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Generalization of the single-site Callen identity within Tsallis statistics
PHY$1CA ELSEVIER
Physica A 218 (1995) 482-486
Generalization of the single-site Callen identity within Tsallis statistics Enaldo F. Sarmento Universidade Federal de Alagoas, Departamento de Fisica, Cidade Universitdtria, 57.072-340 Maceir. AL, Brazil Received 10 March 1995
Abstract
Within the framework of a generalized thermostatistics introduced by Tsallis, we establish the generalization of the single-site Callen identity. It is used to calculate the critical temperature of the Ising ferromagnet within a consistently generalized mean field approximation.
On a multifractal basis, a generalized entropy has been recently introduced with the aim of generalizing Statistical Mechanics [1] and Thermodynamics [2]. This entropy possesses the usual properties of positivity, equiprobability, concavity, irreversibility and generalizes the standard additivity as well as the Shannon additivity [1-3]. It reads [1] s, =
1 -
p7
/1)
i=l
where k is a positive constant whose value depends on the particular units to be used, q e IR characterizes the particular statistics, Pi is the probability of occurrence of the ith microscopic configuration of the system, and w is the total number of such configurations. This entropy is related with the Rrnyi entropy [4] of information theory given by "SqR- 1 -qk (lni=l ~ PT)
(2,
and, in the limit q ~ i, $1 = S~ = - k ~ i =wl p i l n p i . For a discussion of some important properties of the above entropies, we address the reader to Refs. [1,2, 5, 6]. Tsallis entropy has been the subject of much recent work [7-17], and a direct application of the formalism has been reported [181, in the field of astrophysics. Indeed, this new entropy has overcome the inability of Boltzmann-Gibbs statistics to 0378-4371/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 3 7 1 ( 9 5 ) 0 0 1 2 6 - 3
E.F. Sarmento / Physica A 218 (1995) 482 486
483
yield a finite mass for the polytropic model of stellar dynamics. Also, a direct connection between the Tsallis index q and the relevant fractal dimension has been recently established for Levy flights [19]. Finally, the time-dependent solutions of the Vlasov equations associated with some galaxy models have been shown to correspond to q = - 1 [203. Generally speaking, the relevance of this generalization for nonextensive physical systems has been discussed in [21]. The basic point is that, if we have a d-dimensional system with long-range interactions (characterized by a potential oc 1/r ~ with d/ct > 1), standard (extensive) thermodynamical quantities (entropy, internal and free energies per particle) diver qe. The index q(d, ~) has been proposed to take into account the non-extensive scalings that appear: the general function q(d, ct) is still unknown, but we do know some particular values: (i) d/e ~< 1 of course yields q = 1; (ii) q(3, 1) = - 1 [20]. In order to address the connection with thermodynamics we follow along the lines w of [1,2], i.e., we extremize Sq with the conditions y,~= 1 Pi = 1 and
uq =
Pi ei
(3)
i=l
where {el } are the associated energies, and Uq is the generalized internal energy. The entropy definition (1) leads to the Tsallis distribution law [1, 2] Pi =
[1 - fl(1
-
q)l;i] 1/(1 -q)
Zq
(4)
with
Zq = ~ [1 - fl(1 - q)el] 1/~1-q~
(5)
i=1
In the present work we wish to generalize the single-site Callen identity [22]. The Hamiltonian for the Ising model is given by 1
H = - ~ ~,.. JijPiPj
(6)
l,J
where the sum runs over N identical spins,/~i is the dynamical variable which can take two values _ 1, and Jij is the exchange interaction between sites i a n d j (Ji~ could be of the long-range type, i.e., Ji~ oc 1/r~). The standard expectation value (i.e., q = 1) of the spin variable at the lattice site is given by Tr #i e - aH =
Z
(7)
where Z = Tr e - an and Tr means the sum over all allowed states of the system. Here fl = 1/kBT, where kn is the Boltzmann constant, and T is the absolute temperature. By
484
E.F. Sarmento / Physica A 218 (1995) 482-486
using Eqs. (6) and (7), it is easy to obtain the single-site Callen identity, namely (Pl) = (tanh(flEi))
(8)
with El ~- ~ j JijPj. Note that the standard molecular field approximation (MFA) is obtained from (8) by approximating the thermal average of the hyperbolic tangent by the hyperbolic tangent of the thermal average, i.e., (Pi)--_ t a n h [ f l ~ , dij(l@].
