Legendre transforms general principles

P. W. Ayers, S. Golden, and M. Levy, Generalizations of the Hohenberg—Kohn theorem I. Legendre transform constructions of variational principles for density matrices and electron distribution functions. J. Chem. Phys. 124, 054101 (2006). 

In conclusion, let us summarize the main principles of the equilibrium statistical mechanics based on the generalized statistical entropy. The basic idea is that in the thermodynamic equilibrium, there exists a universal function called thermodynamic potential that completely describes the properties and states of the thermodynamic system. The fundamental thermodynamic potential, its arguments (variables of state), and its first partial derivatives with respect to the variables of state determine the complete set of physical quantities characterizing the properties of the thermodynamic system. The physical system can be prepared in many ways given by the different sets of the variables of state and their appropriate thermodynamic potentials. The first thermodynamic potential is obtained from the fundamental thermodynamic potential by the Legendre transform. The second thermodynamic potential is obtained by the substitution of one variable of state with the fundamental thermodynamic potential. Then the complete set of physical quantities and the appropriate thermodynamic potential determine the physical properties of the given system and their dependences. In the equilibrium thermodynamics, the thermodynamic potential of the physical system is given a priori, and it is a multivariate function of several variables of state. However, in the equilibrium... 

The Legendre transformation could be formulated as an extremum-task as well [i ]. The Onsager-Machlup function (OM-function) play a central role in the extremum principles of the irreversible thermodynamics based on generalized Onsager constitutive theory. The OM-function could be introduced by the spontaneous entropy production and one of dissipation potentials using the Legendre transformation. 

Considerable effort will be made in this chapter to derive the criteria of stability and criticality from basic principles. In order to appreciate these concepts, we first derive the criteria of stability for a pure component separately and then proceed briefly to binary mixtures. The stability criteria for multicomponent systems will be derived in detail. The derivation for multicomponent systems is very general and relies on the Legendre and Jacobian transformations discussed in Chapter 1 and the mathematical principles for definiteness that will be discussed in this chapter. 

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