1549380323-Statistical Mechanics Theory and Molecular Simulation

652 Critical phenomena

g. Show thatK′can be estimated by

K′≈K 1 +K 2.

h. Derive the RG equation that results, and show that it predicts the exis-

tence of a critical point. What is the value ofKc?

i. By expanding the free energy in a Taylor series aboutK=Kc, calculate

the critical exponentαand compare your value to the exact solution of

Onsager.

j. Finally, compare the critical temperature you obtain to the Onsager re-

sult:

J

kTc

= 0. 44069.

k. Devise an analog of the Maris-Kadanoff scheme for the one-dimensional

free-field Ising model by summing over every other spin on the one-

dimensional spin lattice. What is the RG equation that results? Show

that your one-dimensional equation yields the expected fixed point.

16.8. The Wang–Landau method of Section 7.6 can be easily adapted for spin

lattices. First, since the spin variables take on discrete values, thetotal energy

Eof the spin lattice also takes on discrete values. Therefore, we canwrite the

canonical partition function as

Q(β) =

∑

E

Ω(E)e−βE,

where Ω(E) is the density of states. A trial move consists of flipping a ran-

domly chosen spin and then applying eqn. (7.6.3) to decide whether the move

is accepted. Finally, the density of states at the final energyEis modified

by the scaling factor using ln Ω(E)→ln Ω(E) + lnf. Write a Wang–Landau

Monte Carlo code to calculate the partition function and free energy per spin

at different temperatures of a 50×50 spin lattice in the absence of an external

field. Take the initial state of the lattice to be a randomly chosen setof spin

values, and apply periodic boundary conditions to the lattice.

16.9. Write a simple M(RT)^2 Monte Carlo program to sample the distribution

function of a two-dimensional Ising model in the presence of an external field

h. Observe the behavior of your algorithm for temperaturesT > Tcand

T < Tcat different field strengths. For each case, calculate and plot the

spin-spin correlation function〈σiσj〉−〈σi〉〈σj〉.