# lecture12 - Molecular-Dynamics Simulation E q u ilib r iu m...

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Molecular-Dynamics Simulation Equilibrium Simulation Prof. Patrick Schelling Department of Physics University of Central Florida Florida Materials Simulators Meeting and Workshop, May 8-9, 2006

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Some good references… 1. “Computer Simulation of Liquids”, Allen and Tildesley 2. “The Art of Molecular Dynamics Simulation”, Rapaport
r F i = m i d 2 r r i dt 2 MD simulation Integrate to obtain ! r r i ( t )

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r F i = m i d 2 r r i dt 2 Is that all there is to it? Simulation can in some sense be regarded as intermediate between experiment and theory • Need a model for forces, interactions • Provides a test of theoretical predictions • Comparison to experiment tests models and theory
What kinds of problems? • Complex systems where theory is difficult • Many particles • Sample a large number of configurations • Statistical averaging A eq = d " P ( " ) A ( " ) # Γ represents a point in a 6N dimensional phase space (coordinates and momenta of N particles)

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• Classical dynamics result in Γ (t) • Time averages can be made instead of ensemble averages Connection to molecular dynamics A time = 1 t A ( " ( t ')) dt ' 0 t # If averaging time is long enough the system should have time to sample the phase space and then, A time " A eq
Ensembles A eq = d " P ( " ) A ( " ) # What configurations Γ does the simulation sample? In other words, what is P( Γ )? The two most common are : • Microcanonical ensemble (constant energy) • Canonical ensemble (constant temperature) P ( " ) = # ( E \$ H ( " )) P ( " ) = A exp # E ( " ) k B T \$ % ( )

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r F i = m i d 2 r r i dt 2 d 2 r r i dt 2 " r r i ( t + # t ) \$ 2 r r i ( t ) + r r i ( t \$# t ) # t 2 r r i ( t + " t ) = 2 r r i ( t ) # r r i ( t #" t ) + " t 2 m i r F i ( t ) Verlet Algorithm: numerical integration More complicated methods like the Gear predictor- corrector algorithm exist, but Verlet works pretty well
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lecture12 - Molecular-Dynamics Simulation E q u ilib r iu m...

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