Computational Physics

308 The Monte Carlo method

that the acceptance probability for each such new configuration is proportional to
exp[−βE(X→X′)]. This can be done using induction – see Problem 10.7 and
Ref.[15].
The heat-bath method (see above) can be implemented straightforwardly for the
Ising model. Suppose the spinsiselected by the random generator is surrounded
byn++spins andn−= 4 −n+−spins. In that case the Hamiltonian for the spin
siin the fixed neighbour configuration is given by

H(si|S−si)=−J( 2 n+− 4 )si. (10.25)

Therefore,siis given the value+or−with probabilities

P+=

e(^2 n+−^4 )βJ

e(^2 n+−^4 )βJ+e−(^2 n+−^4 )βJ

and (10.26a)

P−=

e−(^2 n+−^4 )βJ

e(^2 n+−^4 )βJ+e−(^2 n+−^4 )βJ

. (10.26b)

Using the heat-bath instead of the Metropolis method results in a substantial
decrease of the correlation time. We shall encounter this method again in Chapter 15,
where we present results for the correlation times for the scalar lattice field model.
We shall return to the Ising model in Section 15.5.1 where we discuss algorithms
that are much more efficient near the critical point than the standard MC methods
discussed here.

10.3.2 Monte Carlo simulation of a monatomic gas

The Metropolis MC technique enables us to calculate averages of static quantit-
ies. Therefore, the momentum degrees of freedom are irrelevant and we integrate
these out as in Section 7.2.1, so that the Boltzmann factor depends only on the
configurational potential energy:

ρ(R)=exp[−βU(R)], (10.27)

whereRis the combined position coordinater 1 ,...,rN. The configurational poten-
tial energyU(R)is usually written as a sum of pair-potentials as in the previous
chapters.
The Monte Carlo procedure for a monatomic gas proceeds as follows. The matrix
ωXX′is chosen such that only one particle may be moved to a new position – it is
selected at random and the remaining particles are kept fixed. The new position
of the particle is chosen at random with a homogeneous distribution within a cube
centred at the old position of the particle.^3 The energy difference is calculated and

(^3) The cubic shape is not essential – it is chosen for convenience.