arXiv:cond-mat/0105542v1 [cond-mat.supr-con] 28 May 2001
Generalized Statistics and High Tc Superconductivity H. Uys ∗, H. G. Miller
†
Department of Physics, University of Pretoria, Pretoria 0002, South Africa
16 May 2001
Abstract If the generalized statistics suggested by Tsallis are used in statistical mechanics, the fluctuation-dissipation theorem no longer holds . Only in the limiting case where Boltzmann statistics are recovered is the theorem applicable. In spite of the fact that this allows for the possibility of condensation in any dimension, it is demonstrated that this is not always realized.
In statistical mechanics with the standard Boltzmann entropy, a simple relationship may be obtained between the canonical or grand canonical ensemble averages of commutators and anticommutators of two dynamical operators[1]. This relationship is often referred to as the fluctuationdissipation theorem since the anticommutator is used to describe time dependent correlations or fluctuations in the system and the commutator is related to transport coefficients or dissipation[2, 3]. It is simply due to the fact that the Boltzmann factor or distribution function is exponential in nature and therefore factorizable. A consequence of this aforementioned theorem is to rigorously rule out the existence of superconductivity or superfluidity in one and two dimensions[4]. Recently, however, it has been pointed out by Tsallis[5] that the BoltzmannGibbs statistics may be generalized such that the entire Legendre-transform structure of thermodynamics is preserved[6]. Although the resulting statistical mechanics is non-extensive, quantal as well as classical applications of the Tsallis statistics have been suggested (see for example the references in Ref [7]). Unlike in the Boltzmann case, however, the generalized distribution function is not simply factorizable, except in the limiting case where Boltzmann-Gibbs statistics are recovered. Hence, as we shall show the fluctuation-dissipation theorem as stated above no longer holds. This ∗ †
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1
allows for the possibility of forming a condensate in two dimensions, provided these generalized statistics are realized as we have suggested is the case of high Tc superconductivity[8]. In the Tsallis formulation, the entropy is given by Sq =
q−1 ) kB Σm (ρm − ρm q−1
(1)
where kB is the Boltzmann constant (which will be set equal to 1), q is any real number (characterizing a particular statistics), and the sum runs over all microscopic configurations (whose probabilities are {ρm }) and Σm ρm = 1.
(2)
The ensemble average of the internal energy, for example, is given by < E >q = Σm ρqm ǫm .
(3)
Furthermore, the generalized Fermi-Dirac (upper sign) and the generalized Bose-Einstein (lower sign) distributions are given by fq (ǫk ) =
1
(4)
1
[1 + β(q − 1)(ǫk − µ)] q−1 ± 1
and in the Maxwell-Boltzmann case by 1
fq (ǫk ) = [1 + β(q − 1)(ǫk − µ)] q−1
(5)
where β = 1/T and µ is the chemical potential [9, 10]. In the limit q=1, the standard Boltzmann-Gibbs expressions are recovered. In the Maxwell-Boltzmann case (with µ = 0) consider the simplest form of canonical correlation function[1] ˆ B(t ˆ ′ ) >= T r{ˆ ˆ B(t ˆ ′ )} < A(t) ρA(t)
(6)
where ˆ
ˆ
ρˆ ≡ ρˆq=1 = e−β H /T r{e−β H } ˆ ˆ = fq=1 (H)/Z q=1 ≡ f (H)/Z,
(7) (8)
Z is the partition function and fq is given by eq. (5). The spectral function for this correlation function may be written as JAB (ω) = =
Z
∞
Z−∞ Z
ˆ B(0) ˆ eiωt < A(t) > dt dEdE ′ ρ(E)2π¯hδ(E + h ¯ ω − E ′ )jAB (E, E ′ ) 2
(9) (10)
where
′ ˆ Aδ(E ˆ ˆ B} ˆ jAB (E, E ′ ) = T r{δ(E − H) − H)
(11)
ρ(E) = e−βE /Z
(12)
and which yields for the correlation function ˆ B(t ˆ ′ ) >= 1 < A(t) 2π
Z
∞ −∞
′
dωe−iω(t−t ) JAB (ω).
(13)
Interchanging the order of the product and E and E’ yields ˆ ′ )A(t) ˆ >= < B(t
1 2π
Z
∞
−∞
′
dωe−iω(t−t ) eβ¯hω JAB (ω)
(14)
since ρ(E + h ¯ ω) = ρ(E)e−β¯hω . This leads to a simple relationship between ˆ the ensemble average of the commutator and anticommutator of Aˆ and B which is referred to as the fluctuation-dissipation theorem[1, 4]. Unfortunately this factorization is not possible over the complete integration range if ρ in eq. (8) is replaced by ρq6=1 [10, 11]. Hence, in principle, for q 6= 1 condensation may occur in dimensions d ≤ 3. Consider an ideal Bose gas for which the number of bosons is given by N (µ, T ) = ∼
Z Z
Z
0
∞
dd rdd pfq (ǫ) d
dǫǫ 2 −1 fq (ǫ)
(15) (16)
where d is the dimensionality and fq is given by eq( 4). For q=1 it is easy to show that limµ→0 Nq=1 (µ, T ) is divergent for d=1,2[12, 4] and condensation only occurs for d=3. On the other hand for q=2, limµ→0 Nq=2 (µ, T ) is not convergent for d=1,2 or 3 which in spite of the absence of the fluctuationdissipation theorem rules out the possibility of condensation.
References [1] R. Kubo, M. Toda, and N. Hashitsume, Spinger Series in Solid-State Sciences volume 31, Statistical Physcs II Nonequilibrium Statistical Mechanics, second edition (Springer-Verlag, Heidelberg, 1995). [2] H. Nyquist, Phys. Rev. 32, 110 (1928). [3] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951). [4] P. C. Hohenberg, Phys. Rev. 158, 383 (1967). [5] C. Tsallis, J. Stat. Phys 52, 479 (1988). 3
[6] E. M. F. Curado and C. Tsallis, J. Phys. A: Math. Gen. 24, L69 (1991). [7] A. R. Plastino, A Plastino, and C. Tsallis, J. Phys.A: Math Gen. 27, 5707 (1994). [8] H. Uys, H. G. Miller, and F. C. Khanna, Preprint, Univerity of Pretoria (2001). [9] F. Bykkili¸c, D. Dermirhan, and A. Gle¸c, Phys. Lett. A 197, 209 (1995). [10] F. Pennini, A. Plastino, and A. R. Plastino, Phys. Lett. A 1995, 309 (1995). [11] Q-A. Wang and A. Le M´ehaut´e, Phys. Lett. A. 235, 232 (1997). [12] R. K. Patria, Statistical Mechanics (Pergamon Press, Oxford, 1972).
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