ABSTRACT
We need to know the time evolution of configurations, how
physical quantities change under external time-dependent
disturbances, and how the equilibrium probability distribu-
tion is reached a fter disturbances are turned off. Dynamic
phenomena are much richer in variety than static phenom-
ena, including, for example, diffusion, wave propagation,
damping, inelastic scattering of neutrons or light, etc. In
the study of c r it ica l dynamics, we are mainly interested in
the time variations of the large-sca le fluctuations of the
order parameter and other slowly varying physical quantities
near the crit ica l point. Qualitatively it is easy to understand
why the order parameter varies slowly in time. Imagine a
spin system. Near its critica l point, configurations with
large spin patches are favorable. In each patch there is a
net fraction of spins pointing in the same direction. As we
INTRODUCTION 421
mentioned in Chapter I, thermal agitations w i l l flip spins at
random. Owing to the large sizes of the patches, it would
take a long time for the thermal agitations to turn a whole
patch of spins around. In a more formal language, we say
that long wavelength modes of spin fluctuations have very
long relaxation t im es . Often this is re fe r red to as
"cr it ica l slowing down. " There are also other reasons
for long relaxation times, mainly conservation laws, for
various quantities. Observed dynamic phenomena near
critical points are all characterized by long relaxation
times. A theory of crit ica l dynamics must explain how
these long relaxation times come about in term s of small-
scale interactions among spins, how they depend on the
temperature, and how they are affected by conservation
laws and other features. Of course, i f we w ere able to
derive cr it ica l dynamics from microscopic models via
f irst principles, then we would have a theory. The criteria
for dynamic models are subject to the same kind of discus-
sion as those for static ones in Sec. II. 1 (which should be
read again i f forgotten), except that many new com plica-
tions arise for dynamics. In statics, the m odel provides a
422 INTROD UCTION T O DYNAMICS
basis for statistics, which is essentially the counting of
configurations. In going from model (1) to model (5) of
Sec. II. 1, the configurations of spins are more and more
coarse-grained, but there is no essential difference among
the models. For dynamics, we need to do much more than
count configurations ~ w e need equations of motion for
studying time evolutions. At the leve l of models (1) and (2),
the equations of motion are furnished by time-dependent
Schrödinger equations. Because such equations are time-
reversa l invariant and do not d irectly describe any dissipa-
tion, they are impractical for studying crit ica l dynamics.
