# Massieu function

In thermodynamics, Massieu function is a state equation that represents all the properties of a body of invariable composition which are concerned in reversible processes by means of a single function.  Such equations were introduced by French engineer Francois Massieu in 1869 and called by him “characteristic functions”. 

Massieu functions are partial Legendre transforms of fundamental relations, in entropy representation S(U,V,N), that provide additional alternative fundamental relations.  The Massieu functions are rarely used in thermodynamics, often discussed for their historical importance.

First characteristic function
In 1876, American engineer Willard Gibbs credited Massieu as having invented the concept of the characteristic function, the first of which, according to Gibbs, given by Massieu is Ψ (Psi), defined in Gibbs notation as: $\frac{-\epsilon + t \eta}{t} \,$or $\frac{-\psi}{t} \,$

which in modern notation reads, according to French thermodynamicist Pierre Perrot: $J = \frac{-F}{T} \,$

which is the negative of the Helmholtz free energy F divided by the temperature T of the body. The symbol J is referred to as the Massieu function. 

Second characteristic function
The second characteristic function as well as the expression Ψ' (psi prime), cited by Gibbs as: $\frac{-\epsilon + t \eta -pv}{t} \,$or $\frac{-\zeta}{t} \,$

which in modern notation reads: $Y = \frac{-G}{T} \,$

which is the negative of the Gibbs free energy G divided by the temperature T of the body. The symbol Y is sometimes called the Planck function. 

Other Massieu functions
Beyond the first and second Massieu functions, given above, there are other Massieu functions. One is called the Kramer function, symbol omega Ω, and another unnamed function, given by symbol Gamma Γ. 

Conjugate variables
Massieu, according to German scientist Gerhard Inden, was the one who introduced the logic that the a generic function, symbol Ψ (Psi), for a given body, could be expressed as a function of conjugate variables, as defined by the following relation: $\Psi = \Psi \big( X_1, \dots, X_i, Y_{i+1}, \dots Y_r \big) \,$

where for every system with degree of freedom r one may choose r variables, i.e. $\big( X_1, \dots, X_i, Y_{i+1}, \dots Y_r \big) \,$, to define a coordinate system, where X and Y are extensive and intensive variables, respectively, and where at least one extensive variable must be within this set in order to define the size of the system, such that the (r+1)-th variable, Ψ, is then called the Massieu function, a modified equation form or representation of the energy or potential of the body or system. 

References
1. Gibbs, Willard. (1876). "On the Equilibrium of Heterogeneous Substances" (Massieu, pgs. 86, 358),Transactions of the Connecticut Academy,III. pp. 108-248, Oct., 1875-May, 1876, and pp. 343-524, may, 1877-July, 1878.
2. (a) Massieu, Francois. (1869). “Sur les Functions Caracteristiques des Divers Fluides et Sur la Theorie des Vapeurs (On the Various Functions Characteristic of Fluids and on the Theory of Vapors)”, Comptes Rendus, 69: 858-62, 1057-61.
(b) Massieu, Francois. (1876). Thermodynamique: Mêmoire sur les fonctions catactéristiques des divers fluides et sur la théorie des vapeurs. 92-pgs. Académie des Sciences de L'Institut National de France.
3. Perrot, Pierre. (1998). A to Z of Thermodynamics (Massieu function, pg. 190). Oxford University Press.
4. Inden, Gerhard. (2008). “Introduction to Thermodynamics”, Materials Issues for General IV Systems, pgs. 73-112. Springer.
5. Tschoegl, Nicholas W. (2000). Fundamentals of Equilibrium and Steady-State Thermodynamics (ch.9: Massieu Functions, pgs. 62-). Elsevier.
6. Rao, Y.V.C. (1994). Postulational and Statistical Thermodynamics (4.4: Massieu functions, pgs. 89-). Allied Publishers.