Molecular Dynamics

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Molecular Dynamics

• Basics (4.1, 4.2, 4.3)
• Liouville formulation (4.3.3)
• Multiple timesteps (15.3)

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Molecular Dynamics

• Theory
• Compute the forces on the particles
• Solve the equations of motion
• Sample after some timesteps

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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

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Periodic boundary conditions
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Lennard Jones potentials

• The Lennard-Jones potential

• The truncated Lennard-Jones potential

• The truncated and shifted Lennard-Jones potential

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Phase diagrams of Lennard Jones fluids
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Saving cpu time
Cell list
Verlet-list
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Equations of motion
Verlet algorithm
Velocity Verlet algorithm
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Lyaponov instability
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Liouville formulation
Depends implicitly on t
Beware this solution is equally useless as the
differential equation!
Liouville operator
Solution
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Let us look at them separately
Taylor expansion
Shift of coordinates
Shift of momenta
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Call force(fx)
vxvxdeltfx/2
xxdeltvx Call force(fx)
vxvxdeltfx/2
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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!
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Multiple time steps

• What to use for stiff potentials
• Fixed bond-length constraints (Shake)
• Very small time step

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Multiple Time steps
Introduce dt?t/n
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First
Now n times
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Call force(fx_long,f_short)
vxvxdeltfx_long/2
vxvxddeltfx_short/2
xxddeltvx Call force_short(fx_short)
vxvxddeltfx_short/2
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