Application of projection operator method to coarse-grained dynamics with transient potential.

Area of Science:

• Statistical Mechanics
• Computational Physics
• Theoretical Chemistry

Background:

• The transient potential, a fluctuating potential, has been used phenomenologically in coarse-grained dynamics.
• Its connection to underlying microscopic dynamics was previously unclear, unlike the established generalized Langevin equation.
• Clarifying this relationship is crucial for validating and applying coarse-grained models.

Purpose of the Study:

• To rigorously derive the coarse-grained dynamics model featuring a time-dependent transient potential from microscopic Hamiltonian dynamics.
• To establish the theoretical foundation for using transient potentials in coarse-grained simulations.
• To explore approximations for simplifying the derived dynamic equations.

Main Methods:

• Derivation of transient potential dynamics from microscopic Hamiltonian dynamics for a coupled oscillator model.
• Application of the projection operator method to coarse-grained variables and the transient potential.
• Formal derivation of dynamic equations for the transient potential itself.

Main Results:

• The dynamic equations with the transient potential were derived without approximations for the coupled oscillator model.
• The transient potential dynamics were shown to be equivalent to a generalized Langevin equation with memory effects.
• Approximations were proposed, leading to simplified Markovian dynamic equations for potential parameters and Markovian Langevin-type equations.

Conclusions:

• The study provides a rigorous theoretical justification for employing transient potentials in coarse-grained dynamics.
• The derived model offers a viable alternative to the generalized Langevin equation for describing complex systems.
• The proposed approximations facilitate the development of computationally efficient coarse-grained models.