Topics in Statistic Mechanics

Adv. Sta. Phy. Homework 8 Markovian-chain Monte Carlo Li,Zimeng PB06203182
Markovian-chain Monte Carlo
Edited by Li, Zimeng
Contents:
1.Introduction
1.1 Simple Sampling
1.2 Importance Sampling
2.Markov Process
3.MCMC application on 2D Ising model
First, we have learned the MCMC method in Computational Physics, therefore I would
only give a brief review of the method and, second, an extension to the 2D Ising model.
1.Introduction
Partition functions and functional integrals (path integrals) reduce many-body problems
to complicated multidimensional integrals or sums.
The Monte Carlo technique has its origins in the numerical evaluation of integrals. The
technique was later generalized to calculate the partition function and the mean value
of observables in classical systems.
The key of numerical integration is sampling. There are many sampling methods in
Monte-Carlo method. We will only introduce two of them here.
1.1 Simple Sampling
We first generate random numbers
probability density of these points:
uniformly distributed in [a,b], and get the
We can calculate the expectation value of a function f with respect to the distribution P:
E(f(x))=
In order to calculate the above integration I in a numerical way, we define the Monte-
Carlo Integral