Position-dependent memory kernel in generalized Langevin equations: Theory and numerical estimation.

Area of Science:

• Statistical Mechanics
• Computational Physics
• Physical Chemistry

Background:

• Generalized Langevin equations model complex systems in molecular dynamics.
• Non-linear forces and position-dependent friction kernels are crucial for accurate simulations.
• The Mori-Zwanzig formalism provides a rigorous framework for deriving such equations.

Purpose of the Study:

• To rigorously derive generalized Langevin equations with specific complexities.
• To establish the relationship between noise and memory kernel via a fluctuation-dissipation theorem.
• To develop a method for numerical parametrization of these models.

Main Methods:

• Application of the Mori-Zwanzig formalism.
• Derivation of generalized Langevin equations.
• Analysis of fluctuation-dissipation relations.
• Formulation of Volterra-type equations for memory kernels.

Main Results:

• Rigorous derivation of generalized Langevin equations with non-linear forces and position-dependent friction kernels.
• Demonstration of a fluctuation-dissipation theorem connecting noise properties to the memory kernel.
• Obtained Volterra equations enabling numerical parametrization from all-atom simulations.

Conclusions:

• The Mori-Zwanzig formalism successfully derives complex generalized Langevin equations.
• A fundamental fluctuation-dissipation theorem is established for these systems.
• The derived Volterra equations offer a practical approach for model parameterization in molecular dynamics.