Extended Curie–Weiss law via Tsallis statistics

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J.Stat.M

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ournal of Statistical Mechanics:J Theory and Experiment

Extended Curie–Weiss law via Tsallisstatistics

R Chakrabarti and R Chandrashekar

Department of Theoretical Physics, University of Madras, Guindy Campus,Chennai 600 025, IndiaE-mail: ranabir@imsc.res.in and chandrashekar10@yahoo.co.in

Received 14 April 2010Accepted 24 August 2010Published 13 September 2010

Online at stacks.iop.org/JSTAT/2010/P09007doi:10.1088/1742-5468/2010/09/P09007

Abstract. In the framework of the Tsallis nonextensive statistical mechanicswe study an assembly of N spins, first in a background magnetic field, and thenassuming them to interact via a homogeneous mean field. To take into account thespin fluctuations the dynamical field coefficient is considered to be temperature-dependent. Specifically, we study a model where the effective field coefficientincludes two terms depending on temperature, of which one varies linearlywhereas the other exhibits an inverse square behavior. The physical quantities areevaluated using a perturbative expansion in the entropic deformation parameter(1− q). The perturbative procedure may be continued up to any arbitrary orderin the parameter (1 − q). The extended Curie–Weiss law in the mean-field casehas been generalized. The critical temperature and the Curie–Weiss constant arefound to be dependent on the entropic index q.

Keywords: classical phase transitions (theory)

c©2010 IOP Publishing Ltd and SISSA 1742-5468/10/P09007+20$30.00

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Contents

1. Introduction 2

2. Spins in a weak background field 4

3. Mean-field model: temperature-dependent effective field coefficient 13

4. Remarks 18

Acknowledgment 19

Appendix 19

References 20

1. Introduction

A generalization of the Boltzmann–Gibbs extensive statistical mechanics was proposed byTsallis [1] via a deformation of the functional form of entropy:

S = k lnq W, q ∈ R+, (1.1)

where k is the Boltzmann constant and W is the weight factor. The stability of Tsallisentropy (1.1) is ensured by maintaining the parameter q as a positive real number [2].The deformed q logarithm and its inverse the q-exponential function for a real variable xare

lnq x =x1−q − 1

1 − q, expq(x) = [1 + (1 − q)x]

1/(1−q)+ , (1.2)

where we have used the notation [x]+ = max{0, x}. The extensive classical Boltzmann–Gibbs entropy is recovered from (1.1) in the q → 1 limit. It has been observed [3]that for a class of strongly correlated subsystems the entropy (1.1) becomes extensive forcertain nontrivial values of the entropic parameter q. For arbitrarily correlated subsystemssubject to generic values of q the entropy (1.1), however, remains nonextensive. TheTsallis entropy (1.1) has been applied to a wide variety of fields in statistical mechanics.A detailed exposition of the subject may be found in [4].

Several mechanisms have been advanced [5] for the maximization of the deformedentropy (1.1) subject to appropriate constraints. The third constraint scenario [5]that employs suitably normalized escort probabilities is now considered as physicallysatisfactory. In the present work we will follow this formalism consistently. The statisticalq-expectation value of an observable, say, the internal energy of a system described by agiven Hamiltonian H is defined [5] as

Uq ≡ 〈H〉q =

∑j (pj(β))q Ej

c(β), c(β) =

∑

j

(pj(β))q, (1.3)

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where pj(β) is the probability corresponding to the microstate j endowed with an energyEj :

pj(β) =1

Zq

expq

(

−βEj − Uq

c(β)

)

, Zq(β) =∑

i

expq

(

−βEj − Uq

c(β)

)

. (1.4)

The generalized partition function Zq(β) and the sum of q-weights c(β) introduced in (1.4)and (1.3), respectively, are interrelated [5]:

Zq(β) = (c(β))1/(1−q) . (1.5)

In the above description the Lagrange multiplier β corresponding to the internal energy isassociated [5] with the inverse temperature: β ≡ (kT )−1. Introducing the notion of quasi-reversibility that only requires the system to return to a microstate in the neighborhood ofthe original state, and simultaneously implementing the first law of thermodynamics, it hasbeen observed [6] that the Tsallis entropy (1.1) becomes identical to the thermodynamicClausius entropy provided the Lagrange multiplier β is held proportional to the inversetemperature. It is argued [6] that, since in the nonextensive case we are not concerned withthe strict equilibrium states but rather the nonequilibrium stationary states that survivea long period of time, there exists a possibility that the temperature in the nonextensivestatistical mechanics may not be defined via the standard zeroth law of thermodynamicswhich necessitates the long-time limit being taken before the thermodynamic limit.Significantly, it has been noticed [7] that in the context of nonextensive systems thethermodynamic limit and the long-time limit do not commute. The notion of temperaturebased on the above arguments is quite different from the proposed [8]–[10] concept of‘physical temperature’ defined through the standard zeroth law and the assumption [6]of divisibility of the entire system into independent subsystems. Following the formerviewpoint we regard the temperature as the reciprocal of the Lagrange multiplier for theconstraint on the internal energy.

Application of nonextensive statistical mechanics to spin systems was initiated in [11],where an assembly of N noninteracting spin-1

2particles in a background field was studied

in the second constraint formalism [5] while associating the inverse temperature with theLagrange multiplier corresponding to the internal energy. A similar study based on thethird constraint framework was also performed [12]. The magnetic susceptibility in thismodel showed [11, 12] the interesting feature referred to as dark magnetism, indicatingthat the apparent number of spins is different from the actual number of spins. Employinga high temperature limit, it was observed in [12] that in the domain q > 1 (q < 1) theeffective number of spins maintains the property Neff > N (Neff < N). In the studyof manganites the Tsallis statistics has been observed [13] to fit the experimental dataon magnetization better than the standard Boltzmann–Gibbs statistics. A numericalanalysis of a multilevel spin model has been done [14] following the Tsallis statistics.Also numerical studies of the two-dimensional Ising model in the nonextensive contexthave been made [15, 16], indicating the occurrence of a continuous phase transition fornontrivial values of q. A collection of spin clusters has been examined [17, 18] in thethird constraint picture. A generalized paramagnetic susceptibility in the noninteractingregime and a nonextensive modification of the Curie–Weiss law in the context of themean-field model have been established [19]. The authors of this body of work [13]–[19]adopt a scaled form of the Lagrange multiplier corresponding to the internal energy as the

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‘physical temperature’. As a consequence of this choice of temperature the phenomenonof dark magnetism is not observed [14]. As discussed earlier, we, in contrast, identifythe temperature with the Lagrange multiplier pertinent to the constraint on the internalenergy.

