Eur. Phys. J. Plus (2011) 126: 86
https://doi.org/10.1140/epjp/i2011-11086-8

Regular Article

General symmetries: From homogeneous thermodynamics to black holes

¹ Dipartimento di Fisica, Università di Milano, 20133, Milano, Italy
² Dipartimento di Matematica e Fisica, Università dell’Insubria, 22100, Como, Italy

^* e-mail: belgiorno@mi.infn.it

Received: 26 July 2011
Accepted: 1 September 2011
Published online: 20 September 2011

Abstract

We consider the topic of symmetries as classifying tools for thermodynamic systems. We adopt the contact geometry approach, in a general framework including standard homogeneous thermodynamics but not limited to it, and we focus our attention on the problem of the existence of a general symmetry, to be defined as a symmetry which is the same for a class of thermodynamic systems. Homogeneity symmetry of standard equilibrium thermodynamics is the paradigmatic example of general symmetry, and we point out its being associated with a multi-class thermodynamics, whose mathematical characterization is taken into account. Furthermore, quasi-homogeneity symmetry, which describes some non-extensive systems, is shown to give rise to a general symmetry, in the above sense, in the case of non-relativistic self-gravitating fermions. In the latter case, it is also conjectured to give rise to a multi-class structure. An analysis of the behavior of transversal symmetries under the partial Legendre involutions enhances a special role of quasi-homogeneity symmetry, as well as the role of special thermodynamic limit is pointed out as a tool for investigating the topic of general symmetries.