. Above the criticaltemperature (α >0), you should findG11=G22, which is a simple consequence ofrotational invariance. Below the critical temperature, (α <0), choose the state withhm1(r)i=q|α|/βandhm2(r)i= 0 in zero field.You should findG116=G22, anddetermine both functions in 3 dimensions.3. We compute physical properties of the van der Waals gas at different points in thephase diagram in theP,T, plane.For all of the questions below, you may use the“Landau” approximation to the Gibbs free energy derived in class, and obtain answersto leading non-vanishing order inx≡T-Tcand/ory≡P-Pc. In this approximation,we can write the Gibbs free energy,G(P, T)G(P, T)N=GcN+y-x2bm+x+ 4by24b2!m2+a1944b5m4(4)2
where the value ofmhas to be chosen at eachPandTto minimizeG. Further, thisoptimum value ofmis related to the particle density,ρby1ρ=1ρc+m+16bm2(5)(a) Sketch the phase diagram in theP,Tplane.Find the equation of the linerepresenting the liquid-gas transition.(b) Compute the discontinuity in the density,ρL-ρG, along this line.How doesthis discontinuity vanish upon approaching the critical end-point atP=PcandT=Tc?