Statistical Mechanics 2024 - 2025

• Sept 23: 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.
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• Sept 24: Open thermodynamics systems, availability. Thermodynamic stability. Thermodynamic responses and thermodynamic stability. Thermodynamic fluctuations. Linear response in thermodynamics, thermodynamic fluctuations and responses.
  
  

• Sept 30: Again on linear response in thermodynamics, thermodynamic fluctuations and responses. The gas liquid transition and the critical point. 1st and 2nd order phase transitions.
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• Oct 01: Mechanical foundations of statistical mechanics: phase space and observables , time averages. Loschmidt and Zermelo “paradoxes”. Ensemble of initial data, ensemble average.
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• Oct 07: Liouville’s theorem and phase space volumes conservation. Stationary distribution.
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• Oct 08: 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. The mixing entropy and the correct Boltzmann’s counting. The canonical distribution.
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• Oct 17: Canonical distribution, partition function and thermodynamic potential. Energy fluctuations. Equivalence between Canonical and Microcanonical ensemble. The perfect gas.
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• Oct 21: Grandcanonical ensemble. Ensembles equivalence. Number fluctuations. Generalised change of ensemble. 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 22: Generalised equipartition. Virial theorem in classical mechanics and in statistical mechanics. Imperfect gases virial theorem.
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• Oct 29: Equation of state of classical interacting gas in the low-density limit. Comparison of virial expansion to the van der Waals equation. Discussion of the results.
  
  

• Oct 31: 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. Equilibrium Ensembles in Quantum Statistical Mechanics. Quantum gases a recap. br/>
  

• Nov 04: The second quantization. Creation and destruction operators and statistics. Field operators. Single particle operators. Examples. Particle operators Examples.
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• Nov 05: Quantum and classical field theories. Decoupling of interactions in the classical and quantum cases. The global gauge and the total number conservation. The grand canonical partition function for interacting particles.
  
  

• Nov 11: 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. Quantum perfect gases. Bosons Fermions Boltzmannions. Degenerate quantum gases: the Fermi gas at $T=0$ the Bose-Einstein condensation.
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• Nov 12: Bose-Einstein condensation. Isotherms equation of state. Off diagonal long range order.
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• Nov 18: Gibbs variational principle. The entropy functional: microcanonical canonical and grand-canonical distribution. Effective Hamiltonians.
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• Nov 19: Mean field theory of Ising model. The mean field equations. The spontaneous symmetry breaking.
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• Nov 25: Again on Ising model and lattice gas models, introduction to exercise session #4. Back to the principles of statistical mechanics: microscopic reversibility vs macroscopic irreversibility.
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• Nov 26: 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. Ergodicity and mixing (definition). Chaos.
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• Dec 02: 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 03: BBGKY hierarchy for $s=1$ and $s=2$. The master equation. The Boltzmann’s equation as a master equation. Molecular chaos approximation.
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• Dec 09: Proof of the H-theorem, the problem of the loss of time reversal. Molecular chaos approximation form BBGKY estimate.
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• Dec 10: Ergodicity in quantum systems, time averages in QM. Thermalization of an isolated system ,Eigenstate Thermalization Hypotesis. Purity and decoherence in open quantum systems.
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• Dec 12: 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 16: 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. 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 17: 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. Correlations and Responses in the Ising model. The Ornstein-Zernike formula for magnetic susceptibility. Long range correlations and the failure of the mean-field theory. The Ginzburg criterion and the upper critical dimension.
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• Jan 07: Exercises sessione #4 hints for solutions. Again on the Ginzburg criterion and the upper critical dimension. The Ising model in d=1 (zero magnatic field) and the lower critical dimensions. Spontaneous symmetry breaking and the Ising model in infinite dimensions. The Mermin-Wagner theorem (no proof).
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• Jan 09: 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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