Topics in Statistic Mechanics

Adv. Sta. Phy. Homework 8 Markovian-chain Monte Carlo Li,Zimeng PB06203182
where (1.1.1)
Due to the Central Limit Theorem, we have
E[ ]=
Therefore we can use (1.1.1) to calculate the Integral
1.2 Importance Sampling
In order to dismiss the dramatic change in functions which might lead to great error in
the numerical integration in the simple sampling method, we introduce in important
sampling to select points according to the trend of function, and this selection is
previously set.
Referring to (1.1.1), we change it to
In importance sampling, P is relevant with , and therefore we have
We can also demonstrate that E[
]=I in the sense of central limit theorem.
An application of importance sampling will be illustrated here. In statistic physics, an
observable is calculated in the following means:
This is easily done by simple sampling method, but the exponent function is such a
dramatic function that we would use important sampling instead to increase efficiency.
Therefore we write:
(1.2.1)
if P is carefully selected so that it is proportional to the distribution e , we will
get instantly that