Making a slight departure from the references [11, 12], we, in the present work,examine a classical arbitrary N -spin system in a weak background magnetic field withoutadopting the high temperature limit. The thermodynamic quantities in the thirdconstraint picture are evaluated as a perturbative series in the parameter (1 − q) viaa disentanglement of the q exponential (1.2) by expressing it as an infinite productseries of ordinary exponentials. This process of perturbative expansion, which may becontinued up to an arbitrary order, is based on the technique developed in [20] and hasbeen previously employed in [21, 22]. In our perturbative expansion for the spins in thebackground field we retain terms at all orders of temperature. Subsequently we studyinteracting spins in an extended form of the mean-field model [23, 24] where the fieldstrength coefficient (the proportionality factor) is assumed to depend on temperatureto accommodate quantum fluctuations [25] among allowed configurations. In particular,following [24], we study a model where the effective field coefficient includes two termsdepending on temperature, of which one varies linearly whereas the other exhibits aninverse square behavior. The critical temperature and the Curie–Weiss constant have beenevaluated in the third constraint framework by retaining terms in the perturbation schemeup to the order of (1−q)2. Specifically, the critical temperature in the nonextensive regimeincreases (decreases) compared to its value given by the Boltzmann–Gibbs statistics forthe domain q > 1 (q < 1). Our observation qualitatively agrees with the results obtainedin [19] where a different definition of temperature for the nonextensive spin system hasbeen used. In the context of extended mean-field theory the critical constants, however,do not exhibit q dependence. The plan of this article is as follows: noninteracting spins inthe presence of a weak external magnetic field are considered in section 2: this is followedby consideration of an extended mean-field model in section 3: our concluding remarksare given in section 4.

2. Spins in a weak background field

The classical Hamiltonian of a system of N spins in the presence of a background magneticfield H is given by

E = −μH

N∑

i=1

cos θi, (2.1)

where μ is the magnetic moment of the spins oriented at polar angles (θi, φi|i = 1, . . . , N)with the field. The generalized partition function [5] defined in (1.4) now assumes theform

Zq (β) = expq (ε)

∫ π

θi=0

∫ 2π

φi=0

[

1 + (1 − q)βN∑

�=1

cos θ�

]1/(1−q)

+

N∏

i=1

sin θi dθi dφi, (2.2)

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where ε = βUq/c(β) and the scaled dimensionless variable β is given by

β = β μH, β =β

c(β) + (1 − q) β Uq. (2.3)

The generalized partition function (2.2), and, consequently, the integrand therein isrequired to be real and non-negative for the allowed phase space. For a sufficiently smalldeformation regime |1 − q| � 1 the criterion of non-negativity is assumed to hold for theentire phase space. The evaluation of the phase space integral in (2.2) may be completedwith the aid of the result (A.4) given in the appendix:

Zq(β) = (2π)NΦ expq (ε) β−NS1(β), Φ =

N∏

�=1

[1 + (1 − q)�]−1 , (2.4)

where the binomial sum S1(x) involving the q exponentials is

S1(x) =

N∑

n=0

(N

n

)

(−1)n(expq((N − 2n)x)

)Λ, Λ = 1 + (1 − q)N. (2.5)

The physical quantities may be evaluated in a systematic order-by-order perturbationseries with the expansion parameter (1− q). The procedure is based upon an observationin [20] and has been employed [21, 22] previously. It hinges on expressing the qexponential (1.2) as an infinite multiplicative series of ordinary exponentials that, in turn,may be expanded up to a desired order in (1 − q):

expq(x) = exp

(1

1 − qln(1 + (1 − q)x)

)

=∞∏

k=1

exp

(

(−1)k−1 (1 − q)k−1 xk

k

)

. (2.6)

Applying this scheme on the generalized partition function (2.4) up to the order of (1−q)2

we obtain

Zq (β) = (2π)N Φ (Z(β))N exp (ε)(1 + (1 − q) �1 + (1 − q)2 �2 + · · ·) , (2.7)

where β = βμH/c(β) and Z(x) = 2 sinh(x)/x. The coefficients appearing in theperturbative series (2.7) are listed below:

�1 = N2 β coth β − N

2β +

N

2(1 − N) β2 coth2 β + Nε − 1

2ε2 − N ε β coth β,

�2 =N2

2β2(N + cosech2 β − N(1 − 2N) coth2 β

)+

N

6β3(4(1 − 3N)cosech2β

− 9N2 + (11 − 3N) coth2 β)− N

8β4(2 − 3N − 2(1 − 7N + 3N2) coth2 β

+ (6 − 11N + 6N2 − N3) coth4 β)− N

2(1 − N) ε2 − 1

6(2 − 3N) ε3 +

1

8ε4

− N

2

(2(1 − N) ε − (2 − 3N) ε2 + 2N ε3

)β coth β

− N2

4(2(5 − 3N) − 3 ε) ε β2 coth2 β +

N

4(4 − 5 ε) ε β2 cosech2β

+N3

2ε β3 coth3 β +

N

2(2 − 3N) ε β3 coth β cosech2β. (2.8)

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Similar expansion schemes may be developed for the sum of the q weights c(β), andthe internal energy Uq(β). As we later employ the relation (1.5) for a consistency checkof our calculation, we now evaluate c(β) directly employing the definition (1.3) whileconverting the sum to an appropriate phase space integral:

c(β) =

(expq (ε)

Zq(β)

)q ∫ π

θi=0

∫ 2π

φi=0

[

1 + (1 − q)βN∑

�=1

cos θ�

]q/(1−q)

