Data-driven parameterization of the generalized Langevin equation.

Area of Science:

• Statistical Mechanics
• Computational Physics
• Physical Chemistry

Background:

• Generalized Langevin equations (GLEs) are crucial for modeling complex systems with memory effects.
• Determining the memory kernel and stochastic noise accurately is essential for reliable simulations.
• Existing methods often face challenges in practical implementation and theoretical rigor.

Purpose of the Study:

• To develop a data-driven framework for parameterizing the memory kernel and stochastic noise in GLEs.
• To enable practical implementation through rational function approximation in the Laplace domain.
• To ensure the exact satisfaction of the second fluctuation-dissipation theorem.

Main Methods:

• Parameterizing the memory kernel using rational functions in the Laplace domain, linked to equilibrium statistics.
• Embedding the approximated GLE into an extended stochastic model without explicit memory.
• Introducing stochastic noise to precisely satisfy the second fluctuation-dissipation theorem.

Main Results:

• The proposed method allows for arbitrarily high-order approximations of the memory kernel.
• The generalized Langevin dynamics are successfully embedded into a memory-free extended stochastic model.
• Numerical tests confirm the effectiveness of the data-driven approach in accurately capturing system dynamics.

Conclusions:

• The presented data-driven approach offers a robust and practical method for analyzing systems described by GLEs.
• This framework facilitates accurate simulations by ensuring the correct statistical properties and fluctuation-dissipation relations.
• The method is broadly applicable to various fields utilizing generalized Langevin dynamics.