updated  March 4, 1998

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DIFFUSION, TRANSPORT AND NON-EQUILIBRIUM STATISTICAL MECHANICS
Winter quarter 1998
Physics D60-0: NU Graduate Chaos Course schedule
Predrag Cvitanovic

Current schedule: http://ChaosBook.org/~predrag/D60-sched.html

Lecture 1           9:00 - 10:30 Tuesday, Jan 6 in Tech 3823
Boltzmann equation
We start with the classic - a heuristic derivation of the 1872 Boltzmann equation for density of a dilute gas.
Reading: chapters 1-3, section 3.1 of Dorfman's notes.
Other literature of interest: Thompson[6] sect. 1.4, Harris[2] chapter 3.

Lecture 2           10:30-12:00 Thursday, Jan 8 in Tech 3829
Boltzmann H-theorem
H - theorem states that even though the microscopics dynamics is reversible, the gas tends irreversibly toward the equilibrium. How can it be? hang on for the rest of the course.
Problem: Dorfman's 1.1 Ehrenfest wind-tree model; due Tue Jan 13.
Solutions: Dorfman 1.1, 1.2, 3.1.

Lecture 3           9:00 - 10:30 Tuesday, Jan 13 in Tech 3823
Kac's ring model
A simple model illuminating aspects of Stosszahlansatz; macroscopic irreversibility from microscopic reversibility and a Poincare recurrence.
We get started with the tagged particle diffusion, an illustration of the Chapman-Enskog solution of the Boltzmann equation.
Problem: Present a derivation of the Maxwell-Boltzmann distribution for a gas in thermal equilibrium (you can follow your favorite statistical mechanics textbook). Check that this is a (the?) solution of the constant H(t) condition. Due Tue Jan 20.

Lecture 4           10:30-12:00 Thursday, Jan 15 in Tech L313
Tagged particle diffusion
An illustration of the Chapman-Enskog method for solving the Boltzmann equation; we derive the diffusion equation and compute the diffusion coefficient.
Problem: derive the Boltzmann collision kernel B(g,k), p. 18 of Dorfman's notes, for 3-d hard-spheres and 2-d hard-disks collisons.
Due Tue Jan 20.

Lecture 5           9:00 - 10:30 Tuesday, Jan 20 in Tech 3823
Liouville's equations
We derive Liouville's equation for the full N-particle phase-space, and sketch the BBGKY hierarchy of equations derived from it, with the Boltzmann equation as the lowest order approximation.
Reading: chapters 4, 5, sects. 4.1-4.2
Other literature of interest: Thompson[6], Appendix A for Liouville theorem.
Problems:
5.1 Derive the first BBGKY hierarchy equation, Dorfman (4.7).
5.2 (K. Huang's 4.6):
A room of volume 3 \times 3 \times 3 cubic meters is under standard conditions (atmospheric pressure and 300 K).
a) Estimate the probability that at any instant of time a 1-cc volume anywhere within this room becomes totally devoid of air because of spontaneous statistical fluctuations.
b) Estimate the same for a 1-\Aangstrom^3 volume.
5.3 (K. Huang's 4.7):
Suppose the the situation referred to in Problem 4.6a has occured. Describe qualitatively the behavior of the distribution function thereafter. Estimate the time it takes for such a situation to occur again, under the assumption that molecular collisions are such that the time sequence of the state of the system is a random sequence of states.
Due Tue Jan 27.

Lecture 6           10:30-12:00 Thursday, Jan 22 in Tech L313
Poincare recurrence theorem
A pasta lover's derivation of the Poincare recurrence theorem - you cannot wind an infinite-length spaghetti through a finite volume.
Reading: Dorfman chapter 5, sect. 5.1 .
Problem: Dorfman sect 5.1 and problem 2.2 - the Ehrenfest urn model.
Due Tue Jan 27.
Solution: in ref. [3], handed out in the lecture.

Lecture 7           9:00 - 10:30 Tuesday, Jan 27 in Tech 3823
Boltzmann's ergodic hypothesis
Boltzmann's hypothesis equates microcanonical ensemble average with the infinite time average. Birkhoff proved that the time average exists for almost every initial condition.  Gibbs made a stronger, mixing assumption to describe the approach to the equilibrium.
Reading: Dorfman chapter 6, chapter 7, sects. 6.1-2, 7.1-3.
Problem: Dorfman 3.1.
Due Tue Feb 3.
Solutions: Dorfman 3.1, 3.2, 4.1 .

Lecture 8           10:30-12:00 Thursday, Jan 29 in Tech L313
The Green-Kubo formulas
Linear response theory  relates the electrical conductivity of a system of charged particles to the  time correlation function. We also recast the Liouville equation in a more general setting - in terms of operators that move the whole phase space around, and try to make sense of them.
Reading: Dorfman chapter 8, sect. 8.1.
Chaos Webbook Chapter 5 - Transporting densities

Problems: 1-2, 1-5, 1-6, 1-7 and 1-8 from ref. [2], handed out in the lecture.
Due Tue Feb 10.

