1549380323-Statistical Mechanics Theory and Molecular Simulation

648 Critical phenomena

∗16.3. The Maxwell construction in the van der Waals theory attempts to fix the

unphysical behavior of the theory resulting from∂P/∂V >0 forT < Tc. The

procedure is shown schematically in Fig. 16.20. The construction introduces

P

V

a 2

a 1

Tie line

Va n der Wa a ls

isotherm

VL VG

Fig. 16.20 Maxwell construction.

atie linefor the discontinuous change in volume resulting from the first-

order gas–liquid phase transition. The location of the tie line must be chosen

so that the two areas enclosed by the tie line and thePVcurve above and

below the tie line are equal (a 1 =a 2 ). Show that the Maxwell construction

together with the van der Waals equation of state in eqn. (4.7.35) leads to a

mean-field value of the critical exponentβequal to 1/2.

16.4. For the spin-^12 Ising model in one dimension withh= 0, recall that the

partition function could be expressed in the form

∆ = Tr

(

PN

)

.

Consider a RG transformation, called thepair celltransformation, in which

∆ is reexpressed as

∆ = Tr

[(

P^2

)N/ 2 ]

.

The transfer matrix is redefined by P′= P^2. Find the RG equation corre-

sponding to this transformation and show that it leads to the expected stable

fixed point.