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Molecular Dynamics

Molecular Dynamics. Basics (4.1, 4.2, 4.3) Liouville formulation (4.3.3) Multiple timesteps (15.3). Molecular Dynamics. Theory: Compute the forces on the particles Solve the equations of motion Sample after some timesteps. Molecular Dynamics. Initialization

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Molecular Dynamics

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

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

  3. 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

  4. Periodic boundary conditions

  5. Lennard Jones potentials • The Lennard-Jones potential • The truncated Lennard-Jones potential • The truncated and shifted Lennard-Jones potential

  6. Phase diagrams of Lennard Jones fluids

  7. Saving cpu time Cell list Verlet-list

  8. Equations of motion Verlet algorithm Velocity Verlet algorithm

  9. Lyaponov instability

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

  11. Let us look at them separately Taylor expansion Shift of coordinates Shift of momenta

  12. We have noncommuting operators! Trotter identity

  13. Velocity Verlet!

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

  15. 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!

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

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

  18. First Now n times:

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

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