Statistical Physics

9.2 Landau Theory 135Every microscopic state that is realized in the system under consideration hassome value for this Ψ. For a system in a heat bath, each microscopic stateis realized with a probability that is given by the canonical distribution. Thevalue of the order parameter that is obtained at a given temperature is themost probable value of the order parameter at that temperature.To calculate the probability distribution of the order parameter, we definethe density of states Ω(E,Ψ) for a given value of the order parameter Ψ.Namely,Ω(E,Ψ)dE dΨ (9.1)gives the number of microscopic states that have an energy between E andE +dE and an order parameter between Ψ and Ψ +dΨ. We then define thepartial partition function z(Ψ) byz(Ψ) ≡∫ ∞0dEΩ(E,Ψ)e −βE . (9.2)The total partition function Z is given by the integral of z(Ψ):Z ==∫ ∞−∞∫ ∞−∞dψ∫ ∞0dEΩ(E,Ψ)e −βEdψz(ψ) . (9.3)The probability of a particular value of Ψ being realized is given byz(Ψ)Z . (9.4)To discuss this probability, Landau expressed the partial partition function interms of a free energy F L (T,V,N,Ψ), whereF L (T,V,N,Ψ) ≡−k B T ln [z(Ψ)] . (9.5)Landau expected that near the phase transition point the order parameter Ψwould be small, and so the Landau free energy F L can be expanded as a powerseries in Ψ. There should be no terms containing odd powers of Ψ, sincewe are considering a system in which states with order parameters +Ψ and−Ψ are equally probable at high temperature, although this symmetry isspontaneously broken below T c . The general form isF L (T,V,N,Ψ)=F 0 (T,V,N)+aΨ 2 + bΨ 4 + ··· . (9.6)The most probable, and almost certainly realized, value of the order parameterat (T,V,N) is determined by the minimum of this free energy. Since Ψ is smallnear the transition point, we can safely neglect the terms of higher order thanΨ 4 in the expansion of (9.6).This Landau free energy has a minimum at Ψ = 0 if the coefficients aand b are positive. Therefore, a>0andb>0 describes the phase above thecritical temperature. On the other hand, if a0, minima occur at