1. Molecular Dynamics(MD)
운동시뮬레이션 제13주
Molecular Dynamics(MD)

2. Introduction Multi-particle system with collisions Phase transition
Temperature concept
Mean free path
Collision cross section
Lenard-Jones potential
Verlet method

3. 2D case Classical Newtonian force law
𝑑 𝑣 𝑖,𝑥 𝑑𝑡 = 𝑎 𝑖,𝑥 , 𝑑 𝑥 𝑖 𝑑𝑡 = 𝑣 𝑖,𝑥
𝑑 𝑣 𝑖,𝑦 𝑑𝑡 = 𝑎 𝑖,𝑦 , 𝑑 𝑦 𝑖 𝑑𝑡 = 𝑣 𝑖,𝑦
Euler method is not appropriate, since we need very long time evolution

4. Verlet method 𝑑 2 𝑦 𝑑 𝑡 2 = 𝑓 2 (𝑦,𝑡)
𝑑 2 𝑦 𝑑 𝑡 2 = 𝑓 2 (𝑦,𝑡)
𝑦 𝑡 𝑖 +∆𝑡 =𝑦 𝑡 𝑖 + 𝑑𝑦 𝑑𝑡 ∆𝑡 𝑑 2 𝑦 𝑑 𝑡 2 ∆𝑡 𝑑 3 𝑦 𝑑 𝑡 3 ∆𝑡 3 +⋯
𝑦 𝑡 𝑖 −∆𝑡 =𝑦 𝑡 𝑖 − 𝑑𝑦 𝑑𝑡 ∆𝑡 𝑑 2 𝑦 𝑑 𝑡 2 ∆𝑡 2 − 𝑑 3 𝑦 𝑑 𝑡 3 ∆𝑡 3 +⋯
𝑦 𝑡 𝑖 +∆𝑡 +𝑦 𝑡 𝑖 −∆𝑡 =2𝑦 𝑡 𝑖 + 𝑑 2 𝑦 𝑑 𝑡 2 ∆𝑡 2 +𝑂 ∆𝑡 4
𝑦 𝑖+1 =2 𝑦 𝑖 − 𝑦 𝑖−1 + 𝑑 2 𝑦 𝑑 𝑡 2 ∆𝑡 2 +𝑂 ∆𝑡 4
𝑦 𝑖+1 ≈2 𝑦 𝑖 − 𝑦 𝑖−1 + 𝑓 2 ( 𝑦 𝑖 , 𝑡 𝑖 ) ∆𝑡 2
𝑣 𝑖 ≈ 𝑦 𝑖+1 − 𝑦 𝑖−1 / 2∆𝑡

5. Lenard-Jones potential
Large separation : Van der Waals attraction
Small separation : repulsion with the overlap of electron cloud
𝑉 𝑟 =4𝜖 𝜎 𝑟 12 − 𝜎 𝑟 6
𝐹=− 𝜕𝑉 𝜕𝑟 = 24𝜖 𝑟 2 𝜎 𝑟 12 − 𝜎 𝑟 6
𝐹 𝑟 𝑚𝑖𝑛 =0, 𝑟 𝑚𝑖𝑛 = 2 1/6 𝜎=1.122𝜎

6. Lenard-Jones potential

7. Lenard-Jones Potential
𝑓 𝑟 =− 𝜕𝑉 𝜕𝑟

8. Periodic Boundary condition
Cu 2D plane simulation

9. Calculate the new position and velocity
MD algorithm
Set Initial condition
Calculate the force
Calculate the new position and velocity No
End?
Update variables
Yes
End

10. Initial positions All atoms are placed near lattice point
All atoms has random velocity with arbitrary direction
Ising model에서는 스핀의 위치가 바뀌지 않았지만 MD에서는 원자의 위치가 변한다.

11. Calculation 𝑥 𝑛𝑒𝑤 𝑖 =2 𝑥 𝑐𝑢𝑟𝑟 𝑖 − 𝑥 𝑝𝑟𝑒𝑣 𝑖 + 𝑎 𝑖,𝑥 ∆𝑡 2
𝑥 𝑛𝑒𝑤 𝑖 =2 𝑥 𝑐𝑢𝑟𝑟 𝑖 − 𝑥 𝑝𝑟𝑒𝑣 𝑖 + 𝑎 𝑖,𝑥 ∆𝑡 2
𝑦 𝑛𝑒𝑤 𝑖 =2 𝑦 𝑐𝑢𝑟𝑟 𝑖 − 𝑦 𝑝𝑟𝑒𝑣 (𝑖)+ 𝑎 𝑖,𝑦 ∆𝑡 2
𝑎 𝑖,𝑥 = 𝑘≠𝑖 𝑓 𝑘,𝑖 cos 𝜃 𝑘,𝑖
𝑎 𝑖,𝑦 = 𝑘≠𝑖 𝑓 𝑘,𝑖 sin 𝜃 𝑘,𝑖
𝑓 𝑘,𝑖 = 𝑟 𝑘,𝑖 13 − 1 𝑟 𝑘,𝑖 7
𝑟 𝑘,𝑖 = 𝑥 𝑘 − 𝑥 𝑖 𝑦 𝑘 − 𝑦 𝑖 2

12. Periodic Boundary Condition
가까운 거리만 계산한다.
같은 위치임

13. Argon2D 실행 결과

14. Velocity distribution
Depends on temperature
𝑃 𝑣 =𝐶 𝑣 2 𝑘 𝐵 𝑇 𝑒 −𝑚 𝑣 2 /2 𝑘 𝐵 𝑇
Equipartition principle
𝑘 𝐵 𝑇= 𝑚 2 𝑣 𝑥 2 + 𝑣 𝑦 2

15. The melting transition
Temperature increase is simulated by increasing velocity
𝑟 𝑃 → 𝑟 𝑐 −𝑅 𝑟 𝑐 − 𝑟 𝑃 , increase 𝑅>1, decrease 𝑅<1
Melting can be decided by checking 𝑟 2 behavior

16. Fermi-Pasta-Ulam problem

17. Equation of motion 𝑚 𝑖 𝑑 2 𝑥 𝑖 𝑑 𝑡 2 =𝑓 𝑥 𝑖 − 𝑥 𝑖−1 +𝑓 𝑥 𝑖 − 𝑥 𝑖+1
𝑚 𝑖 𝑑 2 𝑥 𝑖 𝑑 𝑡 2 =𝑓 𝑥 𝑖 − 𝑥 𝑖−1 +𝑓 𝑥 𝑖 − 𝑥 𝑖+1
𝑓 𝛼 ∆𝑥 =−𝐾 ∆𝑥 −𝛼 ∆𝑥 2
𝑓 𝛽 ∆𝑥 =−𝐾 ∆𝑥 −𝛽 ∆𝑥 3