STATISTICAL PHYSICS
PAPER A1, SECOND YEAR
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(drawing by G. Gamow) |
Alexander Schekochihin and Andrew Boothroyd
NOTE: The course "blog" and some notes from last year (2011-12) are here. However, both the structure and the contents of the course will be somewhat different this time, so last year's course materials are NOT an up-to-date reflection of what will be taught (there may also be errors and imprecisions in the old version of the lecture notes).
 A sketch of students (or, perhaps, fellows) in a manuscript of William of Ockham's commentary on Aristotle's Physics (MS293 from the Merton College library, image courtesy of J. Walwarth). |
Michaelmas
Term
2012 (skip to HT) ESSENTIAL DOWNLOADS: Lecture Plan and Reading List (subject to changes in mid-course!)
LECTURES (notes, reading material and various other info are likely to appear here) NB:
The reading suggestions are only suggestions --- the way material is
presented in those sources is not always identical to our exposition.
See Reading List (above) for full book titles and publication info.
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A sketch of students (or, perhaps, fellows) in a manuscript of William of Ockham's commentary on Aristotle's Physics (MS293 from the Merton College library, image courtesy of J. Walwarth). |
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| PART I: BASIC THERMODYNAMICS |
Lectures
1-7 (Wed 31.10- Wed 14.11.12) will be given by Professor
Andrew Boothroyd |
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| You are ready to do Problem Sets 1 and 2 | |||
| PART II: KINETIC THEORY |
Lecture 1 (12:00 Thu 15.11.12)
Statistical description of a gas. Energy. Thermodynamic limit. Kinetic
calculation of pressure. Particle distribution functions. Pressure vs. energy. Lecture Notes 1
Reading: Blundell Sec 3 (a primer on probability), 6.1; Pauli Sec 24 (pressure); Kardar Chapter 2 (a more advanced primer on probability); Chapman & Cowling Chapter 2 Note:
Please make sure you have a practical command of the notions from
probability theory that I am using: random variables, their averages
(means), probability density function, joint distribution, indeoendent
random variables, change of variables for pdfs. This
may require reading a book! (for example Sinai's book suggested in the
right column) It is better to sort this out in your mind
early, otherwise initial confusion will fester and undermine you
ability to follow the rest of the course.
The billiard balls example I mentioned in the lecture is on p. 5 of the Notes --- but try to work this out on your own before you look! |
Related
mathematics: Probability and Statistics by S. Biller A good (mathematical but accessible) course on probability theory: Ya. G. Sinai, Probability Theory: An Introductory Course An unorthodox but fascinating book on probability: E. T. Jaynes, Probability Theory: The Logic of Science (CUP 2004) A philosophical treatment of what it all means by a man now very much in the news: J. M. Keynes, A Treatise on Probability (1920, reprinted 2008) |
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| Lecture 2 (10:00 Fri 16.11.12)
Isotropic distributions. Classical ideal gas. Maxwell's distribution.
Equation of state and temperature; heat capacity of monatomic ideal gas. Lecture Notes 2
Reading: Blundell Sec 5, 6.2; Pauli Sec 25; Chapman & Cowling Chapter 4 (much more advanced treatment) |
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| You are ready to start working on Problem Set 3 |
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| Lecture 3 (12:00 Wed 21.11.12) Effusion. Collisions. Lecture Notes 3
Lecture Notes 4 Reading: Blundell Sec 7, 8; Pauli Sec 28, 26; Chapman & Cowling Chapter 5 (much more advanced treatment) Note
that there is a "typo" in my notes on p. 30 in the calculation of
<v_r> (thanks to the attentive student who pointed this out). Fix
it!
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A treatment of collisions (and derivation of Maxwell'sdistribution) from the original source: L. Boltzmann, Lectures on Gas Theory (Dover 1995) |
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| Lecture 4 (10:00 Thu 22.11.12) Collisions cont'd. Local Maxwellian equilibrium. Conservation laws. |
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| Lecture 5 (10:00 Fri 23.11.12)
Conservation laws and transport equations. Qualitative derivation of
the diffusion coefficient, thermal conductivity and viscosity. The
diffusion equation.
Reading: Blundell Sec 10;
Pauli Sec 27; Chapman & Cowling Chapter 6 Key thought #1: All diffusion processes are similar (diffusion of particles, momentum, energy...).
