Introduction to Mathematical Physics/Statistical physics/Entropy maximalization

In general, a system is described by two types of variables. External variables $$y^i$$ whose values are fixed at $$y_j$$ by the exterior and internal variables $$X^i$$ that are free to fluctuate, only their mean being fixed to $$\bar{X^i}$$. Problem to solve is thus the following:

Entropy functional maximization is done using Lagrange multipliers technique. Result is:

where function $$Z$$, called partition function, \index{partition function} is defined by:

Numbers $$\lambda_i$$ are the Lagrange multipliers of the maximization problem considered.

Relations on means that:

This relation that binds $$L$$ to $$S$$ is called a {\bf Legendre transform}.\index{Legendre transformation} $$L$$ is function of the $$y^i$$'s and $$\lambda_j$$'s, $$S$$ is a function of the $$ y^i$$'s and $$\bar{X^j}$$'s.