+

N∏

i=1

sin θi dθi dφi. (2.9)

Performing the above integrations via a method discussed in the context of (2.2), andsubsequently substituting the generalized partition function (2.4), the sum of the q weightsassumes the form

c(β) = NΛβ(q−1)NS2(β)(S1(β))−q, (2.10)

where the binomial sum S2(x) is

S2(x) =N∑

n=0

(N

n

)

(−1)n(expq((N − 2n)x)

)Λ−(1−q). (2.11)

The exponential dependence of the numerical constant N:

N = (2π)(1−q)NΦ1−q (2.12)

on q enables it to play an important role in describing possible physical effects. Towardsour proposed use of the exponential relation (1.5) for confirming the consistency of ourperturbative evaluation process, we need to evaluate the sum c(β) up to the order (1−q)3:

c(β) = N(1 + (1 − q) P1 + (1 − q)2 P2 + (1 − q)3 P3 + · · ·) , (2.13)

where the coefficients may be listed as

P1 = N lnZ(β) − N (β coth β − 1),

P2 =N2

2

(lnZ(β) + 2 + β coth β

)lnZ(β) + Nε − N

2β2 cosech2β +

N2

2β2 coth2 β

+ N2 β2 cosech2β − N ε β2 cosech2β − N(N − 1) β3 coth β cosech2β,

P3 = +N3 β coth β − N3

2(1 + coth2 β) +

N

6(8 − 27N + 24N2)β3 coth β

− N

6

((8 − 27N + 25N2) − (170 − 388N + 313N2 − 5N3(22 + 3N))β2

)β3 coth3 β

+N

12(17 − 33N + 12N2) β4 − N

6(34 − 75N + 39N2) β4 coth2 β

+N

4(17 − 39N + 22N2) β4 coth4 β − N

12(136 − 294N + 207N2 − 45N3)β5 coth β

+N3

6

(lnZ(β) + 3 − 3β coth β

) (lnZ(β)

)2+

N2

2

((1 − 2N) − 2(1 − N)β coth β

− (1 − 3N) coth2 β + (1 − 2N)β coth3 β)β2 lnZ(β)

− N(1 − 2N) ε − 2N2 ε β coth β + N(1 − N) ε β

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+ N(1 − 2N − 3N ε − (4 − 6N − Nε) β coth β

+ 3N(1 − N) β2 coth2 β)

ε β2 cosech2β − N(1 − N) ε β cosech2β

+ N2(1 − β2 cosech β

)ε lnZ(β). (2.14)

Continuing the above perturbative procedure we recast the definition (1.3) into thefollowing integral form of the internal energy in the third constraint picture:

Uq(β) = −μH(Zq(β)

)−1 (expq (ε)

)q

∫ π

θi=0

∫ 2π

φi=0

(N∑

�=1

cos θ�

)[

1 + (1 − q)β

N∑

m=1

cos θm

]q/(1−q)

+

N∏

i=1

sin θi dθidφi.

(2.15)

Implementing the above phase space integration we obtain the internal energy as

Uq(β) =

(2π

β

)N

Φ μH(Nβ−1 S1(β) − S′

1(β)) (expq (ε)

)q

Zq

. (2.16)

On substituting the generalized partition function (2.4) the internal energy (2.16) assumesthe form

Uq(β) =μH

1 + (1 − q) ε

(N

β− S′

1(β)

S1(β)

)

(2.17)

that readily yields the perturbative series expansion:

Uq(β) = μH

(N

β− N coth β + (1 − q) U1 + (1 − q)2 U2 + · · ·

)

, (2.18)

where the coefficients up to the order of (1 − q)2 are

U1 = −N(N − 1) β cosech2β − N2 β coth β L(β) + N(N − 1) β2 coth β cosech2β

− Nβ ε + Nβ ε(coth2 β − β−2

),

U2 = N2 cosech2β + N(2 − 7N + 2N2) β2 coth β cosech2β

− N β3 ((1 − N) − (4 − 6N) coth β − 13(2 − 6N + N2) cosech2β

− 2(1 − 3N − 3N2) coth2 β cosech2β − (3 − 5N) coth4 β)

− Nβ4((2 − 3N + N2) coth β cosech2β − (3−5N+2N2) coth3 β cosech2β

)

+ N ε β((1 − N + ε) − (1 − 2N + ε) coth2 β + N cosech2β

)

+ N ε β2((2 − 2N + ε) coth3 β + (2 − 4N) coth β cosech2β

)

+ N(1 − N) ε β3 cosech2β(1 − 3 coth2 β

). (2.19)

In the above expressions L(x) = coth(x) − x−1 denotes the classical Langevin function.Collecting the previous results we obtain a set of coupled implicit perturbative

expansions for the generalized partition function, the sum of q weights and the internalenergy in (2.7), (2.13) and (2.18), respectively. Employing a recursive procedure thesequantities may be obtained explicitly as perturbative series in the parameter (1 − q) up

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to any arbitrary order. The feature that facilitates the solution of these quantities is thatthese equations are uncoupled and explicitly solvable in the lowest perturbative order.A recursive process now produces explicit solutions at an arbitrary nth order from thecorresponding quantities at lower orders. We now quote the explicit expression of thegeneralized partition function to the order (1 − q)2:

Zq = (2π)NΦ exp(−β L(β)

) Z(β)(1 + (1 − q) Z1(β) + (1 − q)2 Z2(β) + · · ·) , (2.20)

where β = βμH/N and the perturbative coefficients are given by

Z1(β) =3 N2

2+

N2

2β coth β +

N

2β

2coth β − N2β

2+ N(1 + N) β

3coth β cosech2β

N2(1 − β

2cosech2β

)lnZ(β),

Z2(β) =10N3

3+

9N4

8− 3N4

4β

2+

N4

4β

4 − N3

4

(10 − 3N + (2 + N)β

2)

β4

coth β

+N3

8β

2((

4 − 5N + (2 + N)β2)

coth2 β + 2(3 − 3N + (13 + N)β

2)

cosech2β)