Key thought #2: Relaxation to equilibrium occurs in two stages: first to local equilibrium (on collision time scale, so very quickly), then to global equilibrium (diffusively, so slowly) --- both processes are collisional, but different in speed and in the nature of the physics involved. Key thought #3: Transport equations are expressions of conservation laws. We use kinetic theory to calculate the fluxes of the conserved quantities (particle number, momentum, energy). Note: Methods to solve the heat diffusion equation were covered in Mathematical Methods. There are some questions on this in Problem Set. Do go back and brush up your understanding of (i) how to find steady solutions given appropriate boundary conditions (temperature or heat flux on two boundaries), (ii) how to find time-dependent solutions given periodic boundary conditions (real frequencies, complex wavenumbers), (iii) how to find time-dependent solutions given initial conditions (real wavenumbers, complex frequencies) [see Blundell Sec 10 and Appendix C.12.]. |
Here are some notes on dimensional analysis (from a summer school in Oxford), with an elementary treatment of a fun example of the role of viscosity: how do bubbles rise in a fluid? On diffusion, learn from the master: A. Einstein, Investigations on the Theory of the Brownian Movement (Dover 1956) [the theory of Brownian motion will be taught in the 3rd year, paper BI, but there's nothing there that you can't understand now; see also Blundell Sec 33.1] |
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| You are ready to start working on Problem Set 4 (vacation work) | |||
| Lecture 6 (12:00 Wed 28.11.12) Kinetic
derivation of the transport equations and transport coefficients
(thermal conductivity and viscosity): the simplified "dodgy"
derivation; the kinetic equation; the collision operator and
conservation laws. Lecture Notes 6
Reading: Blundell Sec 9 (the simplified derivation); Kardar Chapter 3 (more advanced than my treatment, includes general derivation of Boltzmann's equation and various attendant matters, in particular H theorem; treatment of fluid equations similar to mine); Kittel Sec 40, 43; Landau & Lifshitz-Kinetics Chapter 1 (hard-core Russian treatment); Chapman & Cowling Chapters 3, 7, 9, 10 (hard-core Cambridge treatment) |
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| Lecture 7 (10:00 Thu 29.11.12) Local conservation rederived. Solution of the kinetic equation and calculation of fluxes.
Reading: See Lecture 6
What I have shown you is a very simplified version of this calculation. Do read more advanced texts and find out how it is really done!
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Hilary Term
2013 ESSENTIAL DOWNLOADS: Lecture Plan and Reading List (subject to changes in mid-course!)
LECTURES |
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| PART III: FOUNDATIONS OF STATISTICAL MECHANICS |
Lecture 1 (12:00 Wed 16.01.13)
Our programme: from microphysics to macrophysics (what we need to do to
construct the thermodynamics of a given system). Micorostates. SM
definition of pressure. Principle of maximum entropy. Lecture Notes 7
Lecture Notes 8 Reading: Binney Notes Sec 1 Schroedinger Chapter 1-2 Jaynes Sec 11.4 Blundell Sec 14.8, 15.1-15.2, App C.3 (Stirling's formula) My
account of the principle of maximum entropy and subsequent developments
(continued in the next lecture) owes most to J. Binney's notes,
Schroedinger's book and Jaynes' book and papers. This is not
necessarily the standard approach and I invite you to read other books
on the subject and form your own view of what makes sense. There will
be more discussion of the conceptual issues underpinning this approach
in Lectures 6-7.
Regarding today's perceptive question about whether entropy is a human construct. Basically, yes, it is a measure of uncertainty that helps us make the best (unbiased) statistical inference about the state of a largely unknown system. I will discuss this issue a lot more in Lectures 6-7, so hold on! (But no one stops you thinking about it until then.) |
RECOMMENDED: James Binney's 2002 lecture notes on Statistical Mechanics E. T. Jaynes, Probability Theory: The Logic of Science (CUP 2004) C. Shannon's original paper: Bell System Tech. J. 27, 379 (1948) (the first in the linked issue). Also you might enjoy reading this: E. T. Jaynes, "Information Theory and Statistical Mechanics," Phys. Rev. 106, 620 (1957) On adiabatic processes (and pressure), see Binney & Skinner, Quantum Mechanics, sec 11.1-3 |
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| Lecture 2 (12:00 Thu 17.01.13) Method of Lagrange multipliers. Canonical ensemble: Gibbs distribution. Construction of thermodynamics. Lecture Notes 9
Reading: Binney Notes Sec 3.0 Schroedinger Chapter 2-3 Blundell Example 14.7 & App C.13 (Lagrange multipliers), Sec 20 Note
that in order for the results derived from the maximum entropy
principle to be useful, the maximum must be quite sharp and the
resulting distributions should not have too much various (fluctuations
around the mean values should be small). All this can be quite
rigorously demonstrated for large systems and is basically related to
the quality of the "thermodynamic limit." You will find these
discussions in Schroedinger Chapters V-VI and some further relevant
maths in Jaynes Sec 11.7.
