190 Classical equilibrium statistical mechanics
ATL2 L3 L∞Figure 7.5. Typical behaviour of a physical quantityAvs temperature close to the
critical point for various system sizes.At the critical point the correlation time diverges according to
τ=ξz. (7.60)This divergence implies that close to the critical point the simulation time needed to
obtain reliable estimates for physical quantities increases dramatically. This phe-
nomenon is calledcritical slowing down. For most models with a Hamiltonian
containing only short-range couplings, the value of the exponentzis close to 2.
For the Ising model in two dimensions, the dynamic critical exponent has been
determined numerically – its value isz≈2.125[32].
For systems far from the critical point, the correlation length is small, and it is
easy to simulate systems that are considerably larger than the correlation length.
The values of physical quantities measured will then converge rapidly to those
of the infinite system. Close to the critical point, however, the correlation length
of the infinite system might exceed the size of the simulated system; hence the
system size will set the scale over which correlations can extend. This part of the
phase diagram is called thefinite-size scaling region. It turns out that it is possible to
extract information concerning the critical exponents from the behaviour of physical
quantities with varying system size close to the critical point. Of course, for a finite
system, the partition function and hence the thermodynamic quantities are smooth
functions of the system parameters, so the divergences of the critical point are
absent. However, we can still see a signature of these divergences in the occurrence
of peaks, which in the scaling region (ξ L) become higher and narrower with
increasing system size. Also, the location of the peak may be shifted with respect
to the location of the critical point. The general behaviour is shown in Figure 7.5.
These characteristics of the peak shape as a function of temperature are described