Chapter 3
Phase Transitions and Critical
Phenomena
3.1
Mean-Field Solution to the Ising Model, Revisited
Lets return again to the mean-eld theory for the Ising model. Recall that the free energy was
F = kB T ln(Z) =
N Jzm2
N kB T ln cfw_2 cosh [ (Jzm + H
PHYS 611 - Course Notes 2011
1.3
12
Boltzmann Statistics (aka The Canonical Ensemble)
In the preceding section, we have seen the microcanonical ensemble, where the total number
of particles N = i ni and U = i ni i are xed (constant). Suppose now that the
PHYS 611 - Statistical Physics
Assignment 3 (80 points total)
Due Wednesday, Nov. 9, 2011 at 2:00pm
1. (30 points) Consider the Ising model in the ordinary mean-eld approximation, but assume
that there is always a nite external eld.
(a) (5 points) Obtain
Chapter 1
Review of Undergraduate
Statistical Mechanics
1.1
1.1.1
Microcanonical Ensemble
Entropy
Entropy is the fundamental postulate of statistical mechanics (at least, after the ergodic
principle!). Boltzmann suggested that the entropy of a system S is
PHYS 611 - Statistical Physics
Initial Examination Solutions
1. It is easy to gure out that the prices for each piece of pineapple (labeled 1), banana (labeled 2), and apple
(labeled 3) are 7, 4, and 5 cents, respectively. Also we know that the mean price
PHYS 611 - Statistical Physics
Assignment 1 (50 points total)
Due Friday, Oct. 7, 2011 at 2:00pm
1. (15 points) Two atoms interact with each other with a force derivable from the Lennard-Jones potential
U (r) = U0
a
r
12
2
a
r
6
,
where r is the separatio
Chapter 2
Imperfect (non-ideal) Gases
Before getting to a completely quantum treatment of particles, however, lets step back for a moment
and consider real gases. Virtually all textbooks on statistical mechanics focus on ideal gases, because
the math is v
PHYS 611 - Statistical Physics
Midterm Examination (80 points total)
Due Monday, Nov. 14, 2011 at 2:00pm
1. (25 points)
(a) (15 points) Show explicitly that the exponential series involving the two-dot diagram in the cluster expansion
explicitly includes
Assignment 3 Solutions
Question 1
1(a) First lets expand the constitutive equation for the magnetization, m = Tanh[b(Jz*m+B)] around small magnetization and small (but finite) magnetic fields. Note that I am using B for magnetic field because I define H t
PHYS 611 - Statistical Physics
Assignment 1 Solutions
1. The rst thing that we need to calculate is the partition function, because with this we can calculate anything.
Using the equipartition theorem, the partition function is
Z
=
=
=
=
1
h3
4
h3
4
h3
1
PHYS 611 - Statistical Physics
Assignment 4 Solutions
1. Recall that we can express the partition function for the 1D Ising model in an external eld as a product of
transfer matrices:
Z=
T12 T23 TN 1,N =
S1
S1 SN
SN
S2
where
e(J+H)
eJ
T=
S2
T12 T23
SN 1
PHYS 611 - Statistical Physics
Assignment 2 (60 points total)
Due Friday, Oct. 21, 2011 at 2:00pm
1. As usual, we start by evaluating the partition function. To simplify the calculation, lets dene the new variable
Ti Si Si+1 , so that Si Si+2 = Si Si+1 Si