Critical slowing down
Relaxation of the total magnetization in Ising model from the ordered ("cold"
with M_{o} = 1) state to the equilibrium disordered state
(with M_{inf} = 0) for T > T_{c} = 2.2692...
is shown in Fig.1. These dependences are obtained for L = 640 by the
applet with thermostat algorithm (see below), as since this method leads to
smooth M_{t} relaxation. You see, that exept small beginning
region, relaxation is exponential.
To estimate the relaxation time we use
t_{r} =


M(t) dt .

It is evident, that for M(t) = exp(t t) we get
t_{r} = t.
Using the trapezium formula and M_{o} = 1 one can evaluate
t_{r} as
t_{r} = 1/2 +
S_{t=1,2...}M_{t} .
Relaxation time dependences for T > T_{c} are shown
in Fig.2. From this picture it follows, that in the critical region
TT_{c} << 1 as temperature approaches to
T_{c} relaxation time goes to infinity as (the blue curve)
t_{r} =
4.5 (TT_{c})^{1.85}.
This is called the critical slowing down. E.g.
t_{r} ~ 190 for T = 2.4.
Controls Click mouse to get a new spins configuration.
You see magnetization M (the red curve) and ln M (the black
curve) in the right part of the applet and
t_{r} in the Status bar.
"Print" button sends M_{t} to the Java console.
Press "Enter" to set a new T
value, set Init = +1 and Run new test.
Critical slowing down is observed for T < T_{c} too,
as since it is caused by the infinite growth of the correlation length at
T_{c} . But to estimate the time we have to calculate
accurately equilibrium magnetization. "Hot" (random) state relaxation
with metastable clusters formation was discussed before.
Contents
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Next: Magnetic phase transition
updated 2 Jan 2002