Splitting probabilities as optimal controllers of rare reactive events (2402.05414v3)

Abstract: The committor constitutes the primary quantity of interest within chemical kinetics as it is understood to encode the ideal reaction coordinate for a rare reactive event. We show the generative utility of the committor, in that it can be used explicitly to produce a reactive trajectory ensemble that exhibits numerically exact statistics as that of the original transition path ensemble. This is done by relating a time-dependent analogue of the committor that solves a generalized bridge problem, to the splitting probability that solves a boundary value problem under a bistable assumption. By invoking stochastic optimal control and spectral theory, we derive a general form for the optimal controller of a bridge process that connects two metastable states expressed in terms of the splitting probability. This formalism offers an alternative perspective into the role of the committor and its gradients, in that they encode forcefields that guarantee reactivity, generating trajectories that are statistically identical to the way that a system would react autonomously.

• The paper introduces a theoretical framework connecting the committor function with stochastic optimal control for simulating rare reactive events.
• It demonstrates that a time-dependent committor can generate driven processes that statistically mirror natural reaction pathways in both analytical and numerical models.
• The research offers practical insights for enhanced sampling in chemical systems, paving the way for improved simulation of metastable transitions.

Splitting Probabilities as Optimal Controllers of Rare Reactive Events

The paper explores a sophisticated approach to understanding and advancing controlled sampling processes in the context of chemical kinetics and reaction pathways. The primary focus is on the use of the committor, or splitting probability, as an ideal descriptor of reaction coordinates for rare events. This work bridges theoretical insights and practical methodologies, offering a robust framework for the generation of reactive trajectories in chemical systems characterized by complex reaction dynamics.

Core Contributions

1. Theoretical Framework: The paper establishes a correspondence between the committor function and stochastic optimal control. The committor, a fundamental quantity in transition path theory, is shown to encode force fields which, when applied, create driven processes statistically equivalent to naturally reactive pathways.
2. Generative Utility of the Committor: By demonstrating the committor's ability to reproduce ensemble statistics identical to the original transition paths, this work provides a concrete use-case of the committor as a generative model. The approach involves formulating a time-dependent analogue of the committor, addressing a generalized bridge problem through an optimal control formulation.
3. Stochastic Optimal Control Insight: Utilizing advances in stochastic optimal control theory and spectral analysis, the authors derive solutions to a boundary value problem under bistable assumptions. This framework underscores the potential of control-based sampling in examining rare reactive events, offering a statistical physics foundation for understanding reaction dynamics.

Numerical and Theoretical Insights

• Spectral Representation and Eigenvalue Analysis: The work is grounded in a spectral decomposition of the relevant stochastic processes, specifically the Fokker-Planck and backward Kolmogorov equations. The paper highlights the importance of the second eigenvalue in describing reaction rates for metastable transitions, aligning with classical rate theories by Kramers and others.
• Practical Validations: Through demonstrative applications to both analytically tractable models and complex systems like colloidal clusters, the approach is validated numerically. The results indicate near-perfect alignment between driven and natural pathways, with minor discrepancies attributed to numerical approximations.
• Role of Metastability and Boundary Conditions: The authors provide rigorous validation using Feynman-Kac theory, particularly for systems with clearly defined metastable states, ensuring reliable estimation of eigenfunctions and their role in bridging domains via splitting probabilities.

Implications and Future Directions

This work suggests several theoretical and practical implications for future research in computational chemistry and physics:

• Enhanced Sampling and Rare Event Simulations: The methodology can potentially transform computational strategies for simulating rare events, particularly in large-scale molecular systems, by providing a more efficient pathway sampling technique without exhaustive state-space exploration.
• Interdisciplinary Applications: While the focus is on chemical kinetics, the principles outlined have broader applications in statistical mechanics, machine learning, and dynamical systems, especially for problems involving complex energy landscapes and metastable systems.
• Pathway Prediction and Reaction Mechanisms: By leveraging splitting probabilities, researchers can gain insights into reaction pathways, potentially aiding in the design of novel materials and drugs through simulated exploration of reaction mechanisms.

In summary, this paper provides a rigorous analysis and an innovative perspective on utilizing committor functions within the field of controlled stochastic processes. The implications of this work extend beyond theoretical advancements, offering practical tools for addressing some of the most challenging problems in simulating and understanding the kinetics of rare events.