Liouville formulation (4.3.3)

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Presentation transcript:

Liouville formulation (4.3.3) 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 Force calculations 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

Periodic boundary conditions

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

Phase diagrams of Lennard Jones fluids

Saving cpu time Cell list Verlet-list

Equations of motion Verlet algorithm Velocity Verlet algorithm

Lyaponov instability

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

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

We have noncommuting operators! Trotter identity

Velocity Verlet!

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

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!

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

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

First Now n times:

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