A Variational Perturbation Approximation Method in Tsallis Non-Extensive Statistical Physic.pdf
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A Variational Perturbation Approximation Method
in Tsallis Non-Extensive Statistical Physics
Wen-Fa Lu?
Institute for Theoretical Physics, Department of Physics,
Shanghai Jiao Tong University, Shanghai 200030, the People’s Republic of China
(Dated: February 1, 2008)
Abstract
For the generalized statistical mechanics based on the Tsallis entropy, a variational perturbation
approximation method with the principle of minimal sensitivity is developed by calculating the
generalized free energy up to the third order in variational perturbation expansion. The approxi-
mation up to the first order amounts to a variational approach which covers the variational method
developed in Phys. Rev. Lett. 80, 218 (1998) by Lenzi et al, and the approximations up to higher
orders can systematically improve variational result. As an illustrated example, the generalized
free energy for a classical harmonic oscillator (considered in the Lenzi’s joint work) are calculated
up to the third order, and the resultant approximations up to the first, second, and third orders
are numerically compared with the exact result.
PACS numbers: 05.70.Ce; 05.10.-a; 05.30.Ch
Keywords: non-extensive effect, Tsallis entropy, variational perturbation theory
?Electronic address: wenfalu@
1
I. INTRODUCTION
Tsallis non-extensive statistical physics (TNESP) is one of theories for generalizing the
Boltzmann-Gibbs statistical mechanics and thermodynamics [1]. Its formalism is based
on the Tsallis entropy with a parameter q, the index of non-extensiveness for a system.
It takes the conventional statistical mechanics as its special case of q → 1. A distinct
feature of it is the power-law distributions (q 6= 1) instead of the exponential law in the
conventional statistical mechanics. Since Tsallis’ pioneered work in 1988 [2], the TNESP
has greatly developed, and successfully been used for investigating many sy
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