Critical slowing down
Relaxation of the total magnetization in Ising model from the ordered ("cold"
with Mo = 1) state to the equilibrium disordered state
(with M∞ = 0) for T > Tc = 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 Mt relaxation. You see, that exept small beginning
region, relaxation is exponential.
To estimate the relaxation time we use
It is evident, that for M(t) = exp(-τ t) we get
τr = τ.
Using the trapezium formula and Mo = 1 one can evaluate
τr = 1/2 +
∑t Mt .
Relaxation time dependences for T > Tc are shown
in Fig.2. From this picture it follows, that in the critical region
|T-Tc| << 1 as temperature approaches to
Tc relaxation time goes to infinity as (the blue curve)
This is called the critical slowing down. E.g.
τr ~ 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
τr in the Status bar.
"Print" button sends Mt 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 < Tc too,
as since it is caused by the infinite growth of the correlation length at
Tc . But to estimate the time we have to calculate
accurately equilibrium magnetization. "Hot" (random) state relaxation
with metastable clusters formation was discussed before.
Previous: The Monte-Carlo method
Next: Magnetic phase transition
updated 2 Jan 2002