Brownian bridges for stochastic problems in chemical sciences

Purdue University, West Lafayette IN

Investigators

Abstract

Continuous random walks – noisy processes that drift in time – are omnipresent in a wide range of chemical fields such as in studying polymer molecule dynamics, identifying chemical reaction pathways, and quantifying heat and mass transfer processes at the molecular scale. In many practical situations, one is interested in examining random walks whose paths start and end within specified sets of values that represent distinct chemical species or molecule configurations. Such ideas would allow one to quantify rare events in a chemical process, or conversely, quantify the most probable configurations and reaction pathways. One promising methodology to systematically generate these random walks is to make use of a concept known as a stochastic bridge, an idea that has been used in chemical process control theory to guide noisy processes to safe and profitable end states. However, this idea has yet to be adapted to applications in chemistry despite its potential for a wide range of applications in polymer physics and molecular simulations. This proposal formulates stochastic bridges for two chemistry applications where traditional techniques to account for the noise are computationally inefficient. The first application considers the growth of semi-flexible polymer chains of a specific configuration, which is useful when studying DNA molecule behavior during biochemical processes. The second application focuses on efficiently examining different reaction pathways during crystallization, which is vital for manufacturing pharmaceuticals. The investigators will disseminate their results through national conferences and will provide science demonstrations to a local science museum that will revolve around the theme of random walks. To computationally study the wide range of molecular-scale chemical processes with dynamics governed by continuous random walks, two primary research objectives will be pursued by the research team. The first objective is to develop highly efficient numerical techniques for stochastic bridges that condition continuous random walks to end in a specified region, stay in a given region, or reach one region before another. Currently, all stochastic bridges are created by solving a Backwards Fokker Planck (BFP) equation, a partial differential equation (PDE), and then using this solution to compute an effective drift that guides paths towards the desired region of phase space. The largest roadblock behind this approach occurs when one cannot readily compute the PDE solution exactly, which is the case for complex or high dimensional systems. The first objective will use the asymptotic properties of the BFP equation to generate an approximate drift that guides random walks to the correct regions of phase space, and determine ways to correct (i.e., re-weight) any errors incurred from this approximation. This procedure will create a computationally efficient method to generate conditioned random walks that can scale to higher dimensions and can be used for complex molecular systems. The second objective will employ these ideas in two applications. The first example will focus on simulating the dynamics of a continuous polymer chain in an external field - a canonical problem in all polymer field theories. The proposed research program will demonstrate that a stochastic bridge can effectively sample polymer 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. The second example will consider nucleation pathways during crystallization and will show that a bridge can efficiently sample paths in a free energy landscape that reach one crystal conformation (polymorph) before others, ultimately making it possible to control selectivity between crystal product polymorphs. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.