− N3

8

(4 + 2N − N β

)β

3coth3 β +

N

24(34 − 21N + 54N2 + 3N3) β

4cosech4β

− N

4

((2 − 5N − 7N2)N − 2(4 + 4N + 5N2 + N3) β

2)

coth β cosech2β

− N

4(17 − 27N + 17N2 + N3) β

4coth2 β cosech2β

− N2

2(1 − 4N − N2)β

5coth β cosech4β − N

2(6 − N2 + N3) β

5coth3 β cosech2β

+N2

2(1 + 4N + N2) β

6coth2 β cosech4β

+ N3(

14(4 + 3N + 2β

2) − 1

2(3 − N + N β

2) lnZ(β) − 1

6(lnZ(β))2

)lnZ(β)

− N3

2

(3 − N − lnZ(β)

)β coth β lnZ(β) +

N3

2(1 − N) β

2coth2 β lnZ(β)

− N2

2

(8 + 6N − N2

)β

3coth β cosech2β lnZ(β)

− N2

2

(3 − 6N + 2N2 − N2 β

2+ 2(1 − N) lnZ(β)

)β

2cosech2β lnZ(β)

− N2

2(6 − 2N + N2) β

4coth2 β cosech2β lnZ(β)

− N3

2

(1 − N + N lnZ(β)

)β

4cosech4β lnZ(β)

− N3(1 − N) β5

coth β cosech4β lnZ(β). (2.21)

Proceeding similarly the explicit expression corresponding to the sum of the q weightsc(β) is obtained up to the order (1 − q)3. As we have already remarked in the context ofequation (2.13) our proposal of employing the exponential relation (1.5) for confirming theconsistency of our evaluation process requires obtaining the sum c(β) up to a perturbative

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order higher than the expansions for the partition function and the internal energy. Wequote the result up to O((1 − q)3):

c(β) = N(1 + (1 − q) P1(β) + (1 − q)2 P2(β) + (1 − q)3 P3(β) + · · ·) , (2.22)

where the perturbative coefficients may be listed as

P1(β) = N lnZ(β) − N (β coth β − 1),

P2(β) =N2

2

(lnZ(β) + 4 − 2 β

2cosech2β − 2 β coth β

)lnZ(β)

+N2

2

(4 − 4β coth β − β

2)

+ N(1 + N)β3

coth β cosech2β

+N

2(1 + N) β

2cosech2β,

P3(β) = 5N3 − N3

2β (β + 8 cothβ) +

N2

2(1 − 2N − 6Nβ

2) β

2cosech2β

+N

12(17−12N + 24N2) β

4cosech4β − 2N(1+N + N2) β

5coth β cosech2β

− N(3 − N2) β5coth3 β cosech2β − N2(1 + 3N) β

5coth β cosech4β

− N3

6β

3coth3 β−N

4(17−23N − 12N2) β

4coth2 β cosech2β + 4N3 lnZ(β)

− N2 (1 − 2N) β2cosech2β lnZ(β) − 3N3 β coth β lnZ(β)

− N2 (3 + 2N) β3

cothβ cosech2β lnZ(β) +N3

2β

2coth2 β lnZ(β)

− N2(3 − N) β4

coth2 β cosech2β lnZ(β). (2.23)

Similarly the above recursive scheme provides us with an explicit perturbative expansionof the internal energy that allows us to obtain the magnetization. To the desired orderthe expression is

Uq(β) = μH

(N

β− N cothβ + (1 − q) U1(β) + (1 − q)2 U2(β) + · · ·

)

, (2.24)

where the perturbative coefficients are given by

U1(β) = −N2 (2β2 − 1) L(β) + N2 β lnZ(β) L′(β),

U2(β) = −2N3 L(β) − 4N3 β +N2

2β

3+ N β

3+ (2N + 5N2) β

3coth2 β cosech2β

− N

2

(4 + 10N − (4 + 9N2) β

2)

β2

coth β cosech2β

− 2N(6 + N + N2) β4coth3 β cosech2β +

N

2(9N2 + 4Nβ − 8β

2)β coth2 β

+3N

2(2 + N) β

3coth4 β − N2

6

(9 − 36N − (4 − 6N) β

2)

β cosech2β

− N β4

coth β cosech4β. (2.25)

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The internal energy and the magnetization are related via the standard thermodynamicexpression:

U = −MH (2.26)

that enables us to extract the magnetization from the already known internal energy. Theperturbation series corresponding to magnetization follows from (2.24) and (2.26):

Mq(β) = μ(N L(β) − (1 − q) U1(β) − (1 − q)2 U2(β) + · · ·) , (2.27)

where the relevant coefficients U1(β) and U2(β) are listed in (2.25). Magnetic susceptibilityis computed from the definition

χq(β) ≡ ∂Mq

∂H, (2.28)

and the series expansion (2.27) of the magnetic moment:

χq(β) =μ2 β

N

((N

β

)2

− N cosech2β + (1 − q) ϑ1(β) + (1 − q)2 ϑ2(β) + · · ·)

. (2.29)

The above perturbative coefficients up to the order O((1 − q)2) are given below:

ϑ1(β) =N2

β2 lnZ(β) + N

(1 + N + N lnZ(β)

)cosech2β +

2N2

β2 − 2N2 cosech2β

+ N(1 + N)β2cosech4β + 2N(1 + N)β

2coth2 β cosech2β − N2

βcoth β

− N β(4N2 + lnZ(β)

)coth β cosech2β,

ϑ2(β) =4N3

β2 +

3N3

βcoth β +

N2

2

(N coth2 β − cosech2β

)+ N(4 + N + 17N2)β coth β

+N

2

((6 − 5N + 30N2) cosech2β − (18 + 47N2) coth2 β

)β

2cosech2β

+ 4N2(N − 1 + (1 + 2N) coth2 β

)β

3coth β cosech2β − N2(1 − N)β

4cosech2β

+ N3(7 + 22N) β coth β cosech2β − N(2 + 3N + 3N2) β4coth4 β cosech2β

− N((12 + 25N − 19N2) cothβ + N(3 − 13N)

