Statistical Physics of Non-Interacting and Interacting Systems

ABOUT THE COURSE: The course introduces the reason to study statistical mechanics when thermodynamics yields an equilibrium description of the systems. The basic concepts of the subject are explained from a formal viewpoint that deals with the concept of ensembles and ergodic hypothesis. Quantum statistical mechanics appears as an upgradation of the classical ideas that incorporates the indistinguishability of the particles, and thus is applicable to the ‘microscopic world’ of bosons and fermions. Several examples are discussed in this context, such as Bose-Einstein condensation, magnetism of fermionic systems. Besides, there are special topics, such as, Chandrasekhar limit, thermodynamics of black holes etc Further, as interacting systems are ubiquitous in nature, and hence require an introduction to the masters and the PhD students. Different examples of interacting systems are discussed through Ising model, Heisenberg model etc and specialized techniques such as Bethe ansatz are discussed.INTENDED AUDIENCE: Senior undergraduate, Masters’ students, PhD students, researchers of condensed matter physics, teachers in colleges and universities.PREREQUISITES: Preferably Quantum Mechanics and Classical Mechanics at the introductory levelINDUSTRY SUPPORT: Data analysis for complex systems

Week 1: Introduction, Why study Statistical Physics, Review of Thermodynamics, Entropy, Gibb’s paradox, concept of Temperature, Free energies and their interrelationships, Legendre transformation, Maxwell’s relations, Micro and Macrostates, Examples.
Week 2:Ensembles, Different Ensembles: Microcanonical, Canonical and Grand Canonical Ensembles, concept of Phase Space, Phase Space density, Liouville’s Theorem, Lagrange’s undetermined multipliers, Partition Function.
Week 3:Application of ensemble theory to classical ideal gas, non-interacting spins in a magnetic field, Magnetization, Curie’s law, Classical Harmonic Oscillators, Specific Heat, concept of Negative Temperature, Realization of negative temperature in lab, Cold atomic systems.
Week 4:Quantum Statistics: Distribution functions for indistinguishable particles, occupation numbers, Quantum Ideal Gases, Ideal Bose and Fermi systems, resolution of Gibb’s paradox.
Week 5:Ideal Bose Gas, Bose-Einstein Condensation (BEC), properties of an ideal Bose gas, Black Body Radiation, Specific heat of solids, Einstein’s and Debye’s theories.
Week 6:Ultracold atomic systems, Recent experiments on achieving negative temperature in labs, liquid Helium, Superfluidity
Week 7:Non-interacting Fermions, Pauli Exclusion Principle, Slater determinants, Free Fermi Gas, Fermi surface and Fermi temperature of Metals, Shubnikov de Haas and de Haas van Alphen (dHvA) oscillations
Week 8:Electronic Specific heat, Magnetic Properties of Solids, Pauli paramagnetism, Semiconductors, Law of mass action.
Week 9:Special topics: White dwarfs, Chandrasekhar limit, classical and quantum black hole thermodynamics, Temperature and surface gravity.
Week 10:Interacting Systems, Formalism, Virial equation of state, Virial coefficients, different kinds of interparticle interactions, introduction to linked cluster expansions.
Week 11:Ising model in one dimension, Exact solutions, Transfer matrix, Mean field theory, Bragg-William’s approximation
Week 12:Renormalization Group, Phase transitions, fixed points, Two-dimensional Ising Model, Bethe Anstaz.