1 Introduction Molecular Dynamics Method: Theory and Implementation Department of Applied Physics and Physico-Informatics, Keio University Hiroshi Watanabe

2 Molecular Dynamics Simulation Integrate equations of motion numerically • Calculate Force • Update momenta • Update position

3 q = 0.0 p = 1.0 h = 0.01 t = 0.0 for i in range(1000): q += p * h p -= q * h t += h print(t, q, p) Python code of MD

4 Particle methods ARE NOT MD. Governing Eq. MD Equations of Motion SPH Navier-Stokes Eq. CFD Navier-Stokes Eq. Specification Eulerian Lagrangian Lagrangian

5 Range Force Ab Initio (DFT) Classical Short-Range (LJ, WCA, etc) Long-Range (Coulomb, Gravity etc)

6 Boiling from Wikipedia from Wikipedia Cavitation Bubbles are important. • Bubbles affect efficiency of power plants. • Bubbles cause noise and damage on propeller. Understand behavior of bubbles and control them.

7 Spontaneous Phase Transition Stable for creating/annihilation of Phase Boundary Size of Simulation is highly limited. ~ um^3

8 Bubbles Gas/Liquid Surface Atoms Micro ~nm Macro ~cm Huge-scale Molecular Dynamics Simulation Bubbles are difficult. • Multi-scale and multi-physics problem • Moving, creation, and annihilation of gas-liquid surface Investigate the behavior of bubbles from the atomic scale

9 F = ma Very simple Easy to understand

10 F = ma Very simple Easy to understand Really?

11 What is temperature? What is pressure? Observables Time Evolution What is time evolution? What is time integration?

12 Extended Hamiltonian Methods Andersen’s method to control pressure Nose’s method to control temperature How can we control extensive variables? What is physical meaning of an extended system? We have the least action principle for original dynamics. What we have for the extended dynamics?

13 Variable We KNOW variables a priori. Used without definition. Observable We DEFINE observables. Defined by using variables.

14 Heat Equation = ∆ Temperature is Variable Navier-Stokes Equation Ԧ = − + ∆ Ԧ Pressure is Variable Equation of Motion ሶ = − , ሶ = Temperature is Observable Pressure is Observable

15 Equation of Motion ሶ = − , ሶ = Temperature is Observable Pressure is Observable Momenta are Variables. Coordinates are Variables. Observables should be defined by Variables.

16 We usually adopt the following temperature. = 2 3 ෍ 2 2 We usually adopt the following pressure. = 2 3 + 1 3 ෍ < ∙ …But Why?

17 We usually use Symplectic Integrators. Symplectic Integrator is better than other scheme. … But Why? Can we apply SI for extended systems?

18 We want to control pressure and/or temperature. We can control pressure by Andersen’s method. We can control temperature by Nose’s method. … But Why? Why can we control pressure and/or temperature? What happen in the extended system? Extended System

19 We consider the fundamental aspects of MD. We will consider • What is the time evolution? • What is the time integration? • What is pressure? How can we control? • What is temperature? How can we control?

20 We consider the implementation of MD. We will consider • Memory optimization • SIMD vectorization • Programming Design

21 0. Introduction 1. Classical Mechanics 2. Pressure 3. Temperature 4. Numerical Integration 5. Nose-Hoover method 6. Langevin Thermostat 7. Integration scheme for non-Hamiltonian systems 8. Generalized Liouville Theorem 9. Implementation and Optimization 10. Programming Design Theory Implementation

22 https://kaityo256.github.io/md2019/ "md2019" Lecture Note (written in Japanese)