Perturbation and Variational Methods in Nonextensive Tsallis Statistics

A unified presentation of the perturbation and variational methods for the generalized statistical mechanics based on Tsallis entropy is given here. In the case of the variational method, the Bogoliubov inequality is generalized in a very natural way following the Feynman proof for the usual statistical mechanics. The inequality turns out to be form-invariant with respect to the entropic index q. The method is illustrated with a simple example in classical mechanics. The formalisms developed here are expected to be useful in the discussion of nonextensive systems. PACS number(s): 05.70.Ce, 05.30.-d, 05.20.-y, 05.30.Ch Typeset using REVTEX 1 Nonextensive effects are common in many branches of the physics, for instance, anomalous diffusion [1–4], astrophysics with long-range (gravitational) interactions [5–9], some magnetic systems [10–12], some surface tension questions [13,14]. These examples indicate that the standard statistical mechanics and thermodynamics need some extensions. In this direction, a theoretical tool based in a nonextensive entropy (Tsallis entropy) [15] has successfully been applied in several situations, for example, Lévy-type anomalous superdiffusion [16], Euler turbulence [17], self-gravitating systems [17–21], cosmic background radiation [22], peculiar velocities in galaxies [23], linear response theory [24] and eletronphonon interaction [25], and ferrofluid-like systems [26]. Lavenda and coworkers [27] stress that any newly proposed entropy [28] must have “concavity” property for it to be correct. Tsallis [15] and Mendes [29] have shown that the Tsallis entropy indeed satisfies this criterion and hence meets this requirement of concavity. The above features of Tsallis entropy thus make it unique among other forms for entropy suggested in the literature. In this context, it is very important to understand more deeply the properties of the generalized statistical mechanics based on the Tsallis entropy. In particular, a generalization of the approximate methods of calculation of its thermodynamical functions is of great value, as for instance, the semiclassical approximation [30], perturbation and variational methods. The present work deals with the last two questions. We develop here the perturbation and variational methods in a unified way. This approach provides the generalization of the Feynman proof [31] of the Bogoliubov inequality, which appears to be a natural generalization of this inequality. Indeed, we shall prove that the original form of the inequality is preserved (see inequality (9)). This inequality does not coincide from that proposed in ref. [32], except for q = 1. This is due to the fact that we use different mathematical inequalities to derive our final results. The Tsallis entropy and a q-expectation value for an observable A [33] is defined respectively as Sq = k Tr ρ (1− ρ ) /(q − 1) and 〈A〉q = Trρ A, where ρ is the density matrix, q ∈ R gives the degree of nonextensivity, k is a positive constant. Without loss of generality, we employ k = 1 in the following analysis. By using the above definitions with Trρ = 1 one 2 obtains the canonical distribution [33], pn = p(En) = [1− (1− q)βEn] 1/(1−q) Zq , (1) where {pn} are the probabilities, β is the inverse of the temperature, {En} is the set of eigenvalues of the Hamiltonian, and Zq = ∑ n [1− (1− q)βEn] 1/(1−q) (2) is the generalized partition function. In the expressions (1) and (2) we assumed that 1 − (1− q) βEn ≥ 0. When this condition is not satisfied we have a cut-off. For instance, when a classical partition function is calculated, the integration limits in phase space are given by the condition 1− (1− q)βH ≥ 0. ¿From Eqs. (1) and (2) several relations can be obtained, for instance, the generalized free energy becomes Fq = Uq − TSq = − 1 β Z q − 1 1− q , (3) where Uq = ∑ n p q nEn is the generalized internal energy, and T is the temperature which satisfies the relation 1/T = ∂Sq/∂Uq. Note that the previous expressions are reduced to the usual ones in the limit case q → 1. To develop the perturbation method in this generalized statistical mechanics we assume that the Hamiltonian of the system is H = H0 + λHI . (4) In this expression, H0 is the Hamiltonian of a soluble model, λHI is small enough so that it can be considered as a perturbation on H0 (H0 and HI need not necessarily commute), and λ is the perturbation parameter. Thus, the perturbative expansion of the free energy can be written as Fq(λ) = F (0) q + λF (1) q + λ2 2 F (2) q + . . . . To understand how the corrections F (n) q are calculated it suffices to evaluate the first three terms. The first term is the free energy for the case without perturbation, F (0) q = Fq(0). The second term is obtained from the first derivative of Fq(λ) at λ = 0, 3 F (1) q = ∂Fq (0) ∂λ = − 1 βZ q ∂ ∂λ ∑ n [1− (1− q)βEn] 1/(1−q) ∣