Statistical Physics

138 9 Second-Order Phase Transitions∫ ∞[Z(T,V,N) ≃ d(∆Ψ) exp −β−∞√ [ π=−2aβ exp −β(F 0 (T,V,N) − a24b − 2a (∆Ψ)2 )](F 0 (T,V,N) − a24b)](9.15)andF (T,V,N)=F 0 (T,V,N) − a24b − 1 ( ) π2 k BT ln−2aβ≃ F 0 (T,V,N) − a2 04b (T c − T ) 2 . (9.16)The last approximation is valid except close to the transition temperature,where a ≃ 0. In this approximation, the free energy changes continuouslythrough the critical temperature.9.2.2 Entropy, Internal Energy, and Heat CapacityHere we consider how the thermodynamic variables behave in the orderedphase compared with the normal phase that exists when T > T c .TheentropyS is given by the derivative of F .ForT>T c ,( ) ( )∂F(T,V,N)∂F0 (T,V,N)S(T,V,N)=−= −∂T∂T≡ S 0 (T,V,N) . (9.17)The internal energy U is given byU(T,V,N)=F (T,V,N)+S(T,V,N)T= F 0 (T,V,N)+S 0 (T,V,N)T . (9.18)The constant-volume heat capacity C is given by( ) ( )∂U∂S(T,V,N)C(T,V,N)== T∂T∂TV,N≡ C 0 (T,V,N) . (9.19)The concrete behavior of these quantities above T c depends on the actualsystem.Next we calculate these variables for the low-temperature phase. The entropyand the internal energy have the following forms below T c :( ) ( )∂F∂F0S(T,V,N)=−= −− a2 0∂T∂T 2b (T c − T )V,NV,NV,NV,NV,N= S 0 (T,V,N) − a2 02b (T c − T ) (9.20)