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 $\overline{X^i}$. Problem to solve is thus the following: Template:IMP/prob Entropy functional maximization is done using Lagrange multipliers technique. Result is: Template:IMP/eq where function $Z$, called partition function, \index{partition function} is defined by: Template:IMP/eq Numbers ${\lambda }_i$ are the Lagrange multipliers of the maximization problem considered. Template:IMP/exmp Template:IMP/exmp Relations on means that: Template:IMP/eq 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 $\overline{X^j}$'s.