)β

2coth β cosech2β

− N

3

(40 − 39N − 48N2 − (32 − 54N + 15N2) coth2 β

)β

3coth β cosech2β

+ N(2N(1 + N) + (13 − 12N − 7N2) coth2 β) cosech4β

+3N3

β2 lnZ(β) − N3

βcoth β lnZ(β)

− N2(1 − N) cosech2β lnZ(β) − N2(10 − N) β coth β cosech2β lnZ(β)

+ N2(5 + 2(1 − 3N) cosech2β − (19 + 10N) coth2 β

)β

2cosech2β lnZ(β)

+ 2N2((3 + 7N) coth2 β − 5N − 1 + (3−N) cosech2β

)β

3coth βcosech2β lnZ(β)

+N3

2β2

(lnZ(β)

)2 − N3

2cosech2β

(lnZ(β)

)2+ 3N3β cothβcosech2β

(lnZ(β)

)2

+ N3(1 − 3 coth2 β

)cosech2β

(lnZ(β)

)2. (2.30)

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0.5

1

1.5

2

2.5

3

3.5

4

0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 1.02

Nef

f / N

q

"N=5""N=10"

"N=15"

"N=25"

Figure 1. Dependence of the ratio Neff/N on q for various values of N .

The specific heat in the third constraint framework Cq(β) ≡ (∂Uq(β))/(∂T ) is obtainedvia the corresponding internal energy (2.24):

Cq(β) = k(βμH)2

N

((N

β

)2

− Ncosech2β + (1 − q) ϑ1(β) + (1 − q)2 ϑ2(β) + · · ·)

. (2.31)

The perturbative coefficients ϑ1(β) and ϑ2(β) appearing in the above expression may beread from (2.30).

The thermodynamic quantities in the weak field limit βμH � 1 follow directly fromequations (2.24) and (2.31):

Uq(β) = −Neff

3βμ2H2, Cq =

Neff

3k (βμH)2, (2.32)

where the q-dependent effective number of spins Neff is

Neff =N Γ

N⇒ Neff

N=

Γ

N. (2.33)

The perturbative series Γ in (2.33) up to the order (1 − q)2 is given below:

Γ = 1 − (1 − q) (1 + N ln 2) − (1 − q)2 N

2

(1 + N − 2 ln 2 − N (ln 2)2

)+ · · · . (2.34)

Substituting the value (2.12) of the q-dependent scale factor N in (2.33), we notice thatup to the perturbative order (1 − q)2 it follows that Neff > N (Neff < N) for the regionq > 1 (q < 1). This is evident from figure 1.

Turning to the magnetization (2.27) in a weak-field βμH � 1 regime, we obtain

Mq =Neffμ2H

3kT, (2.35)

where the corresponding susceptibility may be viewed as a nonextensive generalization ofCurie’s law:

χq =Ceff

T, Ceff =

Neffμ2

3k. (2.36)

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The thermodynamic quantities evaluated above ((2.32)–(2.36)) show a nonlineardependence on the number of spins N , and the entropic deformation parameter (1 − q).This effect of nonextensivity, manifest in the aforesaid inequality between Neff and N ,leads to the phenomenon of dark magnetism discussed in [11, 12]. On the other handthe authors of [14] do not observe dark magnetism as a consequence of their choice of the

inverse of the quantity β defined in (2.3) as the ‘physical temperature’. We, however, thinkthat association of the Lagrange multiplier for the constraint on internal energy with theinverse temperature has the merit that it allows [6] identification of Tsallis q-entropy (1.1)with thermodynamic Clausius entropy. To settle the issue it is imperative to explorepossibilities where alternative choices of the temperature lead to distinct experimentalsituations. In this context it would be extremely interesting to search for experimentalvalidation of the phenomenon of dark magnetism, say, for nanoclusters that may bepotential candidates for nonextensive statistics.

The exponent property (1.5) in conjunction with our perturbative evaluation (2.22)of the sum of the q weights, now allows us to employ the formulation specified in [5]as a consistency check on our results. Examining the generalized partition function adifferential equation involving the internal energy has been established in [5]:

β∂Uq

∂β=

∂

∂βlnq Zq(β). (2.37)

Now the above relation (2.37), the exponent property (1.5) and the definition (1.2) of theq logarithm produce a powerful tool of verification for the consistency of our perturbativeexpansion procedure:

β∂Uq

∂β=

∂

∂β

c(β) − 1

1 − q. (2.38)

Substituting the internal energy (2.24) and the sum of the q weights given in (2.22),it may be explicitly verified that the differential equation (2.38) holds order by orderin our perturbation theory. This is a nontrivial consistency check on our results forphysical quantities. It has been noted earlier [22] that, as a consequence of the exponentrelation (1.5), evaluation of the thermodynamic quantities such as specific heat, say up tothe second order in (1 − q), necessitates computing the sum of the q weights c(β) till thethird order. That counting has been maintained in (2.23).

Furthermore, the differential equation (2.37) may be used for a direct extraction ofthe susceptibility in our model. Towards this, we recast the magnetization (2.27) as

Mq = μ f(β), (2.39)

where the perturbative expansion of the function f(β) is readily obtained bycomparing (2.27) with (2.39). The relations (2.26), (2.38) and (2.39) now produce themagnetic susceptibility:

χq = − μ

N H

∂

∂β

(c(β) − 1

1 − q

)

. (2.40)

The derivation following from (2.40) and the series (2.22) precisely agrees with themagnetic susceptibility obtained earlier in (2.29). This is an additional confirmation onthe consistency of our perturbative procedure.