(9)
J
By following the Tsallis statistics, Eq. (7) is written as follows: Tr{#i[1 - fl(1 - q)H] q/(1-q)} (lai)q = {Tr[1 - fl(1 - q)H] TM-q)}q "
(10)
Let us here separate the Hamiltonian (6) into two parts, one (denoted by Hi) which includes all contributions associated with the site i, and the other (denoted by H') which does not depend on the site i. Then one has H = Hi + H'
(11)
where Hi = - p i E i . We know that for all lattice sites i and j the spin variables commute, i.e., [Pi, Pj] = 0, and therefore [Hi,
H'] = [Hi, H] = 0
(12)
which plays a fundamental role in the derivation. By writting the trace in the following form: (13)
Tr = l-I trtk) = (klJ i tr(k))tr") where tr(i) = y,+~_ 1, Eq. (10) may be rewritten as 1
(Pi)q = Z~q {Tr' tru)Pi[1 - fl(1 - q)(Hi + H')] q/(1 q~} .
(14)
Inserting in the bracket tqo[l -- fl(1 - q)(Hi + H')] q/° -q) =1 tr(i)[1 -- fl(1 - q)(Hi + H ' ) ] q/(1 -q)
(15)
we obtain 1
=
{Tr[1 -- fl(1 - q)(Hi + H')]q/(1
q)
tr(i)Pi[1 ~ fl(l ~ q)(nL +- U')]q/(1-q)~ x tr(i)[1 -- fl(1 -- q)(Hi + H')] q/(1-q) j
(16)
485
E.F. Sarmento / Pttw~ica A 218 (1995) 482 486 or
/tr(i)#xiE1 - fl(1 - q)(Hi + H')] q/(1 -q'\
(17)
By using that Pi = -I- 1, hence Hi = T- Ei, Eq. (17) reduces to
(#i)q
\1 + f,/q
(18)
with (19)
fq---- 1 - f l ( 1 - - q ) ( - - E i + H ) ]
which generalizes, for arbitrary q, Eq. (8). Depending on the particular interest, further processing of Eq. (18) could be needed in order to approximate the present expression by more tractable ones. There is, however, a simple calculation for which no other processing is needed, and which illustrates the kind of modifications that might be expected in the general case. We refer to the calculation of the critical temperature Tc for a z-coordinated spin-½ Ising ferromagnet with coupling constant J. In the T --+ T~ limit (and assuming a second order phase transition), ( ~ i ) q --~ 0 and ( H ' ) q ((,((~li)q) , i.e., ( H ' ) q becomes neglectable with respect to (#i)q. Consequently, Eq. (18) yields =
(~li) q
=
(20)
1 +fql T ~ o
with
] --(1 -fqlr~ r~ =
+ (1
q ) z J ( l i i ) q / k T c ] q/tl-q)
q)zJ(pi)q/kT~J
'
(21)
or (l~i)q = qzJ (#i),Jk T~
(22)
where we have used the fact that (/~i)q is independent of the particular site i. Finally, Eq. (22) implies
kTc J
-
qz
(231
which extends to arbitrary q the well known q = 1 result (kBTc/J = z). This result is perfectly consistent, within a mean field picture, with Tc(q) obtained, for the square lattice, within a recent renormalization group calculation [23]. In particular, it reproduces the linear dependence on q exhibited, for q >> 1, by the renormalization group calculation of T~.
