Statistical Mechanics - Physics at Oregon State University

9.7. RENORMALIZATION GROUP THEORY. 221
which gives us ˜ J ′ . Finally, we divide the first and third equation to get
e 2˜ h cosh(2 ˜ J + ˜ h)
cosh(−2 ˜ J + ˜ h) = e2˜ h ′
which gives us ˜ h ′ . The solutions are
˜J ′ ( ˜ J, ˜ h) = 1
4 log
cosh(2 ˜ J + ˜ h) cosh(−2 ˜ J + ˜ h)
cosh 2 ( ˜
h)
˜h ′ ( ˜ J, ˜ h) = ˜ h + 1
2 log
cosh(2 ˜ J + ˜ h)
cosh(−2 ˜ J + ˜
h)
g( ˜ J, ˜ h) = 1
16 log
16 cosh(2 ˜ J + ˜ h) cosh(−2 ˜ J + ˜ h) cosh 2 ( ˜
h)
(9.159)
(9.160)
(9.161)
(9.162)
We are now able to relate the original partition function to a new partition
function describing half the particles on twice the length scale. Because of our
redefining the variables, the partition function depends on the variables N, ˜ J,
and ˜ h. This is equivalent to the original set N, T, h. We have redefined the
temperature scale by using βJ and redefined the field by using βh. Hence the
partition function is a function Z(N, ˜ J, ˜ h). We have expressed this in the form of
a partition function with new coupling constants and fields at half the number
of particles, We needed to add an energy shift. This translates to
Z(N, ˜ J, ˜ h) = e Ng( ˜ J, ˜ h) Z( 1
2 N, ˜ J ′ , ˜ h ′ ) (9.163)
This equation relates the properties of the original system to the properties of
a system with double the length scale, and renormalized interaction strengths.
Because the partition functions are directly related, thermodynamic properties
are similar! Our goal is now to repeat this process over and over again. We
have
Z( 1
2 N, ˜ J ′ , ˜ h ′ ) = e 1
2 Ng( ˜ J ′ , ˜ h ′ ) Z( 1
4 N, ˜ J”, ˜ h”) (9.164)
Note that the functional forms g(x, y), ˜ J”(x, y), and ˜ h”(x, y) are the same as
before!
When we apply this procedure repeatedly we therefore find
Z(N, ˜ J, ˜ h) = e Ng( ˜ J, ˜ h) e 1
2 Ng( ˜ J ′ , ˜ h ′ ) · · · Z(0, ˜ J ∞ , ˜ h ∞ ) (9.165)
The free energy follows from log(Z) = −βG. Therefore we have
−βG = N
g( ˜ J, ˜ h) + 1
2 g( ˜ J ′ , ˜ h ′ ) +
2 1
g(
2
˜ J”, ˜
h”) + · · ·
(9.166)