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| Lecture 3 (12:00 Fri 18.01.13)
Additivity of entropy. Thermal equilibrium and the validity of the SM
definition of temperature. Heat bath and the physical interpretation of
the canonical ensemble. Mechanical and dynamical equilibria. Lecture Notes 10
Reading: Binney Notes Sec 3.0.3 Schroedinger Chapter 2 Blundell Sec 20.4 Landau & Lifshitz Sec 10, 12 |
On composite systems in QM, see Binney & Skinner, Quantum Mechanics, sec 6.1 If you want some intellectual stimulation of an abstract kind, see what mathematicians can do to thermodynamics: axiomatic thermodynamics by E. Lieb & J. Yngvason, Physics Reports 310, 1 (1999) (open at your own risk; you have to be over the age of consent!) An older account of Caratheodory's axiomatic thermodynamics is in Pauli Sec 11 |
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| Lecture 4 (12:00 Wed 23.01.13) Thermodynamic stability, positivity of temperature and pressure. SM of the classical monatomic ideal gas. Lecture Notes 11
Reading: See readings above on stability Blundell Sec 21.1-21.5 Best
way to understand the stability arguments is to think about cases when
the conclusions don't apply. For example, Problem Set 5 contains some
examples of negative temperatures: why are negative tempeartures
possible there despite my arguments to the contrary in today's lecture?
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| Lecture 5 (12:00 Thu 24.01.13) SM of the calssical monatomic ideal gas. More on entropy: Shannon's theorem. Reading: See readings above on ideal gas
Binney Notes Sec 2 Shannon's paper Sec 6 and App 2 Jaynes Sec 11.3 We
have now encountered three sets of definitions of temperature and
pressure, viz., thermodynamical, kinetic and statistical-mechanical,
and proved they are all equivalent. Ponder how this worked and see if
you really understand the logic involved. This is a good way to
revise.
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| You are ready to start working on Problem Set 5 | |||
| Lecture 6 (12:00 Fri 25.01.13) Boltzmann entropy.
Meaning of probabilities. Boltzmann vs. Gibbs. Microcanonical vs.
canonical ensemble. Lecture Notes 12
Reading: Schroedinger Sec 1-2 Blundell Sec 4.4-4.6 Kardar Sec 4.2, 4.6 Binney Notes Sec 5.0 Jaynes Sec 11.8 Landau & Lifshitz Sec 3, 4, 7, 8 At
the beginning of today's lecture, I referred to the intriguing concept
of the "thermal death of the Universe" possibly without sufficient
explanation. The idea is that if the entropy of the Universe keeps
increasing, we are moving towards an eventual state where it is
globally maximal and so there are no gradients of any kind --- it all
ends up in the ultimate boring state of deadly homogeneity --- which
means, sadly for us, that no structures and so no life can survive.
This realisation, the (possibly apocryphal) story goes, caused
Boltzmann great distress and ultimately led to his suicide. In
contrast, Schroedinger, whose understanding of statistical mechanics
was (as evidenced by his book) possibly deeper and more compelling than
Boltzmann's, was quite a positive fellow, worried less about death, and
indeed, among other things, wrote a book "What is Life?" which I highly
recommend. He was also a Fellow of Magdalen here in Oxford until his
insistence on having two wives became untenable and he moved to
catholic Ireland, which, interestingly, was more tolerant.
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If you are interested in the history of ideas, here's an interesting read about Einstein's struggles with entropy, probability and the second law: A. Pais, "Subtle is the Lord..." (OUP 1982) Ch. 4 |
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| Lecture 7 (12:00 Wed 30.01.13) Second law. Density matrix and entropy in quantum mechanics. How information is lost. Lecture Notes 13
Reading: Binney & Skinner Sec 6.3-6.4 Blundell Sec 15.4 Landau & Lifshitz Sec 5-6 Kardar Sec 6.5 Some
of you have asked me about subjectivity of the a priori probabilities:
if two observers have different information, will they obtain different
statistical mechanics and so different predictions? Does the heat
capacity of a box of gas depend on who is looking?! Although I did not
have time to discuss this in lectures, I have provided some comments on
this in Lecture Notes 12 (pp. 130-131) --- or you can read a more
extented discussion in Jaynes Sec 11.8. There is a question in PS-5
which provides an example of what happens when a superfluous constraint
is imposed.