486
E.F. Sarmento / Physica A 218 (1995) 482 486
References [1] C. Tsallis, J. Stat. Phys. 52 (1988) 479. [2] E.M.F. Curado and C. Tsallis, J. Phys. A 24 (1991) L69; Corrigenda: J. Phys. A 24 (1991) 3187; 25 (1992) 1019. [3] C.E. Shannon, Bell Sys. Tech. J. 27 (1948) 379. [4] A. R6nyi, Probability Theory (North-Holland, Amsterdam, 1970). I-5] A.M. Mariz, Phys. Lett. A 165 (1992) 409. I-6] J.D. Ramshaw, Phys. Lett. A 175 (1993) 169, 171. [7] R.F.S. Andrade, Physica A 175 (1991) 285. [8] A.R. Plastino and A. Plastino, Phys. Lett. A 177 (1993) 177. [9] A. Plastino and C. Tsallis, J. Phys. A 26 (1993) L893. [10] D.A. Stariolo, Phys. Lett. A 185 (1994) 262. [11] E.P. Silva, C. Tsallis and E.M.F. Curado, Physica A 199 (1993) 137; Errata: A 203 (1994) E 160. 1-12] A. Chame and E.V.L. Mello, J. Phys. A 27 (1994) 3663. 1-13] F. Buyukkilic and D. Demirhan, Phys. Lett. A 181 (1993) 24. [14] A.R. Plastino and A. Plastino, Physica A 202 (1993) 438. [15] R.F.S. Andrade, Physica A 203 (1994) 486. [16] C. Tsallis, Chaos, Solitons and Fractals, Vol. 6 ed. by G. Marshall (Pergamon Press, Oxford, 1995) p. 539. [17] P.T. Landsberg, Springer Series in Synergetics, Vol. 61 (Springer Verlag, Berlin) ed. by R.K. Mishra, D. Maab and E. Zwicrlein. [18] A.R. Plastino and A. Plastino, Phys. Lett. A 174 (1993) 384; J.J. Aly, in: N-Body Problems and Gravitational Dynamics, Proceedings of the Meeting held at Aussois-France (21-25 March 1993), eds. F. Combes and E. Athanassoula (Publications de l'Observatoire de Paris, 1993), p. 19. [19] P.S. Alemany and D.H. Zanett, Phys. Rev. B 49 (1994) 956. [20] A.R. Plastino and A. Plastino, Phys. Lett. A 193 (1994) 251. [21] C. Tsallis, in: New Trends in Magnetism Magnetic Materials and their Applications, eds. J.L. Mor/m-L6pez and J.M. Sanchez (Plenum, New York, 1994), p. 451; F.D. Nobre and C. Tsallis, Physica A 213 (1995) 337. [22] H.B. Callen, Phys. Lett. 4 (1963) 161. [23] S.A. Cannas and C. Tsallis, preprint (1994).
Physica A 218 (1995) 482-486
Generalization of the single-site Callen identity within Tsallis statistics Enaldo F. Sarmento Universidade Federal de Alagoas, Departamento de Fisica, Cidade Universitdtria, 57.072-340 Maceir. AL, Brazil Received 10 March 1995
Abstract
Within the framework of a generalized thermostatistics introduced by Tsallis, we establish the generalization of the single-site Callen identity. It is used to calculate the critical temperature of the Ising ferromagnet within a consistently generalized mean field approximation.
On a multifractal basis, a generalized entropy has been recently introduced with the aim of generalizing Statistical Mechanics [1] and Thermodynamics [2]. This entropy possesses the usual properties of positivity, equiprobability, concavity, irreversibility and generalizes the standard additivity as well as the Shannon additivity [1-3]. It reads [1] s, =
1 -
p7
/1)
i=l
where k is a positive constant whose value depends on the particular units to be used, q e IR characterizes the particular statistics, Pi is the probability of occurrence of the ith microscopic configuration of the system, and w is the total number of such configurations. This entropy is related with the Rrnyi entropy [4] of information theory given by "SqR- 1 -qk (lni=l ~ PT)
(2,
and, in the limit q ~ i, $1 = S~ = - k ~ i =wl p i l n p i . For a discussion of some important properties of the above entropies, we address the reader to Refs. [1,2, 5, 6]. Tsallis entropy has been the subject of much recent work [7-17], and a direct application of the formalism has been reported [181, in the field of astrophysics. Indeed, this new entropy has overcome the inability of Boltzmann-Gibbs statistics to 0378-4371/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 3 7 1 ( 9 5 ) 0 0 1 2 6 - 3
