PHYSICAL
BEVIE%
8
VOLUM
E
10,
NUMBER
9
NOVEMBER
1974
Calculation
of
fiymamic
critical
yroperties
from
a
cluster-reaction
theory
K. Binder and D.
Stauffer
Physics
Department;
Technical University
Munich,
8046
Garching,
S'est
Germany
H. Muller-Krumbhaar
IBM Zurich Research
Laboratory, 8803 Ruschlikon,
Switzerland
(Received
1
April 1974)
Critical fluctuations
in kinetic
Ising
models are
interpreted
in terms of
cluster reactions. The
basic
assumption,
that clusters
with
l
spins
grow
at a
rate
ac
l",
is
tested
by
Monte Carlo
computations
in
the
single-spin-flip
case. The
dynamic
susceptibilities
associated with
order
parameter
and
energy
are
then
calculated
also
for nonxero
magnetic field,
and are shown to fulfill
dynamic
scaling.
The
exponent
(2
—
r)PS
of the relaxation times
can
be
different from the
susceptibility
exponent
y.
I.
INTRODUCTION
II.
CLUSTER-REACTION
MODEL
The
critical
dynamics
of kinetic
Ising
models'
has
recently received
great
attention:
(i)
In
con-
trast
to models where the
dynamics
is
governed
by
critical
propagating
modes,
the
dynamic
critical
exponents
cannot in
general
be
expressed
only
by
static ones
by
use of
dynamic
scaling.
'
In
par-
ticular, dynamic
scaling
does not determine
4„„,
the
exponent
of
the
order-parameter relation
time
Qv&
p„„(t)
dt,
0
&g(o)
p(t))
—
&p&'
(ii)
Renormalization-group
techniques,
'
high-
temperature
series,
6
and
Monte
Carlo studies
'
have shown
that
b„„differs
from the
susceptibility
exponent
y
in
single-spin-flip (Glauber
)
models.
(iii)
This result
is
in
cont".
ast
to
mode-mode
cou-
pling
theories of
anisotropic
magnets,
'
which
predict
The
kinetic
Ising
model is described
by
a master
equation
for
the
probability
distribution
I
P~(t)&
of
the
spin
where
the
operator
J.
„can
be
specified
explicitly
in terms
of
spin-flip
transition
probabilities.
De-
noting
a
group
of
$
reversed
spins
linked
together
by
nearest-neighbor bonds as
a
"cluster,
"
one
may
describe the state
of
the
system
by
its cluster
con-
figuration.
Spin flips
then
produce
cluster
reac-
tions
(we
also count the creation
of a
single
re-
versed
spin-a
I
=
1
cluster
—
as a cluster reaction).
Discussing
the
nonequilibrium
relaxation"'
it was
pointed
out that
it is most
important
to derive the
averaged
cluster
concentrations
n,
(t).
Considering
now the
response of
the cluster concentration
to
a
small
change
6e
of an
external
parameter
e,
we
denote
by
n& the cluster distribution
which is in
thermal
equilibrium
with the
applied
parameter
e+5e,
and
n)
=pgt
.
It
was shown
'
that it
is
then
reasonable to
replace
Eq.
(4)
by
which is also
the
result of
the
conventional
theory.
'
No
simple physical
interpretation
for
the
result
4„„&y
and
no
detailed satisfactory theory
of
the
slowing
down
exists.
In
the
present
work
we treat
this
problem
with a cluster
reaction
madel.
"
In
Sec.
II,
we introduce the
model,
and
briefly
dis-
cuss
the
basic
approximations
involved in
it. In
Sec.
ID,
we derive
the
dynamic
critical
properties
of
this model.
For
an
explicit
calculation
of
criti-
cal
amplitudes
we
use the
Fisher
droplet
model'
as an
example,
and
we
compare
these
explicit
re-
sults to
corresponding
Monte Carlo
calculations.
Section IV
then contains
our
conclusions.
where
)
P,
(t)&
means a distribution
of
cluster
con-
centrations
fH,
(t)j,
and
L,
is
given
by
The
growth
rate
D,
is
given
by
the
probability
a
(I,
I')
that
a cluster with
I'
spine
is incorpo-
rated"'
into the
I
cluster,
We
expect
D,
to
have
asymptotically
a
power-law
10