei Uq;r Þg Zq;r ¼
ð14Þ
i 1
and expðxÞ ½1 þ ð1 qÞx1q . Moreover, we have X q pi ¼ ðZq;r Þ1q :
ð15Þ
i
It is interesting to note that the probability and partition function do not depend, explicitly, on the parameter r and they are equal to those obtained in [4]. Following the same reasoning given in [28,29], we can conclude that temperature also retains its ordinary meaning in this extended OLM approach, i.e. b ¼ 1=T .
4. Ideal gas system P Non-interacting system is characterized by the Hamiltonian H ¼ i Pi2 =ð2mÞ. We consider this system in the threedimensional space. The motivation for this study is that the ideal gas system can be analytically worked and it serves to study some important statistical properties of non-extensive statistics. In particular, we are interested in the quantities like generalized variance, covariance and correlation coefficient in order to analyze the separability of a system. Here, we limit our analysis in the range 0 < q < 1 [20,27]. The generalized partition function and the normalized internal energy of a classical ideal gas [27] are given by Zq;r ¼
Uq;r
VN N!h3N
Z Y N
( d3 Pi 1 þ ðq 1Þb
i¼1
VN ¼ 3N cN !h ðZq;r Þq
X Pj2 Uq;r 2m j
1 !)1q
;
q ( !)1q 2 2 X X P P j i 1 þ ðq 1Þb Uq;r d3 Pl ; 2m 2m j i l¼1
Z Y N
ð16Þ
ð17Þ
230
K.S. Fa, E.K. Lenzi / Chaos, Solitons and Fractals 20 (2004) 227–233
and VN c¼ 3N N !h ðZq;r Þq
Z Y N
( 3
d Pi
i¼1
X Pj2 1 þ ðq 1Þb Uq;r 2m j
q !)1q
;
ð18Þ
where h is the linear dimension of the elementary cell in phase space. Eqs. (16)–(18) can be explicitly evaluated by using the following integral representation of the gamma function [30] 8 < 1 b 1z Z 1 ibt eb e for b > 0; dt ð19Þ CðzÞ z ¼ : 2p ð1 þ itÞ 1 0 for b < 0: The results are
N
Zq;r ¼
V 2pm N !h3N ð1 qÞb VN
Uq;r ¼
h
3N 2b
i
cN !h3N ðZq;r Þq
3N2 ½1 þ ð1 qÞbUq
2pm ð1 qÞb
1 3N 1q þ 2
3N2
C 2q 1q ; 2q C 1q þ 3N 2 1 3N 1q þ 2
½1 þ ð1 qÞbUq;r
C 2q 1q ; 2q C 1q þ 3N 2
ð20Þ
ð21Þ
and N
c¼
V 2pm N !h3N ðZq;r Þq ð1 qÞb
3N2 ½1 þ ð1 qÞbUq;r
q 3N 1q þ 2
1 C 1q : 1 þ 3N C 1q 2
ð22Þ
From Eqs. (21) and (22) we verify that Uq;r ¼
3N : 2b
ð23Þ
Now, using the fact that b¼
1 ; T
ð24Þ
one has Uq;r ¼
3N T: 2
ð25Þ
Therefore, one has obtained the classical result of an ideal gas which is independent of the parameters q and r. The specific heat reads Cq;r ¼
oUq;r : oT
ð26Þ
From Eq. (25) we immediately obtain that Cq;r ¼
3N : 2
ð27Þ
Other quantities like generalized variance, covariance and correlation coefficient are also important to analyze the statistical properties of an ideal gas. Following the definitions given in [19], we obtain ðMq;r Hi Þ2 ¼ hHi2 iq;r hHi i2q;r ¼
3 2 þ 3ð1 qÞðN 1Þ ; 2b2 4 2q þ ð1 qÞ3N
Cq;r ðHi ; Hj Þ ¼ hHi Hj iq;r hHi iq;r hHj iq;r ¼
9ð1 qÞ ; 2b ½4 2q þ ð1 qÞ3N 2
ð28Þ ð29Þ
K.S. Fa, E.K. Lenzi / Chaos, Solitons and Fractals 20 (2004) 227–233
Cq;r ðHi ; Hj Þ 3ð1 qÞ : qq;r ðHi ; Hj Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 þ 3ð1 qÞðN 1Þ 2 2 ðDq;r Hi Þ ðDq;r Hj Þ
231
ð30Þ
We note that these quantities above are independent of the parameter r. In the thermodynamic limit (N , V ! 1) yield 3 ; 2b2
ð31Þ
Cq;r ðHi ; Hj Þ ! 0;
ð32Þ
qq;r ðHi ; Hj Þ ! 0:
ð33Þ
ðDq;r Hi Þ2 !
