Illinois MatSE485/Phys466/CSE485 - Atomic-Scale Simulation
27 Jan 2009 | Courses | Contributor(s): David M. Ceperley
THE OBJECTIVE is to learn and apply fundamental techniques used in (primarily classical) simulations in order to help understand and predict properties of microscopic systems in materials science, physics, chemistry, and biology.
THE EMPHASIS will be on connections between the simulation results and real properties of materials (structural or thermodynamic quantities), as well as numerical algorithms and systematic and statistical error estimations
FOR WHOM? This class is oriented for the ...
Illinois PHYS 466, Lecture 10: Sampling
20 Mar 2009 | Online Presentations | Contributor(s): David M. Ceperley, Omar N Sobh
Fundamentals of Monte Carlo
What is Monte Carlo?
Named at Los Alamos in 1940’s after the casino.
Any method which uses (pseudo)random numbers> as an essential part of the algorithm.
Stochastic - not deterministic!
A method for doing highly dimensional integrals by sampling the integrand.
Often a Markov chain, called Metropolis MC.
Simple example: Buffon’s needle - Monte Carlo determination of π
MC is advantageous for high dimensional integrals -the best ...
Illinois PHYS 466, Lecture 11: Importance Sampling
20 Mar 2009 | Online Presentations | Contributor(s): David M. Ceperley
Today We will talk about the third option: Importance sampling and correlated sampling
Finding Optimal p*(x) for Sampling
Example of importance sampling
What are allowed values of a?
What does infinite variance look like?
General Approach to Importance Sampling
Sampling Boltzmann distribution
Independent Sampling for exp(-V/kT)?
This presentation was breezed and uploaded by Omar ...
Illinois PHYS 466, Lecture 12: Random Walks
30 Mar 2009 | Online Presentations | Contributor(s): David M. Ceperley
Today we will discuss Markov chains (random walks), detailed balance and transition rules.
These methods were introduced by Metropolis et al. in 1953
who applied it to a hard sphere liquid.
It is one of the most powerful and used algorithms
Equation of State Calculations by Fast Computing Machines
Markov chain or Random Walk
Properties of Random Walk
Random Walks Example from A&T 110-123
What is probability of being up on the second ...
Illinois PHYS 466, Lecture 13: Brownian Dynamics
08 Apr 2009 | Online Presentations | Contributor(s): David M. Ceperley
Let’s explore the connection between Brownian motion and Metropolis Monte Carlo. Why?
Connection with smart MC
Introduce the idea of kinetic Monte Carlo
Get rid of solvent degrees of freedom and have much longer time steps.
Local Markov process
General Form of Evolution
Generalized Trotter Formula
Evaluation of diffusion term
Green’s function for a gradient
Summary of Brownian Dynamics