Monte Carlo Simulation

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Monte Carlo Simulation
Wednesday, 9/11/2002

• Ensemble sampling
• Markov Chain
• Metropolis Sampling

Stochastic simulations consider particle
interactions.
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Deterministic vs. Stochastic
Newtons equation of motion
F m a
Random Walk
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Brownian motion
n 200 s .02 x rand(n,1)-0.5 y
rand(n,1)-0.5 h plot(x,y,'.') axis(-2 2 -2
2) axis square grid off set(h,'EraseMode','xor','
MarkerSize',18) while 1 drawnow x x
srandn(n,1) y y srandn(n,1) set(h,'XData
',x,'YData',y) end
Animations
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Lennard-Jones Potential
force
potential
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Measuring elastic constants
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Replace the time average with ensemble average
An ensemble is a collection of systems. The
probability for a system in state s is Ps.
If you average the velocity of one molecule of
the air in your room, as it collides from one
molecule to the next, that average comes out the
same as for all molecules in the room at one
instant.
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Thought experiment
Lets pretend that our universe really is
replicated over and over -- that our world is
just one realization along with all the others.
We're formed in a thousand undramatic day-by-day
choices.
Parallel universes ensembles
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Canonical Ensemble
Fixed number of atoms, system energy, and system
volume.
Partition function
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Finite number of microstates
Importance sampling
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Metropolis Sampling I
1. Current configuration C(n)
2. Generate a trial configuration by selecting an
atom at random and move it.
3. Calculate the change in energy for the trial
configuration, DU.
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Metropolis Sampling II
If DU lt 0, accept the move, so that the trial
configuration becomes the (n1) configuration,
C(n1). If DU gt 0, generate a random number r
between 0 and 1 If r lt exp( -DU/kBT ), accept
the move, C(n1) C(t) If r gt exp( -DU/kBT ),
reject the trial move. C(n1) C(n). A sequence
of configurations can be generated by using the
above steps repeatedly. Properties from the
system can be obtained by simply averaging the
properties of a large number of these
configurations.
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Markov Chain
A sequence X1, X2, , of random variable is
called Markov if, for any n,
i.e., if the conditional distribution F of Xn
assuming Xn-1, Xn-2, , X1 equals the conditional
distribution F of Xn assuming of only Xn-1.
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Markov Process
Dart hit-or-miss
Random Walk (RW) Self-Avoiding Walk (SAW) Growing
Self-Avoiding Walk (GSAW)
Diffusion Limited Aggregation
http//apricot.polyu.edu.hk/lam/dla/dla.html