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. Maxwell-Boltzmann velocity distribution Partition function

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. Hamiltonian & Lagrangian The equations of motion give the path that starts at t 1 at position x(t 1 ) and end at t 2 at position x(t 2 ) for which the action (S) is the minimum t x t2t2 t1t1 S<S

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

12. Calculus of variation True path for which S is minimum η(t) should be such the δS is minimal At the boundaries: η(t 1 )=0 and η(t 2 )=0 η(t) is small

13. A description which is independent of the coordinates This term should be zero for all η(t) so […] η(t) If this term 0, S has a minimum Newton Integration by parts Zero because of the boundaries η(t 1 )=0 and η(t 2 )=0

14. This term should be zero for all η(t) so […] η(t) If this term 0, S has a minimum Newton A description which is independent of the coordinates

15. Integration by parts Zero because of the boundaries η(t 1 )=0 and η(t 2 )=0

16. Lagrangian Cartesian coordinates (Newton) → Generalized coordinates (?) Lagrangian Action The true path plus deviation

17. Desired format […] η(t) Conjugate momentum Equations of motion Should be 0 for all paths Partial integration Lagrangian equations of motion

18. Desired format […] η(t) Conjugate momentum Equations of motion Should be 0 for all paths

19. Partial integration

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

21. Pendulum Equations of motion in terms of l and θ Conjugate momentum

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

23. Legrendre transformation We have a function that depends on and we would like Example: thermodynamics We prefer to control T: S→T Legendre transformation Helmholtz free energy

24. Example: thermodynamics We prefer to control T: S→T Legendre transformation Helmholtz free energy

25. Hamiltonian Hamilton’s equations of motion

26. Newton? Conjugate momentum Hamiltonian

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

28. Nosé and thermodynamics Gaussian integral Constant plays no role in thermodynamics Recall MD MC Delta functions

29. Nosé and thermodynamics Gaussian integral Constant plays no role in thermodynamics

30. Recall MD MC

31. Delta functions

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

34. Nosé Hoover