Computational Physics

15.5 Reducing critical slowing down 507

of the quadratic term is negative, whereas the coefficientgofφ^4 is positive. This
means that the field has two opposite minima. For this model, the MCMG method
does not perform very well. This can be explained using a heuristic argument. Sup-
pose the field assumes values very close to+1or−1. Consider a block of four spins
on the fine lattice which belong to the same coarse lattice site. Adding a nonzero
amount,φN, to these four spins will only be accepted if they are either all equal
to−1, so that an amount of 2 can be added, or if they are all equal to+1 so that we
can subtract 2 from each of them. The probability that all spins in a block have equal
values becomes smaller and smaller when coarsening the lattice more and more, so
the efficiency of the MCMG method is degraded severely for this case. However, it
still turns out to be more efficient by a factor of about 10 than the standard heat-bath
method.

15.5.5 The Fourier-accelerated Langevin method

We have encountered the Langevin method for field theories in Section 15.4.3. This
method suffered from slow convergence as a result of small, essentially random,
steps being taken, causing the system to perform a random walk in phase space.
In 1985, Batrouniet al.proposed a more efficient version of the Langevin method
in which the fields are updated globally [17]. This is done by Fourier-transforming
the field, and then applying the Langevin method to the Fourier modes, rather than
to the local fields. That this is a valid approach can be seen as follows. We have
seen that MD methods can be applied to fields after assigning fictitious momenta
to the field variables. In the MD method we have assigned a momentumpnto each
field variableφn. It is, however, perfectly possible to assign the momenta not to
each individual field variable, but to linear combinations of the field variables. After
integrating out the momenta we shall again find a Boltzmann distribution of the
field variables.
In addition we have the freedom to assign a different time step to each linear
combination of field variables. As we have seen in Section 9.3.2, this is equivalent
to changing the mass associated with that variable, but we shall take the masses all
equal to 1, and vary the time step.
In the Fourier-accelerated Langevin method, we assign momentapkto each
Fourier componentφkof the field. Furthermore, we choose a time stephkfor each
kindividually. To be specific, we write the actionSin terms of Fourier transformed
fields, and construct the following classical Hamiltonian expressed in terms of
Fourier modes:

HClass=

∑

k

{

p^2 k

2

+S[φk]

}

. (15.102)