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3. Mean-field model: temperature-dependent effective field coefficient

Physical reasoning tells us that effects of nonextensivity are likely to be pronounced forsystems embodying long-range interactions between the constituents. For an interactingspin system a good first approximation is provided by the mean-field model where wereplace the interactions between the spins with an effective interaction between the spinsand a homogeneous magnetic field (Hm) that is considered to be directly proportional tothe magnetization per spin. The Hamiltonian of the system, as envisaged in the mean-fieldmodel, is

E = −μ(H + Hm)

N∑

i=1

cos θi, (3.1)

where Hm is the dynamical field that is assumed to result from a process of ‘averaging’of the interactions between the spins. The generalized partition function in the thirdconstraint is

Zq (β) = expq (ε)

∫ π

θi=0

∫ 2π

φi=0

[

1 + (1 − q)β

N∑

�=1

cos θ�

]1/(1−q)

+

N∏

i=1

sin θi dθi dφi, (3.2)

where the scaled dimensionless variable β is given by

β =βμ(H + Hm)

c(β) + (1 − q) β Uq, (3.3)

and following (2.2) the quantity ε has been defined earlier. In the regime |1 − q| � 1 weassume that the integrand is real and positive in the entire phase space. Applying theintegral (A.4) the generalized partition function (3.2) may be expressed as follows:

Zq (β) = (2π)N Φ expq (ε) β−N S1(β). (3.4)

The phase space integral corresponding to the sum of the q weights may be read offvia (1.3) and (1.4):

c(β) =

(expq (ε)

Zq(β)

)q ∫ π

θi=0

∫ 2π

φi=0

[

1 + (1 − q)β

N∑

�=1

cos θ�

]q/(1−q)

+

N∏

i=1

sin θi dθi dφi. (3.5)

Employing (A.4) we perform the above integral and therein substitute the generalizedpartition function (3.4) to express the above sum of the q weights as follows:

c(β) = N Λ β(q−1)N S2(β) (S1(β))−q. (3.6)

Based on the infinite product decomposition (2.6) of the q exponentials, as done before insection 2, we perturbatively expand the rhs of the expression (3.6) up to second order inthe deformation parameter (1− q). Moreover, as we are interested in the phase transitionof the nonextensive system described by the extended mean-field Hamiltonian, we retain

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terms up to fourth order in the variable βμ(H + Hm):

c(β) = N

(

1 + (1 − q)N ln 2 + (1 − q)2 N

2

(1 + N + N(ln 2)2

)

+ ℘1(ε)

(βμ(H + Hm)

c(β)

)2

+ ℘2(ε)

(βμ(H + Hm)

c(β)

)4

+ · · ·)

, (3.7)

where the perturbative expansion of the coefficients of monomials of the field variable is

℘1(ε) = −N

6

((1 − q) + (1 − q)2 (1 − N ln 2 + 2 ε) + · · ·) ,

℘2(ε) =N

60

(

(1 − q) − (1 − q)2

(

6 − 15N

2− N ln 2 + 4 ε

)

+ · · ·)

.(3.8)

The defining property (1.3) leads to the following integral form of the internal energyin the third constraint picture:

Uq(β) = −μ(H + Hm)(Zq(β)

)−1 (expq (ε)

)q

×∫ π

θi=0

∫ 2π

φi=0

(N∑

�=1

cos θ�

) [

1 + (1 − q)βN∑

m=1

cos θm

]q/(1−q)

+

N∏

i=1

sin θi dθi dφi.

(3.9)

Implementing the above phase space integration via (A.4) we obtain the internal energyas

Uq(β) =

(2π

β

)N

Φ μ(H + Hm)(Nβ−1 S1(β) − S′

1(β)) (expq (ε)

)q

Zq

. (3.10)

On substituting the generalized partition function (3.4) the implicit equation (3.10) maybe recast as follows:

Uq(β) =μ(H + Hm)

1 + (1 − q) ε

(N

β− S′

1(β)

S1(β)

)

. (3.11)

We expand the term in the parenthesis up to second order in the deformation parameter(1− q) and fourth order in the dynamical variable βμ(H + Hm). The resulting expansionassumes the form

Uq(β) = −N βμ2(H + Hm)2

3 c(β)

(

U1(ε) − U2(ε)

15

(βμ(H + Hm)

c(β)

)2

+ · · ·)

, (3.12)

where the perturbative coefficients are given by

U1(ε) = 1 − (1 − q) (1 + 2ε) + (1 − q)2 ε (2 + 3 ε) + · · · ,U2(ε) = 1 − (1 − q) (6 − 10N + 4 ε)

+ (1 − q)2(11 − 25N + 24ε − 40ε + 10Nε2

)+ · · · . (3.13)

Towards obtaining the magnetization we first explicitly obtain the quantities c(β)and Uq(β) by solving the pair of simultaneous implicit equations (3.7) and (3.12). We cansystematically obtain their solutions in an order-by-order perturbation theory where we

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retain terms up to (1− q)2 in the deformation parameter and (βμ(H + Hm))4 in the fieldvariable. Our method of recursive solution of said implicit equations has already beendescribed in section 2. The explicit expression for the sum of q weights is

c(β) = N(1 + (1 − q)N ln 2 + (1 − q)2 N

2

(1 + N + N(ln 2)2

)

+ Π1

(βμ(H + Hm)

N

)2

+ Π2

(βμ(H + Hm)

N

)4

+ · · ·), (3.14)

where the coefficients may be listed as follows:

Π1 = −N

6

((1 − q) + (1 − q)2 (1 − N ln 2) + · · ·) ,

Π2 =N

60

(

(1 − q) − (1 − q)2

(

6 +5N

2+ 3N ln 2

)

+ · · ·)

.(3.15)

The relevant expression for the internal energy assumes the form

Uq(β) = −Nβμ2(H + Hm)2

3 NΓ +

Nβ3μ4(H + Hm)4

45 N3Ξ + · · · , (3.16)

where the q-dependent numerical factor Γ has been given in (2.34) and Ξ is

Ξ = 1 − (1 − q)

(

6 +5N

2+ 3N ln 2

)

+ (1 − q)2

(

11 − 3N

2(N + 1) +

N ln 2

2(36 + 15N + 9N ln 2)

)

+ · · · .(3.17)

As the internal energy and the magnetization are related via the standard thermodynamicexpression (2.26) we may now readily obtain a perturbative expansion for themagnetization by employing the corresponding series (3.16) for the internal energy:

Mq = μ

(N

3 Nβμ(H + Hm) Γ − N

45 N3(βμ(H + Hm))3 Ξ

)

