Topics in Statistic Mechanics

Adv. Sta. Phy. Homework 7 Renormalization Group Li,Zimeng PB06203182
2.2 RG solution on G-L model
[1] Coarse Grain Transformation
Similar to section 1.3, we will divide k and s into short wave (subscript S) and long wave
(subscript L) parts. Therefore (2.1.1) can be rewritten as
where
Here stands for the first term in (2.1.1) and stands for the
second term in (2.1.1)
Because the first term can be divided independently while the second term has
couplings.
where
and
corresponds to the short wave part, which has no singularity (See Sec. 1.3.[1]) and we
can thus take it as constant and omit it.
Note that
distribution:
is a Gauss function and there is a mathematic theorem for the Gauss
Therefore,
So (2.2.0)
The above equation has the short wave part
integrated out and so the integration