Computation Of Essential Molecular Dynamics By Subdivision Techniques I: Basic Concept.pdf

Traditional methods like long-term trajectory simulations and Monte Carlo techniques have limitations. This work introduces a multilevel subdivision algorithm that directly computes these essential dynamics using eigenmodes of the Frobenius-Perron operator.

Key Points:

Problem: Accurately model and simulate molecular processes, particularly focusing on time averages of physical observables rather than individual trajectories.
Approach: Utilizes mathematical theory of dynamical systems to discretize the eigenvalue problem of the Frobenius-Perron operator associated with a molecular system's dynamics.
Advantages:

Doesn't rely on physical assumptions like ergodicity.
Only requires short-term trajectory simulations, not long-term runs.

Applications:

Predicting molecular motions over long time periods.
Understanding conformational changes in molecules.

Methodology:

1. Background:

Discusses Hamiltonian differential equations describing molecular systems.
Explores different approaches to modeling molecular dynamics (MD) including classical MD, ensemble averages via Monte Carlo, and forward/backward analysis of numerical discretizations.
Introduces the concept of "essential dynamics" as almost invariant sets within the phase space of a dynamical system.

2. Relating Molecular Systems to Dynamical Systems:

Shows that conformations of molecules can be mathematically represented as almost invariant sets in the phase space of a dynamical system.
Connects these almost invariant sets to eigenmeasures (eigenvectors with eigenvalue λ=1) of the Frobenius-Perron operator.

3. Discretization and Algorithm:

Proposes a subdivision algorithm based on techniques for hyperbolic systems, adapted to discretize the Frobenius-Perron eigenvalue problem.
This algorithm uses short-term trajectory simulations as building blocks.

4. Numerical Experiments:

Demonstrates the potential of the proposed method through preliminary numerical experiments.
* Compares its performance against other methods and highlights its advantages.

Future Work: A more advanced version of the algorithm will be presented in a subsequent paper.