THERMODYNAMICS
AND
STATISTICAL PHYSICS
Problems
Week 1
1. One kilogram of water at 0 C is brought into contact with a large reservoir at 100 C.
When the water has reached 100 , what has been the change in entropy of the water, of
the heat reservoir, and o
Physics 212: Statistical mechanics II, Fall 2010
Course information sheet
Website: http:/socrates.berkeley.edu/jemoore/Moore group, UC Berkeley/Physics 212.html
Instructor
Joel Moore
549 Birge Hall
(510)642-8313
[email protected]
Lectures: TuTh 2:10-3:
Physics 212: Statistical mechanics II
Lecture XXV
In this lecture we introduce the theory of dynamical critical phenomena, which generalizes the
previous examples of Glauber and Kawasaki dynamics on the d = 1 Ising model. We concluded
from that simple cas
Physics 212: Statistical mechanics II
Lecture XXIV
Many physical systems have a unique ground state in the limit of zero temperature, and hence
zero entropy. However, there are a number of systems where either because of a degeneracy of
ground states or b
Physics 212: Statistical mechanics II
Lecture XXII-XXIII
For the past few lectures, we have been discussing some of the complex dynamical phenomena
that can appear in classical statistical mechanics. In this nal lecture, we turn to quantum systems
where t
Physics 212: Statistical mechanics II
Lecture XXI
In the previous lecture, we discussed how models with continuous order parameters, like the
XY model, show dierent physics than models with discrete order parameters, like the Ising model.
We can think of
Physics 212: Statistical mechanics II
Lecture XX
In the last lecture we used the Flory model of the self-avoiding walk (SAW) to obtain an estimate
of the exponent that governs the typical size of a polymer:
R N,
=
3
.
d+2
(1)
However, we still have little
Physics 212: Statistical mechanics II
Lecture XIX
Now we begin Part III of the course, which will examine a few applications beyond phase
transitions of the ideas and methods already developed.
Lattice models of polymer physics
This rst lecture will focus
Physics 212: Statistical mechanics II
Lecture XVIII
The last lecture had a number of new ideas, so lets quickly review the main ones. The denition
of the scaling dimension of a local operator or eld like (x) at a critical point was through its
correlation
Physics 212: Statistical mechanics II
Lecture XVII
This lecture contains some fairly sophisticated notions from eld theory, but for the case of
the Gaussian model (related to mean-eld theory) the predictions can be understood in terms of
power-counting: t
Physics 212: Statistical mechanics II
Lecture XVI
The Ising model that has been considered in most of the lectures so far has a discrete degree
of freedom on each site (the spin is either up or down), or in other words, a discrete order parameter. Many ph
Physics 212: Statistical mechanics II
Lecture XIII
The rst part of this lecture explains further the behavior of correlation functions near a secondorder critical point. The second part returns to the 1D Ising model in order to understand how
rescaling tr
Physics 212: Statistical mechanics II
Lecture XII
The main result of the last lecture was a calculation of the averaged magnetization in mean-eld
theory in Fourier space when the spin at the origin is xed up,
m(k )
C
.
1 J (1 R2 k 2 )
(1)
Its Fourier tra
Physics 212: Statistical mechanics II
Lecture XI
The next step is to understand correlation functions from mean-eld theory, just on the disordered side of the transition. First we rewrite the correlation function in a simpler form, where T r
denotes the s
Physics 212: Statistical mechanics II
Lecture X
Note on texts: good choices for phase transitions and critical phenomena are Cardy, Goldenfeld,
and Ma; there is also some material in Huang.
We start by doing a few more calculations on the solvable one-dim
Physics 212: Statistical mechanics II
Lecture IX
Let us work out a simple example of the linear-response formula from the last lecture. Suppose
that the starting Hamiltonian describes a single spin-half in a magnetic eld along the z axis:
Sz ,
h
H0 =
(1)
Physics 212: Statistical mechanics II, Fall 2006
Lecture VIII
Now we turn to a quantum-mechanical discussion. There are some dierences in approach: for
classical Brownian motion (an example of an open system: we view the random force as coming
from intera
Physics 212: Statistical mechanics II
Lecture VII
Our rst picture of Brownian motion will be as a stochastic (i.e., random) process. A heavy
particle in a uid undergoes random forces as a result of collisions with the light particles of the
uid. You may b
Physics 212: Statistical mechanics II
Lecture VI
Turbulence
At high Reynolds number, the static solutions of viscous hydrodynamics are no longer physically relevant; real systems show rapidly varying ow congurations across many length and time
scales. The
Physics 212: Statistical mechanics II
Lecture V
In the previous lecture we nished the derivation of the BBGKY hierarchy and started considering
conservation laws in the Boltzmann equation. These conservation laws will enable us to derive the
hydrodynamic
Physics 212: Statistical mechanics II
Lecture IV
Our program for kinetic theory in the last lecture and this lecture can be expressed in the
following series of steps, from most exact and general to most specic and approximate (but also
useful!):
Liouvill
Physics 212: Statistical mechanics II
Lecture III
We begin with a slight detour into classical dynamics to understand the physical content of the
Boltzmann equation and how it can be reconciled with the microscopic equations of motion. Recall
that the Ham
Physics 212: Statistical mechanics II
Lecture II
The most famous example of how entropy increases in a real system is the dilute classical gas.
The rst assumption we will make is that
= na3
1,
(1)
or that the gaseousness parameter is small. Here n is the
Physics 212: Statistical mechanics II
Lecture I
A theory is the more impressive the greater the simplicity of its premises, the more dierent kinds
of things it relates, and the more extended its area of applicability. Therefore the deep impression
that cl
Physics 212: Statistical mechanics II, Fall 2010
Problem set 4: due Tuesday, 11/30/10
1. Imagine that you have a large table of fundamental constants expressed in SI or cgs units and without
leading zeros. What fraction of these constants would you expect
Physics 212: Statistical mechanics II, Fall 2010
Problem set 3: due Tuesday, 11/2/10, 5 pm
1. The point of this problem is to show that the innite-range Ising model, in a certain limit, corresponds
exactly to mean-eld theory. Each spin interacts equally s
Physics 212: Statistical mechanics II
Problem set 2: due on Friday 10/08/10, based on lectures IV-VIII
1. A quick numerical problem: Suppose that a sphere of radius 1 mm moves at 1 mm/sec
through a monatomic helium gas at room temperature and atmospheric
Physics 212: Statistical mechanics II, Fall 2010
Problem set 1: assigned 9/14/10, due 9/24/10, based on lectures I-V
1. Start from the formula for the entropy of a discrete probability distribution
k
S=
pi log pi .
(1)
i=1
Prove that the entropy for a sys