Advanced Molecular Dynamics

1. Advanced Molecular Dynamics Velocity scaling Andersen Thermostat Hamiltonian & Lagrangian Appendix A Nose-Hoover thermostat

2. Naïve approach Velocity scaling Do we sample the canonical ensemble?

3. Partition function Maxwell-Boltzmann velocity distribution

4. Fluctuations in the momentum: Fluctuations in the temperature

5. Andersen thermostat Every particle has a fixed probability to collide with the Andersen demon After collision the particle is give a new velocity The probabilities to collide are uncorrelated (Poisson distribution)

7. Velocity Verlet:

8. Andersen thermostat: static properties

9. Andersen thermostat: dynamic properties

10. x t1 t2 t Hamiltonian & Lagrangian The equations of motion give the path that starts at t1 at position x(t1) and end at t2at position x(t2) for which the action (S) is the minimum S<S S<S

11. Example: free particle Consider a particle in vacuum: v(t)=vav Always > 0!! η(t)=0 for all t

12. Cartesian coordinates (Newton) → Generalized coordinates (?) S[q+η] = S[q] Lagrangian Lagrangian Action The true path plus deviation

13. S[q+η] = S[q] Should be 0 for all paths Equations of motion Lagrangian equations of motion Conjugate momentum

14. Newton? Valid in any coordinate system: Cartesian Conjugate momentum

15. Lagrangian dynamics We have: 2nd order differential equation Two 1st order differential equations With these variables we can do statistical thermodynamics Change dependence:

16. Hamiltonian Hamilton’s equations of motion

17. Newton? Conjugate momentum Hamiltonian

18. Lagrangian Nosé thermostat Hamiltonian Extended system 3N+1 variables Associated mass Conjugate momentum

19. Nosé and thermodynamics Recall MD MC Gaussian integral Constant plays no role in thermodynamics

20. Lagrangian Equations of Motion Hamiltonian Conjugate momenta Equations of motion:

21. Nosé Hoover