Monte Carlo: A Simple Simulator’s View

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Presentation transcript:

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