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 $N+N^{\prime }$ internal variables \index{constraint} $n_1,\dots ,n_N,X_1,\dots ,X_{N^{\prime }}$. 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 $F$ : Template:IMP/thm Template:IMP/pf

Template:IMP/rem Let us write a Gibbs-Duheim type relation \index{Gibbs-Duheim relation}: Template:IMP/eq Template:IMP/eq At thermodynamical equilibrium $S^F=S^L$, so: Template:IMP/eq Template:IMP/exmp Template:IMP/exmp Template:IMP/rem Template:IMP/exmp