# Critical slowing down in Monte Carlo (MC) simulations

Here is what I understood from critical slowing down.

When we are near a phase transition, the autocorrelation time $\tau$ is very long.

Imagine we are doing MC simulations on a ferromagnetic near $T_c$ where $T_c$ is the temperature of phase transition. We are studying the observable $M$ which is the total magnetisation. The fact that $\tau$ is long physically mean that if I am in a microstate with magnetisation far from the average magnetization, it will take a long time (many steps) for the simulation to access to microstates that have magnetisation near the average magnetisation.

And, as the microstates that have magnetisation near the average magnetisation have a more statistical importance in the calculation, it means that we will have a very bad statistics if we don't sample long enough.

My questions:

2) Consider the Landau theory of phase transitions. The free energy functional is $$F = \int dx \left[ \kappa(\nabla M)^2 + a M^2 + b M^4 + \ldots \right]$$ and the correlation length (in the disordered phase) is $\xi \sim 1/\sqrt{a}$. At the phase transition $a\to 0$ and $\xi\to\infty$. Fluctuations modify the simple mean field scaling, and $\xi\sim 1/t^\nu$ where $t$ is the reduced temperature and $\nu$ is a critical exponent.
3) In order to study the relaxation time we have to make the Landau theory time dependent. This leads to a hydrodynamic theory known as model A. The equation of motion is $$\partial_t M = -D \frac{\delta F}{\delta M}$$ which has eigenmodes of the form $\omega_k=iD(k^2+a)$. Usually the relaxation time is finite, but near the critical point $a\to 0$ and $\omega_k\sim ik^2$, so that modes with wave number $k\sim 1/\xi$ relax over a time $\tau \sim \xi^2$. Again, this is mean field and a more sophisticated analysis gives $\tau \sim \xi^z$ with a critical exponent z.