The course

This page describes my notes for the winter 2013 session of the University of Toronto Basic Statistical Mechanics course (PHY452H1S), taught by Prof. Arun Paramekanti, which I took as a non-degree student.

The official course description at the time was:

Classical and quantum statistical mechanics of noninteracting systems; the statistical basis of thermodynamics; ensembles, partition function; thermodynamic equilibrium; stability and fluctuations; formulation of quantum statistics; theory of simple gases; ideal Bose and Fermi systems.

Mathematica notebooks associated with the class are also available.

The notes.

My notes (393 pages, 6×9″) for this class are available in a few forms:

In all cases, these notes are essentially free (I took about $0.70 USD royalty for the paper version), so if you feel undercharged, please send me some bitcoin.

Contributing.

Should you wish to actively contribute typo fixes (or additions, editing, …) to these notes, you can do so by contacting me, or by forking your own copy of the associated git repositories and building the book pdf from source, and submitting a subsequent merge request.

git clone [email protected]:peeterjoot/latex-notes-compilations.git peeterjoot
cd peeterjoot

submods="figures/phy452-basicstatmech phy452-basicstatmech mathematica latex"
for i in $submods ; do
git submodule update --init $i
(cd $i && git checkout master)
done

export PATH=`pwd`/latex/bin:$PATH

cd phy452-basicstatmech
make all

I reserve the right to impose dictatorial control over any editing and content decisions, and may not accept merge requests as-is, or at all. That said, I’ll probably not refuse reasonable suggestions or merge requests.

Contents:

  • Copyright
  • Document Version
  • Dedication
  • Preface
  • Contents
  • List of Figures
  • 1 What is statistical mechanics and equilibrium
  • 1.1 Equilibrium
  • 1.2 Probabilities
  • 2 Probability
  • 2.1 Probability
  • 2.2 Central limit theorem
  • 2.3 Binomial distribution
  • 2.4 Stirling formula
  • 2.5 Stirling log approximation
  • 2.6 Problems
  • 3 Random walk, diffusion, and Maxwell distribution
  • 3.1 Spatial Random walk
  • 3.2 Continuity equation and Ficks law
  • 3.3 Random walk in velocity space
  • 3.4 Review: Lead up to Maxwell distribution for gases
  • 3.5 What went wrong?
  • 3.6 Equilibrium steady states
  • 3.7 Problems
  • 4 Phase space
  • 4.1 Phase space
  • 4.2 Motion in phase space
  • 4.3 Liouville’s theorem
  • 4.4 Time averages, and Poincar\351 recurrence theorem
  • 4.5 Liouville’s theorem questions on density and current
  • 4.6 Liouville’s theorem
  • 4.7 Ideal gas and SHO phase space volume calculations
  • 4.8 Entropy
  • 4.9 Gibbs paradox
  • 4.10 Hypersphere volume calculation the easy way
  • 4.11 Problems
  • 5 Thermodynamics
  • 5.1 First law of thermodynamics
  • 5.2 Adiabatic and other processes
  • 5.3 First law
  • 5.4 Examples of work
  • 5.5 Heat
  • 5.6 Adiabatic processes
  • 5.7 Reversible processes
  • 5.8 Equilibrium
  • 5.9 Cyclic state variable verses non-state variables
  • 5.10 Irreversible and reversible processes
  • 5.11 Problems
  • 6 Statistical and thermodynamic connection
  • 6.1 Connections between statistical and thermodynamic views
  • 6.2 Ideal gas
  • 6.3 Non-ideal gas. General classical system
  • 6.4 Canonical ensemble
  • 6.5 Canonical partition and Helmholtz free energy
  • 6.6 Quantum mechanical picture
  • 6.7 Interacting spin
  • 6.8 Problems
  • 7 Grand canonical ensemble
  • 7.1 Grand partition function
  • 7.2 Fermions and Bosons
  • 7.3 Fermi gas thermodynamics
  • 7.4 Bosons
  • 7.5 Phonon modes
  • 7.6 Problems
  • Appendixes
  • A Pondering a question on conditional probability
  • B Outline of midterm I solutions
  • C Phase space trajectories for 1D SHO
  • D Thermodynamic identities
  • E Addition of two one half spins
  • F Langevin small approximation
  • G Relativisitic generalization of statistical mechanics
  • H Relativisitic density of states
  • I Velocity volume element to momentum volume element
  • J Unresolved question on energy distribution around mean energy
  • K Non integral binomial coefficient
  • L Fermi function expansion for thermodynamic quantities
  • M Bernoulli polynomials and numbers and Euler-MacLauren summation
  • N Summary of relations and helpful formulas (cheat sheet fodder)
  • O Mathematica notebooks
  • Bibliography

Changelog

phy452.V0.1.14-12.pdf

  • spelling: s/statististical/statistical/
  • gutter fixes (dmath purge fallout.)

phy452.V0.1.14-8.pdf

  • spellcheck rules, and spell check.
  • prune blank lines surrounding \paragraph
  • purge dmath
  • updated some figures (Mathematica Thick,Black attributes.)  Some hand drawn figures replaced.

phy452.V0.1.13.pdf, Sept 10, 2020 (commit 309bc76da70f3dd7aa38ee343f5f40750f516da0)

  • Dup word check and url fixes.

phy452.V0.1.12.pdf, Wed Aug 5, 2020 (commit 7bbcdf66b26e950fa01ae6cbae86f987bc2c8d49)

  • Fix hyphens in listing, typos in bio.
  • Remove appendix part so that the index and bib aren’t grouped with the appendix.
  • Tweak the preface and backcover
  • Group intro probability text together, and expand on probability distribution definition.
  • Remove singlton part heading so that chapters are the highest level.
  • Fix pdfbookmarks for contents and list of figures (so that they don’t show up under the preface)
  • Streamline FrontBack specialization.

phy452.V0.1.11.pdf, Sun Mar 3, 2019 (commit 5120e0354cfd1ef0303623644c252437556a730f)

  • Fix latex errors that caused NO page numbers!

phy452.V0.1.10.pdf, Thu Feb 28, 2019 (commit fcd648f1724ce1488f134f60c32f0eb3a772377f)

  • Re-export mathematica based figures as .pdf instead of .png
  • Add commas and periods into equation,dmath contexts.
  • remove blank lines between text and equation/dmath that are connected.
  • switch to 6×9 format for kindle direct publishing and reformat various equations that overflowed page boundaries.
  • use tcolorbox for tables
  • suppress page numbers for list of figures, table of contents, and first page in chapters.This is a hack to deal with my hacking of classicthesis.sty, as it wants to put some of those page
    numbers in an unprintable region that kdp doesn’t like (and won’t ignore.)
  • replace crude battery/resistor diagram with an svg based figure (with transparency)
  • shrink various figures.
  • periods on chapter, section, and figure captions.
  • move versioning to separate file: make.revision

phy452.V0.1.9.pdf, Wed Feb 13, 2019 (commit b4d935717084383fcd73e9dead3a65753c4dee84)