Topics in Statistic Mechanics

Adv. Sta. Phy. Homework 8 Markovian-chain Monte Carlo Li,Zimeng PB06203182
However, P cannot not be easily solved on many conditions. We would use Markov
Process instead to solve this problem.
2.Markov Process
Markov process is an example of 'random walk' through phase space in a series of .
We would first give the three properties of Markov chain below:
[1]
[2]
[3]f(x)
means the probability that a point transfer from site x to site x'. [1] is
obvious. [2] says the random walk is not restrict to history, the sites can be visited many
times, and we will finally reach to the border or equilibrium state. We call this
"ergodicity hypothesis". [3] is just the learned "detailed balance" where the equilibrium
distribution f(x) is reached.
How to efficiently converge in the Marcov Chain depends on the selection of ,
which has a large freedom to choose. A simple choice is seen below, also known as the
Metropolis method.
How to pursue the random walk? We use (1.2.1) as an example and use Metropolis
method. We define the transition probability as
Following Metropolis, we choose another transition probability p, which reads
or
Then follow the following steps:
[1] Select a site i, and its first random step to the nearest neighbour i'
[2] Compute
[3] Calculate , if , then p=1, accept the spin value of this site; otherwise,
go to [4]