# 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*_{∞} = 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
It is evident, that for *M(t) = exp(-τ t)* we get
*τ*_{r} = τ.
Using the trapezium formula and *M*_{o} = 1 one can evaluate
*τ*_{r} as

*τ*_{r} = 1/2 +
∑_{t} M_{t} .

Relaxation time dependences for *T > T*_{c} are shown
in Fig.2. From this picture it follows, that in the critical region
*|T-T*_{c}| << 1 as temperature approaches to
*T*_{c} relaxation time goes to infinity as (the blue curve)

*τ*_{r} =
4.5 (T-T_{c})^{-1.85}.

This is called the *critical slowing down*. E.g.
*τ*_{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
*τ*_{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
Previous: The Monte-Carlo method
Next: Magnetic phase transition

*updated* 2 Jan 2002