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arXiv:cond-mat/0508334v2 [cond-mat.stat-mech] 27 Feb 2006Trace inequalities in nonextensive statistical mechanicsShigeru Furuichi 1∗1 Department of Electronics and Computer Science,Tokyo University of Science, Yamaguchi, 756-0884, JapanAbstract. In this short paper, we establish a variational expression of the Tsallis rela-tive entropy. In addition, we derive a generalized thermodynamic inequality and a generalizedPeierls-Bogoliubov inequality. Finally we give a generalized Golden-Thompson inequality.Keywords : Trace inequalities, thermodynamic inequality, Peierls-Bogoliubov inequality,Golden-Thompson inequality and Tsallis relative entropy2000 Mathematics Subject Classif i cation : 47A63, 94A17, 15A391 IntroductionRecently, the matrix trace inequalities in statistical mechanics are studied by Bebiano et.al.in [2]. Their results are generalized by them in [1] via α-power mean. In addition, the furthergeneralized logarithmic trace inequalities are obtained and their convergences are shown via gen-eralized Lie-Trotter formulae in [7]. Inspired by their works, we generalize the trace inequalitiesin [2] by means of a parametric extended logarithmic function ln λ which will be def i ned below.Our generalizations are dif f erent from those in [1, 7]. In the sense of our generalization, we givea generalized Golden-Thompson inequality. In addition, we give a related trace inequality asconcluding remarks.We denote e xλ≡ (1 + λx)1λ and its inverse function ln λ x ≡x λ −1λ, for λ ∈ (0,1] and x ≥ 0.The functions e xλand ln λ x converge to e x and logx as λ → 0, respectively. Note that we havethe following relations:e x+y+λxyλ= e xλ eyλ ,ln λ xy = ln λ x + ln λ y + λln λ xln λ y. (1)The Tsallis entropy was originally def i ned in [20] by −P ni=1 pqiln 1−q p i =P ni=1 (pqi −p i )1−qfor anynonnegative real number q and a probability distribution p i ≡ p(X = x i ) of a given randomvariable X. Taking the limit as q → 1, the Tsallis entropy converges to the Shannon entropy−P ni=1 p i logp i . We may regard that the expectation value E q (X) =P ni=1 pqi x idepending onthe parameter q is adopted in order to def i ne the Tsallis entropy as x i = −ln 1−q p i , while theusual expectation value E(X) =P ni=1 p i x iis adopted in order to def i ne the Shannon entropyas x i = −logp i . In the sequel we use the parameter λ ∈ (0,1] insead of q. There is a relationbetween these two parameters such that q = 1 − λ.The Tsallis entropy and the Tsallis relative entropy in quantum system (noncommutativesystem) are def i ned in the following manner. See [5, 6] for example.Def i nition 1.1 The Tsallis entropy is def i ned byS λ (ρ) ≡Tr[ρ 1−λ − ρ]λ= −Tr[ρ 1−λ ln λ ρ]∗ E-mail:furuichi@ed.yama.tus.ac.jp1