Statistical Mechanics 2023 - 2024

• Sept 25: 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 26: Thermodynamic stability. Thermodynamic responses and thermodynamic stability. Thermodynamic fluctuations. Linear response in thermodynamics, thermodynamic fluctuations and responses.
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• Oct 02: Again on linear response in thermodynamics, thermodynamic fluctuations and responses. Mechanical foundations of statistical mechanics: phase space and observables , time averages. Ensemble of initial data, ensemble average.
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• Oct 03: Ensemble of initial data, ensemble average. Ensembles evolution, Liouville’s theorem and phase space volumes conservation. Stationary distribution.
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• Oct 09: Again on convervation of volumes in phase space. The microcanonical ensemble. The Boltzmann’s Entropy, additivity and the second principle.
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• Oct 10: Correct counting of microscopically different states, Sackur-Tetrode formula. Gibbs paradox. Surface interactions.
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• Oct 16: Canonical distribution, partition function and thermodynamic potential. Energy fluctuations. Exercises.
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• Oct 17: Grandcanonical ensemble. Ensembles equivalence. Number fluctuations. Generalised change of ensemble. Generalised equipartition theorem. 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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• Oct 23:Generalised equipartition. Virial theorem in classical mechanics and in statistical mechanics. Imperfect gases virial theorem and virial expansion. equation of state of classical interacting gas in the low-density limit role of interaction in gas-liquid phase transition.
  
  
• Oct 24: Again on imperfect gases comparison of virial expansion to the van der Waals equation. Quantum statistical mechanics: the density matrix, the Liouville’s theorem in the quantum realm. Density matrix of a pure state and of a mixture. Coherences.
  
  
• Oct 30: Equilibrium Ensembles in Quantum Statistical Mechanics. Quantum gases a recap. The second quantization. Creation and destruction operators and statistics. Field operators. Single particle operators. Examples.
  
  
• Oct 31: Two particle operators Examples. The global gauge and the total number conservation. 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. The cell volume and the correct Boltzmann counting as output of quantum statistics in the classical limit.
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• Nov 06: Quantum perfect gases. Bosons Fermions Boltzmannions. Degenerate quantum gases: the Fermi gas at $T=0$ the Bose-Einstein condensation.
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• Nov 07: Again on BEC. Isotherms equation of state. Off diagonal long range order. Imaginary time and perturbation expansion in statistical mechanics. The Wick rotation. Time-dependent perturbation expansion. Interaction picture.
  
  
• Nov 13: Dyson series and perturbation expansion. Time ordering. Expansion of the thermodynamic potential. Calculation of a generic thermal average in a perturbative framework. Linear response theory at equilibrium homogeneous case. Classical limit. Examples.
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• Nov 14: Gibbs variational principle. The entropy functional: microcanonical canonical and grand-canonical distribution. Effective Hamiltonians. Examples: The mean field treatmanet of the Ising model.
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• Nov 20: Again on Ising model and lattice gas models. The mean field equations. The spontaneous symmetry breaking.
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• Nov 21: Back to the principles of statistical mechanics: microscopic reversibility vs macroscopic irreversibility. Loschmidt and Zermelo’s paradoxes and their resolution. Ergodic trajectory in the Phase Space. The recurrence time and its calculation. Separable Hamiltonians and lack of asymptotic equipartition.
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• Nov 27: Ergodicity and mixing (definition). Chaos. Ergodicity breaking, reduction of d.o.f. reduced probability density for classical systems.
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• Nov 28: Relaxation times. Reduced probability density for classical systems and reduced density matrix for quantum systems. Reduced density matrix evolution. The BBGKY hierarchy.
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• Dec 04: BBGKY hierarchy for $s=1$ and $s=2$. The Boltzmann’s equation: Molecular chaos approximation. H-theorem and the Maxwell Boltzmann distribution.
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• Dec 05: Proof of the H-theorem, the problem of the loss of time reversal. Ergodicity in quantum systems, time averages in QM. Thermalization of an isolated system ,Eigenstate Thermalization Hypotesis.
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• Dec 06: 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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• Dec 11: 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 and how to numerically integrate it.
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• Dec 12: 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 (Kramers equation).
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• Dec 18: 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 Landau theory of phase transitions. Critical indexes.
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• Dec 19: Critical indexes and classes of universality. On the validity of the Mean Field Theory, Ising model in d=1 and in infinite dimensions. The upper and the lower critical dimensions. Correlations and Responses in the Ising model. Correlation and responses in the Ising model. The Ornstein-ernike formula for magnetic susceptibility.
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• Jan 08: Long range correlations and the failure of the mean-field theory. The Ginzburg criterion and the upper critical dimension.
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• Jan 09: The Mermin-Wagner theorem (no proof) and the lower critical dimension… 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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