Introduction to Mathematical Physics/Statistical physics/Constraint relaxing

We have defined at section secmaxient external variables, fixed by the exterior, and internal variables free to fluctuate around a fixed mean. Consider a system $$L$$ being described by $$N+N'$$ internal variables \index{constraint} $$n_1,\dots,n_N,X_1,\dots,X_{N'}$$. This system has a partition function $$Z^{L}$$. Consider now a system $$F$$, such that variables $$n_i$$are this time considered as external variables having value $$N_i$$. This system $$F$$ has (another) partition function we call $$Z^{F}$$. System $$L$$ is obtained from system $$F$$ by constraint relaxing. Here is theorem that binds internal variables $$n_i$$ of system $$L$$ to partition function $$Z^F$$ of system $$F$$ :

Let us write a Gibbs-Duheim type relation \index{Gibbs-Duheim relation}:

At thermodynamical equilibrium $$S^{F}=S^{L}$$, so: