334 The Monte Carlo method
and 1/K^3 are determined in a molecular dynamics simulation at constant energy in
order to calculateαandγaccording to (10.79) and (10.80).
Using a target temperature (according to the equipartition theorem) of 1.0, a
(measured) temperature of 1.043 (using the correct definition (10.71)), and a total
energy of 10.913±0.003 (in reduced units) have been found in our simulation.
This is the energyE∗. In the canonical ensemble, we should add the correction
3 γ^2 /( 4 α^2 ). Adding the correction, we obtain an energy of 11.94+0.03 in natural
units. An MC calculation with a temperature of 1.043 givesE=11.96, in good
agreement with the prediction. The statistical errors inαandγ are difficult to
calculate because the inverse powers of the kinetic energy are correlated quantities.
The best method is to calculate the correction from several independent simulations
and infer the error from these results, or by data-blocking of the correction over
a long run (see Section 7.4). Note that in order to calculate the canonical kinetic
energy,K= 3 (N− 1 )kBT/2 should be used rather than 3NkBT/2 in order to relate
the energy to the molecular dynamics energy which has three degrees of freedom
less (in the canonical ensemble, the equipartition is satisfied even for small particle
numbers).
Obviously, the practical value of this calculation is limited – the main point is to
show that careful analysis is necessary and possible for systems consisting of small
numbers of particles (such as droplets).
Exercises
10.1 Consider a Monte Carlo algorithm for the two-dimensional Ising model in which the
sites are scanned in lexicographic order, that is, each row is scanned from left to
right, starting with the top row and proceeding towards the bottom row. We want to
show that this method satisfies the detailed balance criterion. A sweep through the
entire lattice is considered as a single step in the Markov chain.
(a) Explain why this Markov chain is ergodic.
The proof that the transition probabilities satisfy detailed balance is done by
recursion. Suppose that the lattice containsNsites, and that for the lattice
containingN−1 sites the algorithm satisfies detailed balance.
(b) Show that if theNth spin is flipped according to the usual Metropolis algorithm,
the sweep over the lattice withNsites satisfies detailed balance.
10.2 [C] Code the heat-bath algorithm for the Ising model and analyse the correlation
time (see Section 7.4), in particular close to the critical temperature. Compare the
results with the standard Metropolis algorithm.
10.3 Consider the Norman–Filinov method for a system with a large chemical potential.
From Eqs. (10.37) and (10.38) we see that in that case the acceptance rate for
creation is much smaller than that for annihilation. Therefore we use a generalised