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Possibly the first paper that made the connection between entropy and information (and sucessfully exorcised Maxwell's demon): Leo Szilard, Z. Physik 53, 840 (1929) --- English translation in Behavioral Sci. 9, 301 (1964) Second Law: E. T. Jaynes, "Gibbs vs Boltzmann Entropies," Am. J. Phys. 33, 391 (1965) Entropy and thermodynamics from the QM perspective: E. T. Jaynes, "Information Theory and Statistical Mechanics II," Phys. Rev. 108, 171 (1957) |
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| PART IV: APPLICATIONS OF STATISTICAL MECHANICS |
Lectures 8-10 will be given by Professor
Andrew Boothroyd |
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| PART V: OPEN SYSTEMS AND QUANTUM GASES |
Lecture 11 (12:00 Thu 7.02.13) Grand canonical ensemble. Chemical potential. Thermodynamics of open systems. Particle equilibrium. Lecture Notes 14
Reading: Blundell Sec 22.1-4; Kittel Sec 14; Kardar Sec 4.9; Landau & Lifshitz Sec 35-36 Note
that, like temperature, the chemical potential can be introduced purely
thermodynamically (as energy cost of adding particles to the system)
and then shown to be the same as the parameter that appears in the
grand canonical distribution. Also, instead of maximising entropy, as I
have done, one can derive the grand canonical distribution from the
microcanonical isolated-system set up by considering a small subsystem
of the world exchanging energy and particles with its surroundings.
This is how it is done in most textbooks.
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| Lecture 12 (12:00 Fri 8.02.13)
Chemical potential of a classical ideal gas. Equilibria of systems in
external fields. Chemical potential and the Gibbs function.
Multispecies (multicomponent) systems: generalisation of the grand
canonical ensemble. Reading: Blundell Sec 22.5-6;
Kittel Sec 14, 15; Kardar Sec 4.9; Landau & Lifshitz Sec 24, 25 |
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| Lecture 13 (12:00 Wed 13.02.13) Multispecies systems continued. Gibbs phase rule. Chemical equilibrium. Law of mass action. Lecture Notes 15
Reading: Blundell Sec 28.5, 22.8; Kittel Sec 16; Landau & Lifshitz Sec 101-102, 104-105 Another interesting topic that involves the use of chemical
potential but that I have not covered is solutions. Read Landau &
Lifshitz Chapter IX if you want to find out about that (also see
Blundell & Blundell Sec 22.9 about osmotic pressure).
I realised today, to my great consternation, that Lecture Notes 14 and 15, while uploaded on the page, were not linked from here: profuse apologies! (...but I wish someone had told me...) |
The ionisation-recombination equilibrium (see Problem Set 6) has interesting applications to Early Universe ("the recombination epoch"). Here and here are some lecture notes (from the US) on this subject. |
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| You are ready to do Problem Set 6 |
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| Lecture 14 (12:00 Thu 14.02.13)
Quantum gases. Pauli exclusion principle. Partition function for
fermions and bosons. Occupation number statistics. Preview of
interesting limits. Lecture Notes 16
Reading: Blundell Chapter 29; Kardar Sec 7.1, 7.3; Landau & Lifshitz Sec 53, 54; Schroedinger Sec 8 |
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| Lecture 15 (12:00 Fri 15.02.13) Calculations in the continuous limit. Classical limit. Degeneration. Reading: Blundell Sec 30.1;
Kardar Sec 7.4; Landau & Lifshitz Sec 56; Schroedinger Sec 8 |
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| Lecture 16 (11:00 Wed 20.02.13)
Physical interpretation of degeneration. Degenerate Fermi gas. Fermi
energy. Thermodynamics of Fermi gas at T=0. Finite-temperature
corrections and heat capacity. Lecture Notes 17
Reading: Blundell Sec 30.2; Kittel Sec 19-20; Kardar Sec 7.5; Landau & Lifshitz Sec 57,58; Schroedinger Sec 8(a) |
If you want to learn more about the thermodynamics of high-density, high-mass systems (including stability of neutron stars, which you will encounter in Problem Set 7), see Landau Chapter XI. Chandrasekhar got the 1983 Nobel Prize for his theory of the structure and evolurion of stars. Here is his Nobel lecture on the subject. |
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| Lecture 17 (11:00 Thu 21.02.13) Degenerate Bose gas. Bose-Einstein condensation. Lecture Notes 18
Reading: Blundell Sec 30.3-4; Kittel Sec 21; Kardar Sec 7.6; Landau & Lifshitz Sec 62; Schroedinger Sec 8(b) |
2001 Nobel Prize ("for the achievement of Bose-Einstein condensation...") |
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| Lectures 18-19 (on thermal radiation) will be given by Professor
Andrew Boothroyd Handout 11
Additional handout: Greenhouse Effect |
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| You are ready to do Problem Set 7 | |||
| PART VI: THERMODYNAMICS OF REAL GASES |
Lectures 20-24 will be given by Professor
Andrew Boothroyd Handout 12
Handout 13 Handout 14 Handout 15 Additional handout: Van der Vaals Gas Extracurricular handout: The Third Law |
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| You are ready to do Problem Set 8 (vacation work) |
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