Course Details
Subject {L-T-P / C} : PH4004 : Statistical Mechanics { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Prof. Sanjoy Datta
Syllabus
Quick Ensemble theory: recapitulation of microcanonical, canonical, grand canonical ensemble and connection to thermodynamics, central limit theorem and equivalence of microcanonical and canonical ensemble, equipartition theorem, virial theorem. Entropy of ideal gas, Gibbs paradox, distinguishability, indistinguishability and correct enumeration of microstates, relation between Boltzmann and Shannon entropy, principle of maximum entropy. Localized quantum systems in microcanonical and canonical ensemble: thermodynamics of localized spin-1/2 system, solid of Einstein, system of particles with two levels of energy, negative temperature. Formulation of Quantum Statistics: density matrix and quantum mechanical ensemble theory, density matrix for pure and mixed ensembles, Von Neumann equation, thermodynamic average for a quantum system, density matrix for microcanonical, canonical and grand canonical ensemble, simple examples: an electron in a magnetic field, free particle in a box, quantum harmonic oscillator. Brief discussion on the relation of density matrix and entropy, entropy as measure of ignorance, entropy and information theory. Bose and Fermi-Dirac statistics: many particle quantum system and indistinguishability. Bosons, fermions and symmetry of many particle wavefunction. Ideal quantum gases in microcanonical ensemble, Bose and Fermi distribution, comparison with Maxwell-Boltzmann or the classical gas. Ideal quantum gases in grand-canonical ensemble and mean occupational statistics of bosons and fermions, classical limit of Bose and Fermi-Dirac statistics. Ideal Bose gas: thermodynamic properties and Bose-Einstein (BE) condensation, BE condensation in ultracold atomic gases, detecting BE condensate, thermodynamic properties of BE condensate. Ideal Fermi Gas: thermodynamic properties of completely degenerate and non-degenerate systems, Fermi temperature, Pauli paramagnetism, quick discussion on Landau diamagnetism, white dwarf star and Chandrasekhar limit. Interacting systems: qualitative discussion about statistical mechanics of interacting systems, exact solution of Ising model in 1D, qualitative discussion of 2D Ising model. Phase transition and critical phenomena: random walks and emergent properties. Broken symmetry, order parameter and phases of matter. Continuous phase transition, scale invariance, universality and critical exponents. Mean field approximation and 2D Ising model and critical exponents. Landau’s phenomenological theory of phase transition.
Course Objectives
- To build a solid foundation of the underlying fundamental principles of the thermodynamics of classical and quantum systems in equilibrium.
Course Outcomes
1. Students will be able to appreciate that complete random processes can lead to predictable behaviour. <br />2. Students will be able to understand how classical behaviour appears in a quantum system. <br />3. Fundamental differences between fermions and bosons. <br />4. How phase transition takes place.
Essential Reading
- R. K. Pathria & P. D. Beale, Statistical Mechanics, Elsevier
- Palash B. Pal., Introduction to Statistical Physics, Alpha Science International Ltd
Supplementary Reading
- K. Huang., Statistical Mechanics, Wiley
- Mehran Kardar, Statistical Physics of Particles, Cambridge University Press