Generalization of the Peierls-Bogolyubov inequality by means of a quantum-mechanical variational principle

Gert Roepstorff

João da Providência, Carlos Fiolhais

The principle of least information is used to derive the inequality between the arithmetic and the geometric mean. Some important variational results in quantum statistical mechanics (Peierls and Bogolyubov inequalities) are pedagogically reviewed. Emphasis is put on the distinction between mixed states describing metastable and stable equilibrium.

D. E. Ruiz, I. Y. Dodin

Frank Hansen, Jin Hao Liang, Guanghua Shi

We study the convexity or concavity of certain trace functions for the deformed logarithmic and exponential functions, and in this way obtain new trace inequalities for deformed exponentials that may be considered as generalizations of Peierls–Bogolyubov’s inequality. We use these results to improve previously-known lower bounds for the Tsallis relative entropy.

Guy Rideau

Natália Bebiano, João da Providência, João P. da Providência

Extensions of trace inequalities arising in statistical mechanics are derived in a straightforward and unified way from a variational characterization of the Tsallis entropy. Namely, one-parameter extension of the thermodynamic inequality is presented and its equivalence to a generalized Peierls–Bogoliubov inequality is stated, in the sense that one may be obtained from the other, and vice versa.