Legendre Analysis, Thermodynamic Formalism, and Spectra of Perron-Frobenius Operators

Abstract

It is well known that, in classical equilibrium termodynamic systems, some basic characteristics, such as complete energy, free energy, enthalpy, entropy, etc., are related to each other by (finite-dimensional) Legendre transforms with respect to certain variables determining these characteristics. The first section of this paper considers mathematical objects naturally arising in relation to the general Fenchel-Legendre transform. In the second section, it is shown that, for these objects, thermodynamic characteristics such as internal energy, temperature, entropy, and heat transfer can be defined. In the third section, it is established that the logarithms of the spectral radii of weighted shift and Perron-Frobenius operatos convexly depend on phase potentials.