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Introduction:
Scope and aim of statistical mechanics.
Recontextualize thermodynamics with respect to statistical mechanics
Classical Statistical Mechanics:
Phase space, phase points, Ensembles, Density of phase points.
Liouville’s equation and Liouville’s theorem.
Stationary ensembles: Micro canonical, canonical and grand canonical ensembles.
Partition functions. Fluctuation in energy and particle.
Equilibrium properties of ideal systems: Ideal gas, Harmonic oscillators, Rigid rotators. Paramagnetism.
Quantum Statistical Mechanics:
Density matrix: Idea of quantum mechanical ensemble
Statistical and quantum mechanical approaches
Pure and mixed states
Density matrix for stationary ensembles
Applications: Particle in a box, charged particle in a magnetic field, Spin ½ particles
Construction of the density matrix for different states (pure and mixture) and calculation of the polarization vector
Identical Particles in Statistical Physics:
Introduction to ideas of cooperative and emergent phenomena.
Indistinguishability of identical particles in quantum many body systems
Spin and Statistics:
Length scales and the role of degeneracy in Statistical Mechanics.
Thermal wavelength and interparticle spacing, onset of quantum degeneracy in energy space
Ideal quantum systems:
Planck, Bose-Einstein and Fermi-Dirac statistics
Planck distribution and black body radiation
Ideal Bose gas, Bose-Einstein condensation
Ideal Fermi gas, Fermi energy, Sommerfeld expansion
Strongly Interacting quantum systems:
Bose-Einstein Condensation of interacting gases and the Gross-Pitaevski Equation
Superfluidity in Helium
Classical Ising model: Spin exchange interaction, mean field solutions, exact solution in 1D
Phase Transitions and Critical Phenomena: Landau Theory and Spontaneous Symmetry Breaking