# Why does it become difficult to sample configuration during a phase change in Monte Carlo simulation?

I recently read in a paper that Monte Carlo integration is very slow near phase transitions. I couldn't understand the concept properly. The paper is Phys. Rev. B 95, 035105 (2017).

• I do not have access to the paper. Slow monte carlo would mean many more computations for a given final value. Hand waving, nearing a phase transition from disordered to ordered, the number of microstates in disorder is very large and increases until the critical point, so many more computations of the setup. see web.mit.edu/8.334/www/grades/projects/projects10/… – anna v Aug 9 '18 at 6:56

Near a first-order phase transition, converting one phase into the other typically involves crossing a free energy barrier. Even though the two phases may have the same free energy, the intermediate configurations will involve both phases present at the same time, which implies an interface between them. The contribution of the interfacial free energy will be significant, in general: proportional to the surface tension and the area of the interface (which approaches $\sim L^2$ in a simulation in a box of length $L$). So, crossing this barrier can be an activated process, with a rate $\propto \exp(-\Delta E^\dagger/kT)$, $T$ being the temperature and $k$ Boltzmann's constant, with $\Delta E^\dagger\propto L^2$. This will happen infrequently for large $L$.
Near a continuous phase transition one observes critical slowing down. Long-wavelength fluctuations in the appropriate order parameter (for example, the density in the case of the liquid-vapour critical point) occur. In real experiments, these can be observed, since they scatter light and the system becomes turbid. Very close to the phase transition, in a finite-sized system of length $L$, the wave length of the fluctuations approaches $L$. There is generally a relation between the time scale and the length scale involving a dynamical critical exponent; the bottom line is that the fluctuations become very slow.