Statistical Mechanics 2021 - 2022

• Sept 27: Introduction to the course. Thermodynamics 1st and 2nd principle. State functions, work and heat. The Carnot cycle and the Clausius inequality. Intensive and extensive variables. Thermodynamic potentials. Legendre transforms among thermodynamic potentials. Maxwell’s relations.
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• Sept 28: Thermodynamics of a system in a bath: availability. Thermodynamic stability. Thermodynamic responses and thermodynamic stability.
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• Oct 4: Thermodynamic fluctuations. Linear response in thermodynamics, thermodynamic fluctuations and responses. Divergence of susceptibility at the critical point. Statistical mechanics, Hamiltonian dynamics for classical systems.
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• Oct 5: Phase space and observables, time averages. Microscopic reversibility vs macroscopic irreversibility. Loschmidt and Zermelo’s paradoxes and their resolution.
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• Oct 11: Ensemble of initial data, ensemble average. Ensembles evolution, Liouville’s theorem and phase space volumes conservation. Stationary distribution.
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• Oct 12: Ergodicity and mixing (definition). Ergodic trajectory in the Phase Space. The recurrence time and its calculation.
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• Oct 19: Chaos. Lyapunov exponent. Again of Loschmidt and Zermelo’s paradoxes. Separable Hamiltonians and lack of asymptotic equipartition. Example: interacting harmonic oscillators, normal modes. Typical initial data and time averages. Role of observable in ergodicity.
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• Oct 22: 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.

• Oct 26: Thermalization of an isolated system ,Eigenstate Thermalization Hypotesis. 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 28: BBGKY hierarchy for reduced density for pair interactions. $s=1$ and $s=2$ examples. Time/length scales in the $s=1$ and $s=2$ equation of the hierarchy. Truncation for low density.
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• Nov 02: Different timescales in BBGKY. Truncation of BBGKY leading to Boltzmann equation. Boltzmann’s equation: Molecular chaos approximation.
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• Nov 08: H-theorem and the Maxwell Boltzmann distribution, the problem of the loss of time reversal.
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• Nov 09: An example of thermalization: a classical particle interacting with N harmonic oscillators. Ohmic density and effective Langevin equation.
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• Nov 15: 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.
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• Nov 16: Again on drift and diffusion. A general form for Langevin equation. Averages over histories and it significance. The Fokker-Planck equation. The Boltzmann equation as a Master Equation and it’s relation with Fokker-Planck equation.
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• Nov 22: Again on Fokker-Planck equation, Maxwell distribution from Fokker-Planck equation. Boltzmann’s Entropy, additivity and the second principle.

• Nov 23: Correct counting of microscopically different states, Sackur-Tetrode formula. Gibbs paradox. Surface interactions.
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• Nov 29: Canonical distribution, partition function and thermodynamic potential. Energy fluctuations.
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• Nov 30: Grandcanonical ensemble. Ensembles equivalence. Number fluctuations. Thermodynamic linear response. Generalised change of ensemble.
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• Dec 06: 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).

• Dec 07: 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. Quantum ensembles, introduction.
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• Dec 13: The second quantization. Creation and destruction operators and statistics. Field operators. Single particle operators. Examples.

• Dec 14: Two particle operators Examples. The global gauge and the total number conservation. The grand canonical partition function for interacting particles. Bosons Fermions Boltzmannions.

• Dec 16: Equation of state, density of states. Fugacity expansion for quantum gases, the role of exchanges. The Fermi gas at $T=0$.
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• Dec 20: Gibbs variational principle. Effective Hamiltonians. The Ising model: introduction and symmetries.
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• Dec 21: The linear response theorem, application to the Ising model. Mean field theory in the Ising model from variational principle. Self-consistency and its solution, the second order phase transition and the spontaneous symmetry breaking for scalar order parameter.
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• Dec 10: Again on second order phase transition in the Ising model, susceptibility. The Ising model and the lattice gas. The gas-liquid transition. The order parameter. The Landau theory of phase transitions. Classical critical exponent.
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• Dec 11: 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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