PU-302 · Statistical Mechanics
Statistical mechanics builds the bridge from the reversible microscopic dynamics of many particles to the irreversible, law-like behaviour of macroscopic matter, deriving thermodynamics rather than postulating it. Beginning from microstate counting and equal a priori probability, the unit constructs the three canonical ensembles, proves their equivalence in the thermodynamic limit, and applies them to ideal and quantum gases, radiation, magnetism, and phase transitions where fluctuations and collective behaviour dominate.
Lectures
| L01 | From Mechanics to Probability: Why Statistical Mechanics? — |
| L02 | Microstates, Macrostates and Phase Space — |
| L03 | Equal a Priori Probability and Boltzmann Entropy |
| L04 | Temperature, Pressure and Chemical Potential as Entropy Derivatives |
| L05 | The Microcanonical Ensemble in Practice |
| L06 | Systems in a Heat Bath: The Boltzmann Distribution |
| L07 | The Partition Function and Helmholtz Free Energy |
| L08 | The Maximum-Entropy Principle and the Three Ensembles |
| L09 | Energy Fluctuations and Heat Capacity |
| L10 | The Grand-Canonical Ensemble and Fugacity |
| L11 | Why the Ensembles Agree: The Thermodynamic Limit |
| L12 | The Equipartition Theorem and Classical Heat Capacities |
| L13 | The Classical Ideal Gas and the Gibbs Paradox |
| L14 | The Sackur-Tetrode Entropy and Absolute Counting |
| L15 | Identical Particles: Bose and Fermi Statistics |
| L16 | Occupation Numbers and the Classical Limit |
| L17 | Black-Body Radiation and Planck's Law |
| L18 | Stefan-Boltzmann, Wien and the Cosmic Microwave Background |
| L19 | Phonons and the Debye Theory of Solids |
| L20 | The Degenerate Fermi Gas: Metals and White Dwarfs |
| L21 | The Sommerfeld Expansion and Electronic Heat Capacity |
| L22 | Bose-Einstein Condensation |
| L23 | Interacting Systems and the Ising Model |
| L24 | Mean-Field Theory and Spontaneous Magnetization |
| L25 | Landau Theory of Phase Transitions |
| L26 | Critical Exponents, Universality and the Breakdown of Mean Field |
| L27 | Fluctuations, Response and the Fluctuation-Dissipation Idea |
| L28 | Non-Equilibrium and the Approach to Equilibrium |
| L29 | The Arrow of Time and Irreversibility |
| L30 | Synthesis: The Unity of Statistical Mechanics — |
Derivations homed in this unit
Boltzmann Entropy from Microstate Counting
Derives S = k_B ln Ω for an isolated system by requiring entropy to be additive over independent subsystems whose microstate counts multiply.
Temperature, Pressure and Chemical Potential from Entropy Maximization
Derives the thermodynamic definitions 1/T = dS/dE, p/T = dS/dV, mu/T = -dS/dN by maximizing total entropy of two systems free to exchange energy, volume and particles.
Boltzmann Distribution from a Heat Reservoir
Derives P(state) proportional to exp(-E/k_B T) for a system in contact with a large reservoir by expanding the reservoir entropy to first order in the system energy.
The Partition Function and F = -kT ln Z
Normalizing the Boltzmann distribution defines Z and shows the Helmholtz free energy, energy, and entropy are all derivatives of ln Z.
Gibbs Entropy and the Maximum-Entropy Derivation of the Ensembles
Derives the microcanonical, canonical and grand-canonical distributions as the ones maximizing S = -k_B sum p ln p subject to constraints on average energy and particle number.
Grand-Canonical Ensemble and the Grand Potential
Builds the grand partition function Xi for systems exchanging particles with a reservoir and derives Phi = -k_B T ln Xi together with mean N and its fluctuations.
Energy Fluctuations and Heat Capacity
The second derivative of ln Z gives the mean-square energy fluctuation ⟨ΔE²⟩ = kT²C_V, tying microscopic noise to a macroscopic response.
Equivalence of Ensembles in the Thermodynamic Limit
Proves microcanonical, canonical and grand-canonical ensembles give identical intensive thermodynamics as N goes to infinity because relative fluctuations vanish, via saddle-point evaluation.
The Equipartition Theorem
Derives that each quadratic degree of freedom in the Hamiltonian contributes half k_B T to the mean energy by Gaussian integration of the classical Boltzmann factor.
The Gibbs Paradox and Indistinguishability
Shows classical counting overcounts identical-particle states giving a non-extensive entropy, and that the 1/N! correction restoring extensivity anticipates quantum indistinguishability.
Ideal Gas Law and Sackur-Tetrode Entropy
Computes Z for N non-interacting particles with Gibbs 1/N! counting and the thermal de Broglie wavelength, yielding pV = Nk_B T and the absolute Sackur-Tetrode entropy.
Bose-Einstein and Fermi-Dirac Occupation Numbers
Derives mean n = 1/(exp[beta(eps-mu)] -/+ 1) for bosons and fermions by applying the grand-canonical ensemble mode-by-mode, with the Maxwell-Boltzmann limit at low occupancy.
Planck's Law from the Photon Gas
Treats electromagnetic modes as a gas of massless bosons with mu=0 and integrates over the mode density to obtain Planck's spectrum, recovering Stefan-Boltzmann and Wien's law.
Debye Model of Phonon Heat Capacity
Quantizes lattice vibrations as a phonon gas with a Debye cutoff and derives the T^3 low-temperature heat capacity and the Dulong-Petit high-temperature limit.
Degenerate Fermi Gas and the Sommerfeld Expansion
Derives the Fermi energy, ground-state degeneracy pressure, and the linear-in-T electronic heat capacity via the Sommerfeld low-temperature expansion of Fermi-Dirac integrals.
Bose-Einstein Condensation
Shows that in an ideal Bose gas below a critical temperature a macroscopic fraction of particles occupies the ground state, deriving T_c and the condensate fraction from saturation of the excited states.
Mean-Field Ising Model and Spontaneous Magnetization
Solves the Ising model in the mean-field (Weiss) approximation, deriving the self-consistency equation, the Curie temperature, and spontaneous symmetry breaking below T_c.
Landau Theory and Critical Exponents
Expands the free energy in powers of the order parameter near a continuous transition to derive mean-field critical exponents and the structure of second-order phase transitions.