# 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
τr =
 ∞ ∫ o
M(t) dt
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 as
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)
τr = 4.5 (T-Tc)-1.85.
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.

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updated 2 Jan 2002