We see that the generalized entropies Sq;r , with the use of the constraint (11), are also able to reproduce the ordinary results of a classical ideal gas. Finally, Sq;r of ideal gas is given by rð1qÞ h i3N2 1 3N Cð2qÞ N 2pm 1 NV!h3N ð1qÞb ½1 þ ð1 qÞbUq;r 1q þ 2 C 2q1q ð1qþ3N2 Þ Sq;r ¼ : ð34Þ rðq 1Þ In particular, we take r ! 0 (RenyiÕs entropy) and then take N large. For r ! 0, we have 8 1 þ 3N 1 9 1q 1 2 < C 1q = 1 þ ð1 qÞ 3N 2 3 2pmT Sq;0 ¼ N ln V þ ln ln N ! þ ln : 3N 2 : ; 2 h C 1 þ 3N ð1 qÞ 2 1q
ð35Þ
2
1 large too. Using StirlingÕs formula CðnÞ ffi þ 3N Now, take N large and consider the argument of C 1q 2 p ffiffiffiffiffiffi n we 2pn 1=2 expðnÞ we obtain 8 9 1 1 1 > 1 1q 2 1q > < = ð1 qÞ C e 1q V 3 2pmT 5 ffi : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ln Sq;0 ¼ N ln þ ln þ ð36Þ 2 > N 2 h 2 : ; 2p 1 þ ð1 qÞ 3N > 2
This is the generalization of the standard result. The difference between them is the last term which is absent in the standard expression [31]. As we can see that, the last term of Eq. (36) may represent a problem to the extensivity of RenyiÕs entropy and to establish the zeroth law of thermodynamics in the non-extensive scenario. However, the last term of Eq. (36) is not linear in N and it grows less than the linear term. Therefore, the dominant term is linear and it is an extensive quantity as can be viewed below. To see that, we suppose that a container of gas is divided into two equal 1 2 and Sq;0 the entropies of the two parts, thus parts by a partition. We consider Sq;0 8 9 1 1 1 > 1 1q2 1q > < = C e ð1 qÞ 1q V1 3 2pmT 5 1 2 ffi ; þ ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sq;0 þ ¼ Sq;0 ¼ N1 ln þ ln ð37Þ 2 > h 2 N1 2 : 2p 1 þ ð1 qÞ 3N1 > ; 2
and Sq;0 is the entropy of the whole gas without partition given by 8 9 1 1 1 1 < C 1q ð1 qÞ1q2 e1q = V1 3 2pmT 5 þ ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Sq;0 ¼ 2N1 ln þ ln þ : 2p½1 þ ð1 qÞ3N1 ; h2 2 N1 2 1 yields The difference Sq;0 2Sq;0 8 9 pffiffiffiffiffiffi 1 < = 2p 1 þ ð1 qÞ 3N2 1 e1q 1 Sq;0 2Sq;0 ¼ ln ; 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : C 1 ð1 qÞ1q 2 1 þ ð1 qÞ3N ; 1 1q
ð38Þ
ð39Þ
that is not equal to zero, except for q ! 1. 5. Summary and conclusion P We have considered a generalized entropy function given by Sq;r ¼ ½1 ð i piq Þr =½rðq 1Þ which includes several entropies presented in the literature. We have studied their thermostatistical properties by using the extended OLM
232
K.S. Fa, E.K. Lenzi / Chaos, Solitons and Fractals 20 (2004) 227–233
approach. In this approach we have used the constraints (10) and (11) in order to maintain several properties investigated in [4]. We have shown that the transmutation which occurs from TsallisÕ entropy to RenyiÕs entropy is also verified for Sq;r . Moreover, P it can be shown this transmutation to RenyiÕs entropy also occurs for an arbitrary entropy function given by S ¼ S½ i piq . We have also studied a non-interacting system due to the fact that this system permits us to analyze the separability of the joint probability which was implicitly assumed in order to obtain the generalized thermodynamic relations. In fact, we have two important quantities which are related to the separability of joint probability. First, we have shown that the correlation coefficient of RenyiÕs entropy is suppressed in the thermodynamic limit. However, this result does not guarantee that the probabilities are independent [32,33]. Second, our result (36) presents a certain degree of non-extensivity due to the non-linear term in N . However, this non-linear term grows less than the linear term and, the dominant term is an extensive quantity. Therefore, the equilibrium entropy can be considered as an extensive quantity. This last result seems to be valid for other generalized entropies Sq;r .
Acknowledgement E.K. Lenzi thanks CNPq for financial support.