. (3.18)

Usually in the context of mean-field theory the exchange interaction on a magneticion that manifests as the homogeneous magnetic field Hm is considered to be proportionalto the magnetic moment per spin where the proportionality factor, known as the effectivefield coefficient (λ), is regarded as independent of temperature. For certain weaklyferromagnetic materials, however, the exchange interaction between electrons in differentions depends on temperature [23]–[24]. Spin fluctuations of these ions show a temperaturedependence, introducing a new mechanism for Curie–Weiss susceptibility. Above thecritical temperature the spin fluctuations have mainly long wave components [26], andthey generate a linear variation of the field coefficient with temperature [23, 24]. Below theCurie temperature the dominant contribution of spin fluctuations in weakly ferromagneticmatter comprises short wave components having small spatial range [24]. It is also knownthat, below the Curie temperature, spin fluctuations decrease nonlinearly with increasingtemperature [26], which results in an inverse square temperature-dependent term [24] inthe coefficient of the exchange field. Following [24] we here study a mean-field spin modelwith the field coefficient λ(T ) having a linear as well as an inverse square dependence on

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temperature. To wit, the exchange interaction between the magnetic ions introduces aneffective field given by

Hm = λ(T ) mq, mq ≡ Mq

N, λ(T ) = ξ + ζ T + η T−2, (3.19)

where the coefficients ζ and η characterize the temperature-dependent spin fluctuations.For the vanishing limit of the external field H = 0 we obtain the magnetization bysubstituting the effective field (3.19) in the rhs of the expression (3.18):

mq

(3 Ξ

5

(Θ λ(T )

Γ

)3 m2q

μ2+ (1 − ζΘ) g(T )

)

= 0, (3.20)

where the polynomial g(T ) is defined as follows:

g(T ) ≡ T 3 − ξ Θ

1 − ζ ΘT 2 − η Θ

1 − ζ Θ, Θ =

μ2 Γ

3k N. (3.21)

For typical values of the parameters (ξ, ζ, η) discussed in [24] the cubic polynomial maybe factorized:

g(T ) ≡ (T − Tc) (T − T) (T − T). (3.22)

The equation g(T ) = 0, therefore, has a real solution that is identified with the criticaltemperature Tc, and two other solutions (T, T) complex conjugates to each other exist:

Tc =1

3

ξ Θ

1 − ζ Θ

(1 + (Υ+)1/3 + (Υ−)1/3

),

T =1

6

ξ Θ

1 − ζ Θ

(2 − (Υ+)1/3 − (Υ−)1/3 + i

√3((Υ+)1/3 − (Υ−)1/3

)),

T =1

6

ξ Θ

1 − ζ Θ

(2 − (Υ+)1/3 − (Υ−)1/3 − i

√3((Υ+)1/3 − (Υ−)1/3

)),

(3.23)

where the conjugate radicals Υ± are

Υ± = 1 +27

2

η

ξ3

(1 − ζ Θ

Θ

)2⎛

⎝1 ±√

1 +4

27

ξ3

η

(Θ

1 − ζ Θ

)2⎞

⎠ . (3.24)

It is evident from (3.20) that below the critical temperature T < Tc the magnetization inthe null external field limit is given by two stable real values:

mq|H=0 = ±μ

√5 Γ3 (1 − ζ Θ) Tc

3 Ξ Θ3 (λ(T ))3F(T )

(

1 − T

Tc

) 12

, (3.25)

where the quadratic factor F(T ) may be listed as follows:

F(T ) = (T − T)(T − T) = T 2 − 2 (Re T) T + |T|2. (3.26)

This corresponds to the ferromagnetic transition as observed in the generalized mean-fieldapproach in the nonextensive framework. In the regime T → T−

c the susceptibility in the

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null external field limit may be obtained via (2.28), (3.18) and (3.25):

χq =C/2

Tc − T, C =

N Θ T 2c

(1 − ζ Θ) F(Tc). (3.27)

Above the critical temperature T > Tc, the only real solution for the magnetization in thenull external field condition is given by

mq|H=0 = 0 (3.28)

that corresponds to the paramagnetic phase. Using the standard definition (2.28) inconjunction with the relations (3.18) and (3.28) we now obtain the magnetic susceptibilityin the paramagnetic regime:

χq =C

T − Tc. (3.29)

As is characteristic of the mean-field approach, the divergence of the susceptibilityconsidered above in (3.27) and (3.29) follows the critical exponent law:

χq ∼ 1

|T − Tc| , (3.30)

where the critical temperature Tc in our extended mean-field model depends on theentropic deformation parameter (1 − q). This result is qualitatively similar to theobservation in [19] where these authors have adopted a distinct definition of thetemperature for the nonextensive spin system. The mean-field model studied by theseauthors also differs from the one considered here in that we assume the dynamical fieldstrength to be temperature-dependent in order to accommodate the quantum fluctuationsin spin configurations. It is interesting to note that the critical behaviors for themagnetization and the susceptibility, given in (3.27) and (3.30), respectively, suggest thatunder the mean-field model settings the critical indices do not change with the introductionof the Tsallis q statistics.

It is worth considering the limiting case η = 0 in (3.19), where the effective fieldcoefficient has only a linear dependence on temperature: λ(T ) = ξ + ζ T . Suchtemperature dependence of the dynamical field coefficient has been observed [23], forinstance, in the case of the measurement of magnetic susceptibility of CeF3 crystals.Taking the η = 0 limit in (3.25) we obtain the magnetization in the null external fieldcondition as follows:

mq|H=0 = ± μ

√5 Γ3

3 Ξ(1 − ζ Θ)−1

(

1 +ζ

ξT

)−3/2T

Tc

√

1 − T

Tc

, (3.31)

where the critical temperature in the said limit may be read from (3.23):

Tc =ξ Θ

1 − ζ Θ. (3.32)

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In the regime T → T−c the susceptibility in the null external field limit may be obtained

via (3.18) and (3.31):

