(501f) Brownian Bridge Method for Studying Rare Events – an Approximation Scheme

A stochastic bridge is a continuous random walk that ends in a given region, stays in a region for its entire path, or reaches one region before another region. This concept has been used in control theory to guide random processes to end in a given state, but has not been used in chemistry although it could have wide ranging implications for polymer physics and molecular simulations. This talk will formulate numerical schemes to generate stochastic bridges for a wide range of chemistry applications where traditional sampling techniques are inefficient due to the intrinsic issues such as trial-and-error and loss of information. Our previous work [1, 2] studied a continuous polymer chain in an external field -- a canonical problem in all polymer field theories. One can demonstrate that a stochastic bridge can sample conformations that end with a given topology or end with a range of final energies, the latter of which is important for rare event sampling.

In this work, an approximation method will be developed to extend the stochastic bridge formalism to more complex systems and higher dimensional problems. In our past work [2], a stochastic bridge was created by calculating a drift that constrained random walks to the correct regions of phase space. This drift was determined by a solution of a Backwards Fokker Planck (BFK) equation, which is infeasible for complicated systems. We will show how to systematically generate approximations to the drift without solving the BFK equation, and how to correct (i.e., re-weight) any errors incurred from this approximation.

Using asymptotic properties of the Backwards Fokker Planck equation for different applications, we will develop approximations for the hitting probability needed to develop a stochastic bridge. We will then develop a novel procedure to reweight the approximations so that one is guaranteed to generate the correct conditional statistics. We accomplish this goal by applying the approximation method to one dimensional Wiener processes (i.e. Ornstein Ulenbeck process and Geometric Brownian motion). Next, the approach will be generalized for high dimensional systems. The main outcome of this talk will be to show that one can now generate a stochastic bridge without knowing the solution to a high dimensional Fokker Planck equation, which will allow one to apply this methodology to a wide range of problems. We outline many applications that will be served by this development.

[1] S. Krishnaswami, D. Ramkrishna, and J. M. Caruthers, “Statistical-mechanically exact simulation of polymer conformation in an external field,” The Journal of Chemical Physics, vol.107, no. 15, pp. 5929–5944, 1997.

[2] S. Wang, D. Ramkrishna, and V. Narsimhan, “Exact sampling of polymer conformations using Brownian bridges,” The Journal of Chemical Physics, vol. 153, no. 3, p. 034901, 2020.