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# Entropic variables and Massieu-Planck functions

We consider a simple pure substance under hydrostatic conditions described by the following fundamental equation:

where the extensive variables U, V and N are the internal energy, the volume, and the number of particles respectively, and the intensive variables T, p and are the temperature, the pressure and the chemical potential respectively.

Equation () corresponds to the choice of the variables U, V and N as independent variables of the entropy S(U,V,N). These variables are precisely those which are fixed and determine the macrostate of the members of the Microcanonical Ensemble and consequently S is the relevant potential in this statistical ensemble.

It is useful to define the following quantities: , and so that Eq. () can then be written in the dimensionless form:

In general, for other thermodynamic systems with degrees of freedom, one will have:

where are extensive variables, and the corresponding entropic conjugate variables. Massieu-Planck functions [] are entropic thermodynamic potentials defined as Legendre transformations of the entropy. In the case of a pure substance, the following (dimensionless) potentials can be formally defined:

The function was first introduced by Massieu [], and it is called Massieu's potential. The function was introduced by Planck[] and is called Planck's. potential.

Given the extensivity of , and using Euler's theorem for homogeneous functions, it is easy to see that . Therefore the Legendre transformation of all variables redefines the entropy,

Substituting Eq. () into the differentials of the potentials defined above one gets:

From Eq. () one obtains:

The above equations allow a re-derivation of all the standard thermodynamic equations in terms of , and . For instance, Maxwell relations can be deduced, by imposing that the equations ()-() are exact differentials (equality of crossed derivatives). Moreover, Eq. () is the Gibbs-Duhem equation which states that the complete set of intensive variables of the system are not all independent. On the other hand, the extremal condition of leads us to deduce that , and are homogeneous at equilibrium [].

Next: Statistical Mechanics Up: Entropic Formulation of Statistical Previous: INTRODUCTION

Eduard Vives
Tue Oct 24 11:27:45 CEST 2000