E.F. Sarmento / Physica A 218 (1995) 482 486
483
yield a finite mass for the polytropic model of stellar dynamics. Also, a direct connection between the Tsallis index q and the relevant fractal dimension has been recently established for Levy flights [19]. Finally, the time-dependent solutions of the Vlasov equations associated with some galaxy models have been shown to correspond to q = - 1 [203. Generally speaking, the relevance of this generalization for nonextensive physical systems has been discussed in [21]. The basic point is that, if we have a d-dimensional system with long-range interactions (characterized by a potential oc 1/r ~ with d/ct > 1), standard (extensive) thermodynamical quantities (entropy, internal and free energies per particle) diver qe. The index q(d, ~) has been proposed to take into account the non-extensive scalings that appear: the general function q(d, ct) is still unknown, but we do know some particular values: (i) d/e ~< 1 of course yields q = 1; (ii) q(3, 1) = - 1 [20]. In order to address the connection with thermodynamics we follow along the lines w of [1,2], i.e., we extremize Sq with the conditions y,~= 1 Pi = 1 and
uq =
Pi ei
(3)
i=l
where {el } are the associated energies, and Uq is the generalized internal energy. The entropy definition (1) leads to the Tsallis distribution law [1, 2] Pi =
[1 - fl(1
-
q)l;i] 1/(1 -q)
Zq
(4)
with
Zq = ~ [1 - fl(1 - q)el] 1/~1-q~
(5)
i=1
In the present work we wish to generalize the single-site Callen identity [22]. The Hamiltonian for the Ising model is given by 1
H = - ~ ~,.. JijPiPj
(6)
l,J
where the sum runs over N identical spins,/~i is the dynamical variable which can take two values _ 1, and Jij is the exchange interaction between sites i a n d j (Ji~ could be of the long-range type, i.e., Ji~ oc 1/r~). The standard expectation value (i.e., q = 1) of the spin variable at the lattice site is given by Tr #i e - aH =
Z
(7)
where Z = Tr e - an and Tr means the sum over all allowed states of the system. Here fl = 1/kBT, where kn is the Boltzmann constant, and T is the absolute temperature. By
484
E.F. Sarmento / Physica A 218 (1995) 482-486
using Eqs. (6) and (7), it is easy to obtain the single-site Callen identity, namely (Pl) = (tanh(flEi))
(8)
with El ~- ~ j JijPj. Note that the standard molecular field approximation (MFA) is obtained from (8) by approximating the thermal average of the hyperbolic tangent by the hyperbolic tangent of the thermal average, i.e., (Pi)--_ t a n h [ f l ~ , dij(l@].
(9)
J
By following the Tsallis statistics, Eq. (7) is written as follows: Tr{#i[1 - fl(1 - q)H] q/(1-q)} (lai)q = {Tr[1 - fl(1 - q)H] TM-q)}q "
(10)
Let us here separate the Hamiltonian (6) into two parts, one (denoted by Hi) which includes all contributions associated with the site i, and the other (denoted by H') which does not depend on the site i. Then one has H = Hi + H'
(11)
where Hi = - p i E i . We know that for all lattice sites i and j the spin variables commute, i.e., [Pi, Pj] = 0, and therefore [Hi,
H'] = [Hi, H] = 0
(12)
which plays a fundamental role in the derivation. By writting the trace in the following form: (13)
Tr = l-I trtk) = (klJ i tr(k))tr") where tr(i) = y,+~_ 1, Eq. (10) may be rewritten as 1
(Pi)q = Z~q {Tr' tru)Pi[1 - fl(1 - q)(Hi + H')] q/(1 q~} .
