[A Kinetic Views of Statistical Physics - chap8.4] Mean-field approximation

■ Mean-field approximation

neighboring spins are uncorrelated.

Neglect spatial fluctuation.

■ Glauber dynamics

From transition rate in Glauber dynamics, we can write down the equation of motion for average magnetization

expand tanh function.

- For T>T_c,

cubic and higher-order terms are negligible, and magnetization decays exponentially with time 

with 

- At T_c

 , and thus magnetization has the power-law relaxation

(critical slowing down: the system relaxes much more slowly.)

- For T<T_c,

The magnetization decays to its equilibrium value exponentially in time, 

with 

either from above or from below the critical point.