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

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

12. Calculus of variation At the boundaries: η(t) is small
η(t1)=0 and η(t2)=0
η(t) is small
True path for which S is minimum
η(t) should be such the δS is minimal

13. A description which is independent of the coordinates
This term should be zero for all η(t) so […] η(t)
Integration by parts
If this term 0, S has a minimum
Zero because of the boundaries η(t1)=0 and η(t2)=0
Newton
A description which is independent of the coordinates

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

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

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

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

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

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

20. Hamilton’s equations of motion
Hamiltonian
Hamilton’s equations of motion

21. Newton?
Conjugate momentum
Hamiltonian

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

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

24. Equations of Motion Lagrangian Hamiltonian Conjugate momenta

25. Nosé Hoover