(14)
Inserting in the bracket tqo[l -- fl(1 - q)(Hi + H')] q/° -q) =1 tr(i)[1 -- fl(1 - q)(Hi + H ' ) ] q/(1 -q)
(15)
we obtain 1
=
{Tr[1 -- fl(1 - q)(Hi + H')]q/(1
q)
tr(i)Pi[1 ~ fl(l ~ q)(nL +- U')]q/(1-q)~ x tr(i)[1 -- fl(1 -- q)(Hi + H')] q/(1-q) j
(16)
485
E.F. Sarmento / Pttw~ica A 218 (1995) 482 486 or
/tr(i)#xiE1 - fl(1 - q)(Hi + H')] q/(1 -q'\
(17)
By using that Pi = -I- 1, hence Hi = T- Ei, Eq. (17) reduces to
(#i)q
\1 + f,/q
(18)
with (19)
fq---- 1 - f l ( 1 - - q ) ( - - E i + H ) ]
which generalizes, for arbitrary q, Eq. (8). Depending on the particular interest, further processing of Eq. (18) could be needed in order to approximate the present expression by more tractable ones. There is, however, a simple calculation for which no other processing is needed, and which illustrates the kind of modifications that might be expected in the general case. We refer to the calculation of the critical temperature Tc for a z-coordinated spin-½ Ising ferromagnet with coupling constant J. In the T --+ T~ limit (and assuming a second order phase transition), ( ~ i ) q --~ 0 and ( H ' ) q ((,((~li)q) , i.e., ( H ' ) q becomes neglectable with respect to (#i)q. Consequently, Eq. (18) yields =
(~li) q
=
(20)
1 +fql T ~ o
with
] --(1 -fqlr~ r~ =
+ (1
q ) z J ( l i i ) q / k T c ] q/tl-q)
q)zJ(pi)q/kT~J
'
(21)
or (l~i)q = qzJ (#i),Jk T~
(22)
where we have used the fact that (/~i)q is independent of the particular site i. Finally, Eq. (22) implies
kTc J
-
qz
(231
which extends to arbitrary q the well known q = 1 result (kBTc/J = z). This result is perfectly consistent, within a mean field picture, with Tc(q) obtained, for the square lattice, within a recent renormalization group calculation [23]. In particular, it reproduces the linear dependence on q exhibited, for q >> 1, by the renormalization group calculation of T~.
486
E.F. Sarmento / Physica A 218 (1995) 482 486
References [1] C. Tsallis, J. Stat. Phys. 52 (1988) 479. [2] E.M.F. Curado and C. Tsallis, J. Phys. A 24 (1991) L69; Corrigenda: J. Phys. A 24 (1991) 3187; 25 (1992) 1019. [3] C.E. Shannon, Bell Sys. Tech. J. 27 (1948) 379. [4] A. R6nyi, Probability Theory (North-Holland, Amsterdam, 1970). I-5] A.M. Mariz, Phys. Lett. A 165 (1992) 409. I-6] J.D. Ramshaw, Phys. Lett. A 175 (1993) 169, 171. [7] R.F.S. Andrade, Physica A 175 (1991) 285. [8] A.R. Plastino and A. Plastino, Phys. Lett. A 177 (1993) 177. [9] A. Plastino and C. Tsallis, J. Phys. A 26 (1993) L893. [10] D.A. Stariolo, Phys. Lett. A 185 (1994) 262. [11] E.P. Silva, C. Tsallis and E.M.F. Curado, Physica A 199 (1993) 137; Errata: A 203 (1994) E 160. 1-12] A. Chame and E.V.L. Mello, J. Phys. A 27 (1994) 3663. 1-13] F. Buyukkilic and D. Demirhan, Phys. Lett. A 181 (1993) 24. [14] A.R. Plastino and A. Plastino, Physica A 202 (1993) 438. [15] R.F.S. Andrade, Physica A 203 (1994) 486. [16] C. Tsallis, Chaos, Solitons and Fractals, Vol. 6 ed. by G. Marshall (Pergamon Press, Oxford, 1995) p. 539. [17] P.T. Landsberg, Springer Series in Synergetics, Vol. 61 (Springer Verlag, Berlin) ed. by R.K. Mishra, D. Maab and E. Zwicrlein. [18] A.R. Plastino and A. Plastino, Phys. Lett. A 174 (1993) 384; J.J. Aly, in: N-Body Problems and Gravitational Dynamics, Proceedings of the Meeting held at Aussois-France (21-25 March 1993), eds. F. Combes and E. Athanassoula (Publications de l'Observatoire de Paris, 1993), p. 19. [19] P.S. Alemany and D.H. Zanett, Phys. Rev. B 49 (1994) 956. [20] A.R. Plastino and A. Plastino, Phys. Lett. A 193 (1994) 251. [21] C. Tsallis, in: New Trends in Magnetism Magnetic Materials and their Applications, eds. J.L. Mor/m-L6pez and J.M. Sanchez (Plenum, New York, 1994), p. 451; F.D. Nobre and C. Tsallis, Physica A 213 (1995) 337. [22] H.B. Callen, Phys. Lett. 4 (1963) 161. [23] S.A. Cannas and C. Tsallis, preprint (1994).