Lecture 9           9:00 - 10:30 Tuesday, Feb 3 in Tech 3823
The Green-Kubo formula for diffusion
Standard derivation of the the Green-Kubo formulas for the diffusion coefficient in terms of the time velocity-velocity correlation function.

Lecture 10           10:30-12:00 Thursday, Feb 5 in Tech L313
Symbolic dynamics I
Qualitative dynamics of simple stretching and mixing flows is used to introduce Smale horseshoes and symbolic dynamics. One learns how to count and describe itineraries.
Reading: Chaos Webbook Chapter 2 - Dynamics, qualitative, sect. 2.1
Problem: Chaos Webbook chapter 2 exercises, exer. 2.1
Due Tue Feb 10.

Lecture 11           9:00 - 10:30 Tuesday, Feb 10 in Tech 3823
Symbolic dynamics II
continued: The topological dynamics is incoded by means of transition matrices/Markov graphs, and while computing the topological entropy we encounter our first zeta function.
Reading: Chaos Webbook chapter 2, sects. 2.2-7
Problems: Chaos Webbook chapter 2, exer. 2.11, 2.14, 2.19
Due Tue Feb 17.

Lecture 12           10:30-12:00 Thursday, Feb 12 in Tech L313
Symbolic dynamics III
again...
Problem:
Due Tue Feb 17.

Lecture 13           9:00 - 10:30 Tuesday, Feb 17 in Tech 3823
Symbolic dynamics IV
and again - we now really encounter our first zeta function.
Problem:
Due Tue Feb 24.

Lecture 14           10:30-12:00 Thursday, Feb 19 in Tech L313
Global dynamics I
This is the core chapter: If there is one idea that one should learn about chaotic dynamics, it happens in this chapter. Here one defines the evolution operators and derives the associates trace formulas, spectral determinants, dynamical zeta functions.
Reading: Chaos Webbook Chapter 6 - Global dynamics, sect. 6.1, 6.2
Problems: Chaos Webbook exercises 5.2, 5.3, 6.1, 6.4, 8.2(a)-(f), 8.2(g) optional.
Due Tue Feb 24.

Lecture 15           9:00 - 10:30 Tuesday, Feb 24 in Tech 3823
Trace formulas, spectral determinants
Trace formulas.
Problems: Chaos Webbook exercise 8.2(a)-(f), 8.2(g) optional.
Due Tue Mar 3.

Lecture 16           10:30-12:00 Thursday, Feb 26 in Tech L313
Cycle expansions
In last two lectures we have derived a plethora of periodic orbit trace formulas, spectral determinants and zeta functions. Now we learn how to expanded these as cycle expansions, series ordered by increasing topological cycle length, and evaluate average quantites like escape rates. These formulas are exact, and, for nice flows, highly convergent. The pleasant surprise is that the terms in such expansions fall off exponentially or even faster, so that a handful of shortest orbits suffices for rather accurate estimates of asymptotic averages.

Problem: Chaos Webbook exercise 9.2 (use other paramenter value than the solution, A=9/2)
Due Tue Mar 3.

Lecture 17           9:00 - 10:30 Tuesday, Mar 3 in Tech 3823
Applications of cycle expansions
We apply periodic orbit theory to evaluation of Lyapunov exponents and diffusion constants.
Reading: Chaos Webbook Recyling chapter 9.1, 9.3 (not beyond 9.3.1)

Lecture 18           10:30-12:00 Thursday, Mar 5 in Tech L313
Diffusion confusion
We compute diffusion constant for a simple 1-d model of deterministic diffusion.
Reading: Chaos Webbook Diffusion chapter 11.1, 11.3 (not 11.2.1)
Problem: Chaos Webbook exercises 11.1, 11.2
Due Tue Mar 10.

Lecture 19           9:00 - 10:30 Tuesday, Mar 10 in Tech 3823
Horocycles
Sinai-Bunimovich formula for Lyapunov exponents in terms of horocycles.
Term papers will take 1 week from start to finish. Niels Sondergaard has started his project March 5, delivery date 4 PM Thur March 12. I am still formulating the other 3 and am open to suggestions.
Chaos Webbook section 4.6

Lecture 20 (the last of the course)           10:30-12:00 Thursday, Mar 12 Tech L313
What's wrong with this course?
The grand picture and it's many failings.
Term papers          due 4:30 PM Thursday Mar 19, 1998 - Predrag's office

## References

1
G.E. Uhlenbeck, G.W. Ford and E.W. Montroll, Lectures in Statistical Mechanics (Amer. Math. Soc., Providence R.I., 1963).
2
S. Harris, An introduction to the Theory of the Boltzmann Equation (Holt, Rinehart and Winston, New York, 1971).
3
M. Kac, Random walk and the theory of Brownian motion'', (1946), reprinted in ref. [4].
4
N. Wax, ed., Selected Papers on Noise and Stochastic Processes (Dover, New York 1954).
6
C.J. Thompson, Mathematical Statistical Mechanics (Macmillan, New York 1972).

I liked reading Uhlenbeck's discussion [1] of the Boltzmann equation, as well as most of his lecture notes, and especially his remarks appended to lectures.

Binary collision integrals are worked out in detail in ref. [2].