Umbrella sampling method

The umbrella sampling method is very efficient in sampling low probability events [1,2]. Suppose we need to compute the thermodynamic average of some quantity $A$. Let $<\cdot>$ denote the ensemble average, then

$$<A> = \frac{Tr(\rho A)}{Tr(\rho)}$$ (1)

Where $\rho$ is the density operator, and $Tr(\cdot)$ is the trace. The density operator assumes different form in different ensembles, for example, in microcanonical ensemble, $\rho(E) = \delta(E - E_{0})$; In canonical ensemble, $\rho(E) = g(E)e^{-\beta E}$, where $g(E)$ is the density of states. Suppose we generated a series of configurations in some ensemble(say NVT) according to the appropriate distribution $\rho(E) = e^{-\beta E}$, we can calculate the thermodynamic average of $A$ by using

$$<A> = \frac{1}{n}\sum_{i = 1}^{n}A_{i}$$ (2)

here $A_{i}$ is the value of $A$ calculated based on each configuration. If the probability density for some values of $A$ is low, it is hard to get a good sampling using conventional methods. For instance, if we want to sample the energy of a system in the canonical ensemble, we know that the probability of $E$ is peaked roughly at its mean value $<E>$. Configurations with very low or very high energies are hardly sampled using Metropolis algorithm, which invalidates the final result. To circumvent this, we can use the umbrella sampling technique. The idea is to introduce a weight function $w$ to enhance the probability in the low probability regions, or to force the system to stay in some particular regions in the phase space. With the weight function, the density operator is now $\rho w$, and the average in the unbiased case can be calculated as

$$<A> = \frac{Tr(\rho wA/w)}{Tr(\rho w/w)} = \frac{Tr(\rho wA/w)}{Tr(\rho w)}\frac{Tr(\rho w)}{Tr(\rho w/w)} = \frac{<A/w>_{w}}{<1/w>_{w}}$$ (3)

From this we can see that if the weight function overlaps both with $\rho$ and $A$, we get a reasonable sampling of $A$. Now if the configurations are generated according to the new density operator $\rho w$, the average of $A$ is

$$<A> = \frac{\frac{1}{n}\sum_{i = 1}^{n}\frac{A_{i}}{w_{i}}}{\frac{1}{n}\sum_{i = 1}^{n}\frac{A_{i}}{w_{i}}}$$ (4)