ヘルムホルツの自由エネルギー

${\displaystyle F=U-TS}$

${\displaystyle F(T,V,N)=U(S(T,V,N),V,N)-T\,S(T,V,N)}$

${\displaystyle dF=-S(T,V,N)\,dT-p(T,V,N)\,dV+\sum _{i}\mu _{i}(T,V,N)\,dN_{i}}$

${\displaystyle F=-pV+\sum _{i}N_{i}\mu _{i}}$

${\displaystyle Q\leq T_{\text{ex}}\Delta S}$

${\displaystyle W=Q-\Delta U\leq T_{\text{ex}}\Delta S-\Delta U}$

${\displaystyle \Delta F=\Delta (U-T_{\text{ex}}S)=\Delta U-T_{\text{ex}}\Delta S}$

${\displaystyle W\leq -\Delta F}$

${\displaystyle W(T_{\text{ex}};X_{0}\to X_{1})\leq W_{\text{max}}(T_{\text{ex}};X_{0},X_{1})}$

${\displaystyle W_{\text{max}}(T_{\text{ex}};X_{0},X_{1})=F(T_{\text{ex}};X_{0})-F(T_{\text{ex}};X_{1})}$

${\displaystyle \Delta F\leq -W=0}$

${\displaystyle F(\beta )=-{\frac {1}{\beta }}\ln Z(\beta )}$