References [1] Tsallis C. Possible generalization of Boltzmann–Gibbs statistics. J Stat Phys 1988;52:479–87. [2] Curado EMF, Tsallis C. Generalized statistical mechanics––connection with thermodynamics. J Phys A 1991;24:L69–72, Corrigenda: 1991;24:3187 and 1992;25:1019. [3] Tsallis C, Mendes RS, Plastino AR. The role of constraints within generalized nonextensive statistics. Physica A 1998;261:534–54. [4] Martınez S, Nicol as F, Pennini F, Plastino A. TsallisÕ entropy maximization procedure revisited. Physica A 2000;286:489–502. [5] Plastino AR, Plastino A. Non-extensive statistical mechanics and generalized Fokker–Planck equation. Physica A 1995;222: 347–54. [6] Tsallis C, Bukman DJ. Anomalous diffusion in the presence of external forces: exact time-dependent solutions and their thermostatistical basis. Phys Rev E 1996;54:R2197–200. [7] Malacarne LC, Mendes RS, Pedron IT, Lenzi EK. Nonlinear equation for anomalous diffusion: unified power-law and stretched exponential exact solution. Phys Rev E 2001;63:30101R-1-4. [8] Reis MS, Araujo JP, Amaral VS, Lenzi EK, Oliveira IS. Magnetic behavior of a nonextensive S-spin system: possible connections to manganites. Phys Rev B 2002;66:134417-1-6. [9] Reis MS, Freitas JCC, Orlando MTD, Lenzi EK, Oliveira IS. Evidences for Tsallis non-extensivity on CMR manganites. Europhys Lett 2002;58:42–8. [10] Salazar R, Toral R. Scaling laws for a system with long-range interactions within Tsallis statistics. Phys Rev Lett 1999;83:4233–6; Vollmayr-Lee BP, Luijten E. Comment on Scaling laws for a system with long-range interactions within Tsallis statistics. Phys Rev Lett 2000;85:470; Toral R, Salazar R. Comment on Scaling laws for a system with long-range interactions within Tsallis statistics––Toral and Salazar reply. Phys Rev Lett 2000;85:471. [11] Anteneodo C, Tsallis C. Breakdown of exponential sensitivity to initial conditions: role of the range of interactions. Phys Rev Lett 1998;80:5313–6. [12] Baldovin F, Robledo A. Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics. Phys Rev E 2002;66:045104-1-4. [13] Baldovin F, Robledo A. Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: rigorous nonextensive solutions. Europhys Lett 2002;60:518–24. [14] Koponen I. Thermalization of an electron–phonon system in a nonequilibrium state characterized by fractal distribution of phonon excitations. Phys Rev E 1997;55:7759–62. [15] Salinas SRA, Tsallis C. Nonextensive statistical mechanics and thermodynamics. Braz J Phys 1999;29:1–214. [16] Abe S, Okamoto Y. Nonextensive statistical mechanics and its applications. Heidelberg: Springer-Verlag; 2001. [17] Grigolini P, Tsallis C, West BJ. Classical and quantum complexity and nonextensive thermodynamics. Chaos, Solitons & Fractals 2002;13:367–616. [18] Santos RJV. Generalization of ShannonÕs theorem for Tsallis entropy. J Math Phys 1997;38:4104–7. [19] Abe S. Correlation induced by TsallisÕ nonextensivity. Physica A 1999;269:403–9. [20] Abe S, Martinez S, Pennini F, Plastino A. Classical gas in nonextensive optimal Lagrange multipliers formalism. Phys Lett A 2001;278:249–54. [21] Sasaki K, Hotta M. Generalized entropy composition with different q indices: an attempt. Chaos Solitons & Fractals 2002;13: 513–9. [22] Rajagopal AK, Abe S. Implications of form invariance to the structure of nonextensive entropies. Phys Rev Lett 1999;83:1711–4.
K.S. Fa, E.K. Lenzi / Chaos, Solitons and Fractals 20 (2004) 227–233
233
[23] Lenzi EK, Mendes RS, Silva LR. Normalized Tsallis entropy and its implications for the nonextensive thermostatistics. Physica A 2001;295:230–3. [24] Suyari H. Nonextensive entropies derived from form invariance of pseudoadditivity. Phys Rev E 2002;65:066118-1-7. [25] Czachor M, Naudts J. Thermostatistics based on Kolmogorov–Nagumo averages: unifying framework for extensive and nonextensive generalizations. Phys Lett A 2002;298:369–74. [26] Abe S. Heat and entropy in nonextensive thermodynamics: transmutation from Tsallis theory to Renyi-entropy-based theory. Physica A 2001;300:417–23. [27] Abe S, Martinez S, Pennini F, Plastino A. Nonextensive thermodynamic relations. Phys Lett A 2001;281:126–30. [28] Martınez S, Pennini F, Plastino A. The concept of temperature in a nonextensive scenario. Physica A 2001;295:246–9. [29] Rama SK. Tsallis statistics: averages and a physical interpretation of the Lagrange multiplier beta. Phys Lett A 2000;276:103–8. [30] Gradshteyn IS, Ryzhik IM. Table of integrals, series and products. New York: Academic Press; 1980. [31] Reif F. Fundamentals of statistical and thermal physics. New York: McGraw-Hill; 1985. [32] Feller W. In: An introduction to probability theory and its applications, 2nd ed, vol. 1. New York: John Wiley and Sons; 1965. [33] Risken H. The Fokker–Planck equation. Berlin: Springer-Verlag; 1984.