Generalized Bulgac-Kusnezov methods for sampling of the Gibbs-Boltzmann measure

A wide family of methods is described for sampling in the canonical ensemble. The Bulgac-Kusnezov method is generalized to include a more complicated coupling structure and stochastic perturbations. It is shown that a controlled fluctuation of the potential surface or force field in a molecular model may be used as part of a sampling method (instead of the more standard friction or driving term). Numerical experiments demonstrate that the family includes methods that are effective for recovering canonical averages.

Figure 1

(Color online) Graph of $\tilde{U}\left(q,\xi \right)=\sigma \left(\xi \right)U\left(q\right)$ where $U$ is a double well and $\sigma \left(\xi \right)={\left[\frac{2}{\pi }\text{arctan}\left(100{\xi }^2\right)\right]}^2$. The slice along $\xi =0$ represents the original potential, which is also a shift of the effective free energy.

Figure 2

(Color online) Trajectories produced by three thermostatting methods for the double well model: Nosé-Hoover-Langevin (top), Langevin dynamics (center), and embedded force method (lower).

Figure 3

(Color online) Sampling of the configurational density using each of the three methods. (NHL, top; Langevin, center; and EFM, lower) (Each method exhibits random error, even in 50 M steps, so precise comparisons based on these sample trajectories should not be inferred.)

Figure 4

(Color online) The occupancy of the right basin should be 1/2. Here we calculate the average occupancy time as a function of sampling steps. (NHL, top; Langevin, center; and EFM, lower) The mean is shown as a bold solid line, while the standard error bars are marked by dashed lines.