Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
I I I.F The Gaussian Mo del (Direct solution)
The RG approach will be applied to the Gaussian model in the next section. For
the sake of later comparison, here we provide the direct solution of this problem. The
Gaussian mo del is obtained by keeping only
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
I.C Phase Transitions
The most spectacular consequence of interactions among particles is the appearance
of new phases of matter whose collective behavior bears little resemblance to that of a few
particles. How do the particles then transform from one ma
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
8.334: Statistical Mechanics II
Spring 2008
Test 3
Review Problems & Solutions
The test is closed book, but if you wish you may bring a onesided sheet of formulas.
The intent of this sheet is as a reminder of important formulas and denitions, and not as
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
I. Collective Behavior, From Particles to Fields
I.A Intro duction
The ob ject of the rst part of this course was to intro duce the principles of statistical
mechanics which provide a bridge between the fundamental laws of microscopic physics,
and observe
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
I I I.D The Renormalization Group (Conceptual)
Success of the scaling theory in correctly predicting various exponent identities
strongly supports the assumption that close to the critical point the correlation length
, is the only important length scale
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
I I.C Sp ontaneous Symmetry Breaking and Goldstone Mo des
For zero eld, h = 0, although the microscopic Hamiltonian has full rotational sym
metry, the lowtemperature phase do es not. As a specic direction in nspace is selected
for the net magnetization M
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
IV. Perturbative Renormalization Group
IV.A Exp ectation Values in the Gaussian Mo del
Can we treat the LandauGinzburg Hamiltonian, as a perturbation to the Gaussian
mo del? In particular, for zero magnetic eld, we shall examine
H = H0 + U
dd x
L
t2 K
m
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
III. The Scaling Hyp othesis
I I I.A The Homogeneity Assumption
In the previous chapters, the singular behavior in the vicinity of a continuous transi
tion was characterized by a set of critical exponents cfw_, , , , , , . The saddlepoint
estimates of the
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
I I.D Scattering and Fluctuations
In addition to bulk thermo dynamic experiments, scattering measurements can be used
to probe microscopic uctuations at length scales of the order of the probe wavelength .
In a typical set up, a beam of wavevector ki is i
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
The nal expression can be calculated perturbatively as,
ln eU
= U
+
1 2
U U
2
2
+ +
(1)
th cumulant of U + . (IV.33)
!
The cumulants can be computed using the rules set in the previous sections. For example,
at the rst order we need to compute
d
d q1 dd
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
8.334: Statistical Mechanics II
Spring 2008
Test 1
Review Problems & Solutions
The test is closed book, but if you wish you may bring a onesided sheet of formulas.
The intent of this sheet is as a reminder of important formulas and denitions, and not as
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
IV.F Perturbative RG (Second Order)
The coarse grained Hamiltonian at second order in U is
0
[m] = V fb +
H
/b
0
dd q
( 2 ) d
t + K q2
2
m(q)2 + U
1
2
U2
U
2
+ O (U 3 ).
( IV . 4 7 )
2
To calculate U 2 U we need to consider all possible decompositions
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
I I.G Gaussian Integrals
In the previous section, the energy cost of uctuations was calculated at quadratic
order. These uctuations also mo dify the saddle point free energy. Before calculating
this mo dication, we take a short (but necessary) mathematica
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
8.334: Statistical Mechanics II
Spring 2008
Test 2
Review Problems & Solutions
The test is closed book, but if you wish you may bring a onesided sheet of formulas.
The intent of this sheet is as a reminder of important formulas and denitions, and not as
Statistical Mechanics 2: Statistical Physics of fields
PHYS 8.334

Spring 2008
IV.H Irrelevance of other interactions
The xed point Hamiltonian at O() (from eqs.(IV.55) has only three terms
H =
K
2
dd x (m)2
( n + 2) 2 2
K 4
m +
m,
( n + 8)
2( n + 8) K 4
( IV . 6 5 )
and explicitly depends on the imposed cuto 1/a (unlike the exp