(346bh) Exact Sampling of Stochastic Process from Brownian Bridge

Simulation of macromolecular processes towards the study of phase transitions is seriously deterred by extreme computational tortuosity. We report here a methodology based on the concept of a Brownian bridge that has the promise to contribute significantly to alleviate this problem. We discuss our approach with examples from the statistics of a continuous polymer chain in an external field, a canonical problem of the molecular processes that has tremendous applications in materials science and biology, and is fundamental in field theories describing the phase behavior of concentrated melts and solutions. Using a Brownian bridge (a stochastic process with its start point and end point specified), one can systematically (i) design polymer chains of a given topology, (ii) sample polymer chains in a complicated energy landscape, i.e., a landscape with metastable states separated by large free-energy barriers, and (iii) sample rare conformational states in phase space. Furthermore, one can extend this formalism to condition a polymer to lie a particular region of phase space for its entire path, not just the end points. We note that the bridge formalism is not hampered by restrictive assumptions that are characteristic of other approaches such as Monte Carlo methods and importance sampling techniques such as umbrella sampling. Further, it navigates the process along specific paths by continually employing corrective drifts to the stochastic trajectories. Thus, future applications may be envisaged for computing the so-called minimum free energy paths commissioning the use of the string method in collective variables to study phase transitions. Finally, we propose to solve the Brownian bridge in a high dimensional space for the modeling of complex polymer chains (e.g., semi-flexible polymers).