Problem set 5, Physics 211, University of California, Berkeley
6 problems; due Friday, April 16, 5 pm in Physics 211 box in 251 Le Conte
1. Reif 5.19 (The van der Waals equation.)
2. Reif 5.24 (The latent heat.)
3. Consider the following two distributions
Notes for Physics 211, University of California, Berkeley
Week I: We covered mostly Reif 2.1-2.4 plus a few things (below). The random walk and
Gaussian examples are discussed in detail in Reif chapter 1.
1
Lecture I
:
Examples of successes of statistical
Notes for Physics 211, University of California, Berkeley
The rst lecture (reservoir derivation of the canonical ensemble) is similar to Reif 6.2, and the
example mentioned is discussed in Reif 3.8. The practice with the canonical ensemble in the second
l
Notes for Physics 211, University of California, Berkeley
Week IV: Last week we covered some topics that are not in Reif (e.g., proof of the Central
Limit Theorem via cumulants). Today we return to Reif with a few more words on the canonical
ensemble (cha
Physics 211 (Moore) Spring 2010
Problem Set #1 Solution
February 21, 2010
1. St. Petersburg Lottery
a. The expectation value of your return equals
x =
xn pn
n=1
where xn is the return in outcome n and pn is the probability of outcome n. The outcome is n 1
Physics 211 (Moore) Spring 2010
Problem Set #5 Solution
April 22, 2010
1. Reif 5.19
a. The equation of state reads (p + av 2 )(v b) = RT . We want to solve for the critical point in terms
of a and b. The easiest way to do this is to rst take a derivative
Physics 211 (Moore) Spring 2010
Problem Set #3 Solution
March 14, 2010
1. Gibbs free energy and chemical potential
a. For large systems, entropy is extensive, meaning that if we scale the system with parameter , S =
S. Similarly, the energy, volume, and n
Physics 211 (Moore) Spring 2010
Problem Set #2 Solution
March 5, 2010
1.
a. We have a particle of mass m and a probability distribution for pz of the form
P (pz ) =
1
exp(p2 /2mkT )
z
2mkT
where T = 300 K. Escape velocity is v0 = 11 km/s and g = 9.8 m/s
Physics 211 (Moore) Spring 2010
Problem Set #6 Solution
April 29, 2010
log Z
1
1. Reif 11.1 This is pretty trivial. M = 1
=
H
Z
indeed M .
r
Er Er
1
e
=
H
Z
r
Mr eEr which is
2. Units check The Josephson frequency is 2eV /h = 4.83 1014 s1 for a one volt
Notes for Physics 211, University of California, Berkeley
Week II: We covered mostly topics in Reif, starting with the later part of chapter 2. We started
on chapter 3 but skipped 3.1 and 3.2 now as we will discuss the approach to equilibrium later. We
al
Physics 211 (Moore) Spring 2010
Midterm Solution
April 2, 2010
1.
a. F = kB T log Z, where
Z = eE1 + eE2 .
(1)
b. The mean energy is
E=
log Z
E1 + E2 e(E2 E1 )
.
=
1 + e(E2 E1 )
(2)
The entropy is
S=
EF
.
T
(3)
2
2
2
c. The mean is N (E1 + E2 )/2. The va
Department of Physics
University of California, Berkeley
Physics 211 Final Examination
Monday, May 10, 2010
7 pm - 10 pm
Brief solutions (JEM)
1. (a) S = kB (p1 log p1 + p2 log p2 ), with
p1 =
eB/kB T
,
eB/kB T + eB/kB T
p2 =
eB/kB T
eB/kB T + eB/kB T
(1)
Problem set 4, Physics 211, University of California, Berkeley
6 problems; due Friday, April 2, 5 pm in Physics 211 box in 251 Le Conte
1. Compute a simplied version of the Chandrasekhar limit (the mass above which an ordinary
star collapses under its own
Problem set I, Physics 211, University of California, Berkeley
Due Friday, Feb 5, 5 pm in 251 Le Conte (see box)
1. St. Petersburg lottery: Suppose that you pay a xed fee of x dollars to enter the following
game. A fair coin will be tossed repeatedly unti
Problem set 6, Physics 211, University of California, Berkeley
6 problems; due Friday, April 30, 5 pm in Physics 211 box in 251 Le Conte
This set should be less time-consuming than others.
1. Reif 11.1
2. Units check: (a) compute the Josephson frequency 2
Problem set 3, Physics 211, University of California, Berkeley
7 problems; due Friday, March 5, 5 pm in Physics 211 box in 251 Le Conte
1. Gibbs free energy and chemical potential: (a) Use that entropy S(E, V, N ) is extensive for
large systems to show th
Problem set 2, Physics 211, University of California, Berkeley
7 problems; due Friday, Feb 19, 5 pm in 251 Le Conte (see box)
1. Planetary atmospheres (from Sethna): Treat diatomic oxygen for now as a monatomic ideal
gas with mass 5.3 1023 g, which is twi
Department of Physics, University of California, Berkeley
Physics 211 Final Examination, Monday, May 10, 2010, 7 pm - 10 pm
There are 6 problems; all count equally. No books, notes, or calculators are allowed
for this exam. Please start each problem on a
Physics 211 Spring 2010
Take-home midterm
Due in box (or by electronic submission), 5 pm, Friday March 19
You must complete this midterm in one two-hour period.
Reminder: No lectures Mon, Wed; will make up in RRR period.
There are 4 problems, each worth 2
Physics 211 (Moore) Spring 2010
Problem Set #4 Solution
April 22, 2010
1. Chandrasekhar limit
a. Consider a sphere of density and radius r. We would like to bring in a small mass dm in the form of
a thin spherical shell of radius dr (so that dm = 4r2 dr)