Download Monte Carlo: A Simple Simulator`s View

Monte Carlo: A Simple Simulator’s View
(I use MC regularly and irregularly,
sometimes it’s the only thing that works)
James Grant
MCMC Anonymous
2nd November 2016
Monte Carlo
Numerical method
Uses (pseudo) random numbers
Solve integrals
Sample configurational space
Fitting/Optimisation problems
Buffon’s Needle
"Pi 30K" by CaitlinJo - Own work. This mathematical image was created with Mathematica. Licensed under CC BY
3.0 via Commons - https://commons.wikimedia.org/wiki/File:Pi_30K.gif#/media/File:Pi_30K.gif
Statistical Mechanics
Partition Function:
E(r) is the energy of a configuration
r are the coordinates of the N particles/atoms
T is the temperature, k Boltzmann’s constant
dr is over the 3N dimensional, configurational space
The integral is over all configurations and if known Z
contains all information about the system
Statistical Mechanics
Weight of a particular configuration:
Relative weight of configurations α and β:
Monte Carlo Move(s)
Monte Carlo Move(s)
Monte Carlo Move(s)
Monte Carlo Move(s)
Monte Carlo Move(s)
Correct Sampling
Metropolis Algorithm (other algorithms are available):
Detailed balance:
Sample in accordance with Boltzmann distribution:
Subtleties
Equilibration time, ‘burn in?’
The initial configuration may not be a ‘likely one’, how
long does the system take to relax?
How long must a simulation run to ensure it has visited
sufficient states?
If there are two (or more) regions of configurational
space which are likely, but separated by unlikely
regions, how do you ensure both are sampled correctly.
E.g. Crystal structures, molecule conformations