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FM430 - Meccanica statistica matematica

AA 2018-2019 - II Semestre (Docente: Alessandro Giuliani)

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Programma

Diario delle lezioni

Orari

Bibliografia

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Diario delle lezioni

• Lezioni 1 e 2 [28/2/2019]
  Introduction to statistical mechanics, I: from microscopic (Newton's equations of motion) to macroscopic equations (thermodynamics: phase diagram, isotherms, etc.). The ergodic hypothesis and Boltzmann's proposal.

• Lezioni 3 e 4 [1/3/2019]
  Introduction to statistical mechanics, II: Statistical ensembles (microcanonical and canonical ensembles) as good (and equivalent!) models of thermodynamics. What is known and what is open in the statistical mechanics of interacting particles in the continuum.

• Lezioni 5 e 6 [7/3/2019]
  Introduction to statistical mechanics, III: open problems in statistical mechanics, existence of phase transition for continuum models. A route towards the understanding of phase transitions: lattice models. Some known facts about phase transitions in lattice models for interacting particles. The critical point and critical exponents, the universality hypothesis.

• Lezioni 7 e 8 [8/3/2019]
  A review of thermodynamics, I. The axioms of equilibrium thermodynamics for isolated systems: combination of subsystems, equilibrium of a composite system under constraints. The entropy function and the maximal entropy principle. Temperature, pressure and chemical potential as derivatives of the entropy. Homogeneity and concavity of the entropy.

• Lezioni 9 e 10 [13/3/2019]
  A review of thermodynamics, II.

• Lezioni 11 e 12 [14/3/2019]
  A review of thermodynamics, III. Convexity and Legendre transform.

• Lezioni 13 e 14 [15/3/2019]
  Back to the Boltzmann scheme: the microcanonical and canonical ensembles reloaded. The free (or perfect) gas: computation of the entropy in the microcanonical and of the free energy in the canonical. The notion of thermodynamic limit.

• Lezioni 15 e 16 [27/3/2019]
  Orthodicity of the canonical ensemble. Equivalence of the ensembles: ideas of the proof.

• Lezioni 17 e 18 [28/3/2019]
  Statement of the Fisher's theorem on the existence of the thermodynamic limit, concavity/convexity of the thermodynamic functions, equivalence of the ensembles. The temperedness and stability conditions. Reformulation of the Fisher's theorem for the lattice gas.

• Lezioni 19 e 20 [29/3/2019]
  Lattice gas and Ising model. Reformulation of the Fisher's theorem for the Ising model. Proof of Fisher's theorem, I: the existence of the thermodynamic limit for the grandcanonical pressure.

• Lezioni 21 e 22 [3/4/2019]
  Proof of Fisher's theorem, II: the existence of the thermodynamic limit for the canonical free energy.

• Lezioni 23 e 24 [4/4/2019]
  Proof of Fisher's theorem, III: equivalence of the microcanonical and canonical ensembles.

• Lezioni 25 e 26 [5/4/2019]
  The (spontaneous) magnetization: consequences of Fisher's theorem and general behavior of the magnetization profile. Paramagnetic and ferromagnetic behaviour.

• Lezioni 27 e 28 [11/4/2019]
  Exact solution of the 1D nearest neighbor Ising model.

• Lezioni 29 e 30 [12/4/2019]
  Exact solution of the mean-field (Curie-Weiss) Ising model.

• Lezioni 31 e 32 [17/4/2019]
  Further comments on the mean-field Ising model: relations between its thermodynamical observables and those of the ising model in dimension d. Exercise sheet on the 1D Ising model and the Curie-Weiss model.

• Lezioni 33 e 34 [24/4/2019]
  Infinite volume states and infinite volume Gibbs states. Families of local functions. Translational invariant Gibbs states. Statement of Ruelle's theorem on the simplicial structure of the set of translationally invariant Gibbs states. Pure (or extremal) states: the cluster property. Existence of infinite volume Gibbs states via the extraction of a suitable subsequence. Theorem (statement): the limits of the finite volume Gibbs states with + or - boundary conditions exist and are pure.

• Lezioni 35 e 36 [26/4/2019]
  GKS and FKG inequalities (statement). Proof of the GKS inequality.

• Lezioni 37 e 38 [2/5/2019]
  Proof of the FKG inequality.

• Lezioni 39 e 40 [3/5/2019]
  Proof of the existence of the infinite volume Gibbs states with + or - boundary conditions; translation invariance and cluster property of these states. The DLR condition. Further comments on the structure of the space of infinite volume Gibbs states.

• Lezioni 41 e 42 [8/5/2019]
  Equivalence between different notions of first order phase transition: multiple Gibbs states vs non-differentiability of the pressure.

• Lezioni 43 e 44 [10/5/2019]
  Low temperature expansion. Peierls' argument for the nearest neighbor Ising model at h=0.

• Lezioni 45 e 46 [15/5/2019]
  High temperature expansion. Uniqueness of the Gibbs state at h=0 and high enough temperature. Kramers-Wannier duality for the nearest-neighbor 2D Ising model.

• Lezioni 47 e 48 [16/5/2019]
  The Lee-Yang theorem: statement and implications for the analyticity of the pressure at non-zero magnetic field. Proof of the Lee-Yang theorem (part 1).

• Lezioni 49 e 50 [17/5/2019]
  Proof of the Lee-Yang theorem (part 2): the Asano contraction.

• Lezioni 51 e 52 [23/5/2019]
  Il ferromagnete unidimensionale di Dyson. Assenza di transizione di fase ad alta temperatura (stima dall'alto delle correlazioni in termini di quelle di campo medio). Esistenza di transizione di fase a bassa temperatura: enunciato e schema di dimostrazione. Riduzione al modello gerarchico di Dyson.

• Lezioni 53 e 54 [24/5/2019]
  Esistenza di magnetizzazione spontanea nel modello gerarchico di Dyson.

• Lezioni 55 e 56 [29/5/2019]
  Modelli di spin con simmetria continua. Transizione di fase come fenomeno di rottura spontanea della simmetria. Teorema di Mermin-Wagner (solo enunciato). Stima del costo energetico per produrre uno `spin flip': eccitazioni elementari come `onde di spin'.

• Lezioni 57 e 58 [30/5/2019]
  L'approssimazione Gaussiana (o `spin wave') per il modello O(2) in dimensione d=1,2,3. Calcolo delle correlazioni spin-spin in tale approssimazione. Decadimento polinomiale delle correlazioni spin-spin del modello O(2) in d=2 (stima dall'alto a'la McBryan-Spencer).

• Lezioni 59 e 60 [31/5/2019]
  Esistenza di una transizione di fase per il modello di spin O(N) in tre dimensioni: positivit&\agrave; per riflessioni, dominazione Gaussiana e stima infrarossa.

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Orario lezioni

- mercoledí 11:00-13:00, aula 009; giovedí 16:00-18:00, Aula 009; venerdí 11:00-13:00, Aula 009

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Esercizi proposti

• Foglio di esercizi proposti [1] Esercizi.

• Foglio di esercizi proposti [2] Esercizi.

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Testi di riferimento:


• S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
  Disponibile online in preprint version su https://www.unige.ch/math/folks/velenik/smbook/index.html
• G. Gallavotti, F. Bonetto e G. Gentile: Aspects of the ergodic, qualitative and statistical theory of motion, Springer-Verlag 2004.
  Disponibile on-line su http://ricerca.mat.uniroma3.it/ipparco/pagine/libri.html





Ultima modifica 12/6/2019