Basic Statistical Mechanics
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Basic Statistical Mechanics

Notes and problems from 2013 UofT PHY452H1S

About the Book

This book contains lecture notes taken during the Winter 2013 session of Basic Statistical Mechanics (PHY452H1S), a University of Toronto course that was taught by Prof. Arun Paramekanti.

About the Author

Peeter Joot
Peeter Joot

Peeter Joot is a math and physics enthusiast with a love of geometric algebra.

Peeter's education includes a B.A.Sc from UofT (1997 Engineering Science - computer engineering), 2019 UofT M.Eng (ECE electromagnetics), a lot of self-study, and non-degree study of most of the interesting 4th year UofT undergrad physics courses.

Peeter's day job is software development. He has over 20 years of experience with low level systems programing, operating system and hardware abstraction and exploitation, concurrency, and large scale refactoring. Peeter works for LzLabs and now has fun with PL/I, COBOL, JCL, VSAM, CICS and other legacy subsystems sure to scare away most developers of his generation.

Table of Contents

  • Copyright
  • Document Version
  • Dedication
  • Preface
  • Contents
  • List of Figures
  • 1 What is statististical 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

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