1. 1 Announcements lecture slides online wireless account lifetime group photo Monday morning 1

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3. 33 Molecular Dynamics Theory: Compute the forces on the particles Solve the equations of motion Sample after some timesteps

4. 4 MD Sampling 4

5. 5 Ergodicity 5

6. 66 Lyaponov instability

7. 7 MD Sampling 7

8. 8 Ergodicity bottlenecks time scale representability Congo

9. 9 short MD movie 9 Ergodicity bottlenecks time scale representability

10. 10 Molecular Dynamics Basics (4.1, 4.2, 4.3) Liouville formulation (4.3.3) Multiple timesteps (15.3)

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15. 15 Molecular Dynamics Initialization –Total momentum should be zero (no external forces) –Temperature rescaling to desired temperature –Particles start on a lattice Force calculations –Periodic boundary conditions –Order NxN algorithm, –Order N: neighbor lists, linked cell –Truncation and shift of the potential Integrating the equations of motion –Velocity Verlet –Kinetic energy

16. 16 Periodic boundary conditions

17. 17 Lennard Jones potentials The truncated and shifted Lennard-Jones potential The truncated Lennard-Jones potential The Lennard-Jones potential

18. 18 Phase diagrams of Lennard Jones fluids

19. 19 Saving cpu time Cell listVerlet-list

20. 20 Equations of motion Verlet algorithm Velocity Verlet algorithm

21. 21 Liouville formulation Depends implicitly on t Liouville operator Solution Beware: this solution is equally useless as the differential equation!

22. 22 Shift of coordinatesShift of momenta Let us look at them separately Taylor expansion

23. 23 We have noncommuting operators! Trotter identity

24. 24 Velocity Verlet!

25. 25 vx=vx+delt*fx/2 x=x+delt*vx Call force(fx) vx=vx+delt*fx/2 Call force(fx) Do while (t<tmax) enddo Velocity Verlet:

26. 26 Liouville Formulation Velocity Verlet algorithm: Three subsequent coordinate transformations in either r or p of which the Jacobian is one: Area preserving Other Trotter decompositions are possible!

27. 27 Multiple time steps What to use for stiff potentials: –Fixed bond-length: constraints (Shake) –Very small time step

28. 28 Multiple Time steps Trotter expansion : Introduce: δt=Δt/n

29. 29 Now n times : First

30. 30 vx=vx+ddelt*fx_short/2 x=x+ddelt*vx Call force_short(fx_short) vx=vx+ddelt*fx_short/2 Call force(fx_long,f_short) Do ddt=1,n enddo vx=vx+delt*fx_long/2

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