χq =C/2

Tc − Tfor T → T−

c

=C

T − Tcfor T → T+

c , (3.33)

where C = N Θ/(1 − ζ Θ). We also quote the results obtained in the usual mean-fieldmodel, where the dynamical proportionality factor λ is thought to be independent oftemperature by maintaining the coefficients ζ = 0, η = 0 in (3.19). In this limit thecritical temperature and the Curie–Weiss constant in the nonextensive scenario may beread from (3.32) and (3.33), respectively:

Tc = ξ Θ, C = N Θ. (3.34)

For the choice of temperature-independent field coefficient ζ = 0, η = 0 the ratio of thecritical temperature of the spin system subject to Tsallis q statistics with respect to thatgoverned by Boltzmann–Gibbs statistics assumes the form

Tc

Tc|q=1=

Γ

N, (3.35)

where we have substituted the parameter Θ given in (3.21). The known expression of therhs in (3.35) via (2.34) and (2.12) allows us to infer that the critical temperature increases(decreases) compared to its standard Boltzmann–Gibbs value in the regime q > 1 (q < 1).As remarked before, this result qualitatively agrees with that obtained in [19]. It isinteresting to note that the ratio of the effective number of spins with the actual numberof spins present in a background field obtained in the context of equation (2.33) coincideswith the ratio of the critical temperatures given in (3.35).

4. Remarks

Our main focus in the present work has been to study a system of spins with long-rangeinteractions approximated by a mean-field model governed by the nonextensive Tsallisstatistics. Following the arguments presented in [6] we identify the temperature withthe inverse of the Lagrange multiplier pertinent to the constraint on internal energy.Considering the first law of thermodynamics this choice implies that the Tsallis q-entropycoincides with the thermodynamic Clausius entropy. To incorporate the quantum spinfluctuations the mean-field model investigated here is assumed to have a temperature-dependent dynamical field strength coefficient. To accommodate the spin fluctuationsthe effective field strength coefficient includes [24] a linear and an inverse square termon temperature. The nonextensivity is implemented by using a perturbative technique,where the implicit simultaneous equations involving the internal energy and the sumof q weights were solved explicitly as series expansions up to the order of (1 − q)2

in the entropic deformation parameter. The perturbation method developed here maybe continued to an arbitrary order in the parameter (1 − q). As our method is ofperturbative origin, it is, of course, obvious that the results obtained here are valid inthe regime |1 − q| � 1. The signature of the nonextensivity is evident as the critical

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temperature is found to depend on the number of spins N and the deformation parameter(1 − q). Compared to its standard Boltzmann–Gibbs value the critical temperatureincreases (decreases) for the domain q > 1 (q < 1). The extended Curie–Weiss lawcharacterizing the susceptibility in the regions above and below the critical temperaturehas been generalized to the nonextensive case. Analogous to the critical temperature theCurie–Weiss constant also embodies the effects of nonextensivity. Another interestingfeature reflecting nonextensivity emerges for noninteracting spins in the presence of abackground field, where we observe the presence of dark magnetism. This supports theresults [11, 12] obtained earlier. Parallel to the observation in [12] our analysis indicatesthat the effective number of spins Neff > N (Neff < N) for the regime q > 1 (q < 1).Unlike these authors, we, however, do not adopt a high temperature limit.

The mean-field technique used in our work may be fruitfully employed to investigatethe effects of Tsallis q statistics on the magnetic properties of systems which formclusters [27]. A cluster is typically a macroscopic region consisting of a large numberof spins whose interactions are described via locally homogeneous mean fields in thedomain of each cluster. Using clusters of unrestricted sizes in the context of a generalizedensemble it has been found [27] that the fraction of clusters with a specific amount oforder diverges at Tc. In this context the phenomenon of anomalous clustering, whichrequires [27] the energy absorbed in a selected degree of freedom to remain localized dueto the slow response of the system, is of particular importance as these long-living localizedstates may be looked on as nonequilibrium stationary states in the sense described in [6].From the point of view of the mean-field model subject to Tsallis q statistics discussedhere we wish to visit the issue of anomalous clustering elsewhere.

Acknowledgment

The work of R Chandrashekar is supported by a fellowship offered by the Council ofScientific and Industrial Research (India).

Appendix

Following a simple transformation the integral (2.2) may be reduced to the following form:

I =

∫ +1

ti=−1

[

1 + (1 − q) cN∑

�=1

t�

]1/(1−q) N∏

i=1

dti. (A.1)

For the perturbative regime of the deformation parameter |1 − q| � 1 the integrand isassumed to be real and positive for the entire phase space. The above N -fold integrationis done by successive steps. Integration on the variable t

Nyields

I =1

c (2 − q)

∫ +1

ti=−1

⎛

⎝

[

1 + (1 − q) c + (1 − q) c

N−1∑

�=1

t�

](2−q)/(1−q)

−[

1 − (1 − q) c + (1 − q) c

N−1∑

�=1

t�

](2−q)/(1−q)⎞

⎠N−1∏

i=1

dti. (A.2)

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Continuing this procedure m times we obtain the following sum:

I =1

cm (2 − q) (3 − 2q) . . . ((m + 1) − mq)

×∫ +1

ti=−1

((m

0

)[

1 + (1 − q) mc + (1 − q) c

N−m∑

�=1

t�

](1+(1−q)m)/(1−q)

−(m

1

)[

1 + (1 − q) (m − 2)c + (1 − q) c

N−m∑

�=1

t�

](1+(1−q)m)/(1−q)

. . .

+

(m

m − 1

)[

1 − (1 − q) (m − 2)c + (1 − q) c

N−m∑

�=1

t�

](1+(1−q)m)/(1−q)

+(m

m

)[

1 − (1 − q) mc + (1 − q) c

N−m∑

�=1

t�

](1+(1−q)m)/(1−q) ) N−m∏

i=1

dti.

(A.3)

Completing N integrals on the variables {ti|i = 1, . . . , N} we produce the final result:

I =1

cNΦ

N∑

n=0

(N

n

)

(−1)n [1 + (1 − q) (N − 2n)c](1+(1−q)N )/(1−q) . (A.4)

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doi:10.1088/1742-5468/2010/09/P09007 20