Chapter 6
Ergodicity and the Approach to
Equilibrium
6.1
Equilibrium
Recall that a thermodynamic system is one containing an enormously large number of
constituent particles, a typical large number being Avogadros number, NA = 6.02
1023 . Nevertheless, i
Chapter 11
Shock Waves
Here we shall follow closely the pellucid discussion in chapter 2 of the book by G. Whitham,
beginning with the simplest possible PDE,
t + c0 x = 0 .
(11.1)
The solution to this equation is an arbitrary right-moving wave (assuming c
Chapter 10
Solitons
Starting in the 19th century, researchers found that certain nonlinear PDEs admit exact
solutions in the form of solitary waves, known today as solitons . Theres a famous story of
the Scottish engineer, John Scott Russell, who in 1834
Chapter 9
Pattern Formation
Patterning is a common occurrence found in a wide variety of physical systems, including
chemically active media, uids far from equilibrium, liquid crystals, etc. In this chapter we
will touch very briey on the basic physics of
Chapter 8
Front Propagation
8.1
Reaction-Diusion Systems
Weve studied simple N = 1 dynamical systems of the form
du
= R(u) .
dt
(8.1)
Recall that the dynamics evolves u(t) monotonically toward the rst stable xed point
encountered. Now lets extend the func
Chapter 7
Maps, Strange Attractors, and
Chaos
7.1
7.1.1
Maps
Parametric Oscillator
Consider the equation
2
x + 0 (t) x = 0 ,
(7.1)
where the oscillation frequency is a function of time. Equivalently,
M (t)
d
dt
x
x
0
1
2
0 (t) 0
=
(t)
x
x
.
(7.2)
The form
Chapter 5
Hamiltonian Mechanics
5.1
The Hamiltonian
Recall that L = L(q, q, t), and
L
.
q
p =
(5.1)
The Hamiltonian, H (q, p) is obtained by a Legendre transformation,
n
H (q, p) =
=1
p q L .
(5.2)
Note that
n
dH =
=1
n
=
=1
p dq + q dp
q dp
L
L
dq
dq
Chapter 4
Nonlinear Oscillators
4.1
Weakly Perturbed Linear Oscillators
Consider a nonlinear oscillator described by the equation of motion
2
x + 0 x = h(x) .
(4.1)
Here, is a dimensionless parameter, assumed to be small, and h(x) is a nonlinear function
Chapter 3
Two-Dimensional Phase Flows
Weve seen how, for one-dimensional dynamical systems u = f (u), the possibilities in terms
of the behavior of the system are in fact quite limited. Starting from an arbitrary initial
condition u(0), the phase ow is mo
Chapter 2
Bifurcations
2.1
Types of Bifurcations
2.1.1
Saddle-node bifurcation
We remarked above how f (u) is in general nonzero when f (u) itself vanishes, since two
equations in a single unknown is an overdetermined set. However, consider the function
F
Chapter 1
Dynamical Systems
1.1
1.1.1
Introduction
Phase space and phase curves
Dynamics is the study of motion through phase space. The phase space of a given dynamical
system is described as an N -dimensional manifold, M. A (dierentiable) manifold M is
Chapter 0
Reference Materials
No one book contains all the relevant material. Here I list several resources, arranged by
topic. My personal favorites are marked with a diamond ().
0.1
Dynamical Systems
S. Strogatz, Nonlinear Dynamics and Chaos (Addison-W