Role of Ito's lemma in sampling pinned diffusion paths in the continuous-time limit

Area of Science:

• Statistical Mechanics
• Physical Chemistry
• Computational Physics

Background:

• Particle diffusion in thermal equilibrium is crucial for understanding chemical reactions and material properties.
• The Onsager-Machlup (OM) functional is used to study rare transitions in diffusion processes.
• Existing OM theory relies on discrete-time Brownian dynamics, limiting its application to continuous systems.

Purpose of the Study:

• To explore the continuous-time limit of the Onsager-Machlup functional for diffusion paths.
• To identify and analyze numerical instabilities in simulating rare transition paths.
• To investigate the physical validity of the most probable diffusion path concept.

Main Methods:

• Utilizing the Onsager-Machlup functional as a path probability distribution function.
• Analyzing the continuous-time limit of the OM functional, involving infinite dimensionality.
• Employing the Ito-Girsanov change of measure for regularization.
• Developing and testing numerical algorithms for path sampling in multi-well potentials.

Main Results:

• Numerical schemes based on the regularized continuous-time OM functional exhibit catastrophic failures in multi-well potentials.
• These failures stem from numerical instabilities in the discrete approximation of the continuous-time path measure.
• Instabilities are linked to difficulties in satisfying Ito's lemma conditions in finite dimensions.

Conclusions:

• The most probable diffusion path is not a physically meaningful entity due to a flat thermodynamic action.
• Current numerical methods for simulating rare diffusion events require re-evaluation.
• Understanding the limitations of continuous-time approximations is vital for accurate modeling of diffusion processes.