-Statistical Mechanics 2022 - 2023

• Sept 27: Introduction to the course. Thermodynamics 1st and 2nd principle. State functions, work and heat. Intensive and extensive variables. Thermodynamic potentials. Legendre transforms among thermodynamic potentials. Maxwell’s relations. Open thermodynamics systems, availability.
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• Sept 28: Thermodynamic stability. Thermodynamic responses and thermodynamic stability. Thermodynamic fluctuations. Linear response in thermodynamics, thermodynamic fluctuations and responses.
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• Oct 3: Divergence of susceptibility at the critical point. Mechanical foundations of statistical mechanics: phase space and observables, time averages. Microscopic reversibility vs macroscopic irreversibility. Loschmidt and Zermelo’s paradoxes and their resolution.
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• Oct 4: Ensemble of initial data, ensemble average. Ensembles evolution, Liouville’s theorem and phase space volumes conservation. Stationary distribution.
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• Oct 10: Ergodicity and mixing (definition). Ergodic trajectory in the Phase Space.
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• Oct 11:The recurrence time and its calculation. Chaos. Lyapunov exponent. Again of Loschmidt and Zermelo’s paradoxes. Separable Hamiltonians and lack of asymptotic equipartition. Role of observable in ergodicity.
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• Oct 17: Example: non-interacting and interacting harmonic oscillators, normal modes. Typical initial data and time averages. Kinchin theorem.
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• Oct 18: Ergodicity in quantum systems, time averages in QM. Density matrix for a pure state. Density matrix for a pure and mixtures of states, ensemble averages in QM. Liouville theorem in QM and stationary density matrix. Thermalization of an isolated system ,Eigenstate Thermalization Hypotesis.
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• Oct 24: Thermalization as dissipation and decoherence in quantum systems. Thermalization of an isolated system, ergodicity breaking, reduction of d.o.f. reduced probability density for classical systems and reduced density matrix for quantum systems. Reduced density matrix evolution.
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• Oct 25: Reduced density matrix evolution. BBGKY hierarchy for reduced density for pair interactions. $s=1$ and $s=2$ examples.
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• Nov 03: Time/length scales in the $s=1$ and $s=2$ equation of the hierarchy. Truncation of BBGKY leading to Boltzmann equation. Boltzmann’s equation: Molecular chaos approximation.
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• Nov 07: H-theorem and the Maxwell Boltzmann distribution, the problem of the loss of time reversal.
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• Nov 08: An example of thermalization: a classical particle interacting with N harmonic oscillators. Ohmic density and effective Langevin equation. Momentum relaxation: Ornstein-Uhlembeck process.
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• Nov 21: Langevin equation, velocity relaxation and correlation function. The Ornstein-Uhlembeck process. Brownian motion in the overdaped limit. Diffusion constant. Drift and Drude formula, mobility. Einstein’s relation. Again on drift and diffusion. A general form for Langevin equation. Averages over histories and it significance.
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• Nov 22: The Fokker-Planck equation. The Boltzmann equation as a Master Equation and it’s relation with Fokker-Planck equation. Maxwell-Boltzmann distribution from Fokker-Planck equation.
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• Nov 24: The microcanonical ensemble. The Boltzmann’s Entropy, additivity and the second principle.

• Nov 28: Correct counting of microscopically different states, Sackur-Tetrode formula. Gibbs paradox. Surface interactions. Canonical distribution, partition function and thermodynamic potential.
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• Nov 29: Canonical distribution, partition function and thermodynamic potential. Energy fluctuations. Grandcanonical ensemble. Ensembles equivalence. Number fluctuations.
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• Dec 01: Thermodynamic linear response. Generalised change of ensemble. Generalised equipartition theorem. Virial theorem in classical mechanics and in statistical mechanics. Grandcanonical partition function for a perfect gas. Fugacity and classical limit. Grandcanonical partition function for finite system and Lee-Yang theorem (hints).
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• Dec 05: Virial expansion. Fugacity expansion for a classical gas, equation of state of calssical interacting gas in the low-density limit role of interaction in gas-liquid phase transition. The second quantization. Creation and destruction operators and statistics. Field operators. Single particle operators. Examples.
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• Dec 06: Two particle operators Examples. The global gauge and the total number conservation. Quantum ensembles introduction. The grand canonical partition function for interacting particles.

• Dec 12: The grand canonical partition function for interacting particles. Bosons Fermions Boltzmannions. Equation of state, density of states. Fugacity expansion for quantum gases (exercise), the role of exchanges.
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• Dec 13: Degenerate quantum gases: the Fermi gas at $T=0$ the Bose-Einstein condensation. Isotherms equation of state. Interaction picture, Dyson series and perturbation expansion. Imaginary time and perturbation expansion in statistical mechanics.
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• Dec 19: Time ordering. Expansion of the thermodynamic potential. Calculation of a generic thermal average in a perturbative framework. Linear response theory at equilibrium homogeneous case. Examples. Gibbs variational principle. Effective Hamiltonians.
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• Dec 20: The Ising model: introduction and symmetries. The linear response theorem, application to the Ising model. Mean field theory in the Ising model from variational principle. Self-consistency and its solution.
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• Dec 22: Again on second order phase transition in the Ising model, susceptibility. The mean field solution. The order parameter. The spontaneous symmetry breaking and the thermodynamic limit. The Ising model and the lattice gas. The gas-liquid transition. The Landau theory of phase transitions.
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• Jan 09: The Landau theory of phase transitions. Classical critical exponents. Susceptibility in mean field theory. Correlation length and its classical critical exponent. Long range correlations and the failure of the mean-field theory. The Landau functional for non-homogeneous fluctuations.
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• Jan 10: The Ginzburg criterion and the upper critical dimension. The Mermin-Wagner theorem (no proof) and the lower critical dimension. Ising model in d=1 and in infinite dimensions.
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• Jan 10: Mean field decoupling schemes in quantum interacting systems. Overscreening and effective attraction of electrons in metals. The superconductive transition in the BCS mean field scheme. Landau potential for the superconductive transition. Global gauge symmetry breaking in the superconductive phase.
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