15.5 Reducing critical slowing down 491
ImplementationAll the MD algorithms described can be implemented without difficulty. The details
of the leap-frog and Langevin algorithm can be found in Chapter 8. Moreover,
calculation of the correlation function is described in Section 15.4.2. The programs
can all be tested using the results presented in that section.
15.5 Reducing critical slowing down
As we have already seen in Section 7.3.2, systems close to the critical point suffer
fromcritical slowing down: the correlation timeτdiverges as a power of the cor-
relation length. This renders the calculation of the critical properties very difficult,
which is unfortunate as these properties are usually of great interest: we have seen
in this chapter that lattice field theories must be close to a critical point in order
to give a good description of the continuum theory. In statistical mechanics, crit-
ical properties are very often studied to identify the critical exponents for various
universality classes.
For most systems and methods, the critical exponentz, defined by
τ=ξz, (15.77)is close to 2. For Gaussian models, the valuez=2 of the critical exponent is related
to the convergence time of the simple Poisson solvers, which can indeed be shown
to be equal to 2 (seeAppendix A7.2). The value of 2 is related to the fact that the
vast majority of algorithms used for simulating field theories arelocal, in the sense
that only a small number of degrees of freedom (mostly one) is changed at a step.
For systems characterised by domain walls (e.g. the Ising model), the exponent 2
can be guessed by a crude heuristic argument. The major changes in the system
configuration take place at the domain walls, as it takes less energy to move a wall
than to create new domains. In one sweep, the sites neighbouring a domain wall
have on average been selected once. The domain wall will therefore move over a
distance 1. But its motion has a random walk nature. To change the configuration
substantially, the domain wall must move over a distanceξ, and for a random walk
this will take of the order ofξ^2 steps.
Over the past ten years or so, several methods have been developed for reducing
the correlation time exponentz. Some of these methods are tailored for specific
classes of models, such as the Ising and other discrete spin models. All methods are
variations of either the Metropolis method, or of one of the MD methods discussed
in the previous section. In this section we shall analyse the different methods in
some detail. Some methods are more relevant to statistical mechanics, such as
those which are suitable exclusively for Potts models, of which the Ising model