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Mathematical Physics 1
Corso di Laurea Magistrale in Matematica - A.A. 2014/15 (1 anno, 2 semestre, CFU 6, codice - DT0126)
Docente: Maurizio Serva
”Nissuna umana investigazione si pu`o dimandare vera scienzia s’essa non passa per le matematiche dimostrazioni, e se tu dirai che le scienzie, che principiano e finiscono nella mente, abbiano verit`a, questo non si concede, ma si niega, per molte ragioni, e prima, che in tali discorsi mentali non accade esperienzia, sanza la quale nulla d`a di s´e certezza”. - Leonardo da Vinci (1452-1519)
Course Contents
• PART I - Zermelo and Loschmidt paradoxes. Reversibility of mechanics. Poincar recurrence theorem. Scales of time. Example from probability (N coins toss) explains why Zermelo and Loschmidt paradoxes are irrelevant when the Avogadro number N is involved. The continuity equation and incompressibility of the Hamiltonian fluid. The Boltzmann equation and the Stosszahlansatz. The H function and the Boltzmann H theorem. Validity of the molecular chaos assumption (Stosszahlansatz). Equilibrium and Maxwell-Boltzmann distribution. The limit of Grad. The Ehrenfest model: the H theorem and the molecular chaos.
• PART II - The bi-dimensional Lorentz gas on a lattice. The distribution of obstacles and the motion of electrons.
Expected value of a test function in the limit of low density of obstacles. The expected velocity. Expected value and standard deviation of the position. Random initial conditions and the joint probability of velocity and position at a time τ . The linear Boltzmann equation for the joint probability of velocity and position. The entropy and the non negativity of the entropy production. Reversibility and irreversibility revisited. Equilibrium properties of the Ideal gas from the Maxwell-Boltzmann distribution: the Joule’s law, the ideal gas law and the entropy as a function of density and temperature.
• PART III - Classical statistical mechanics: the micro-canonical ensemble. Entropy and the zeroth law, the first law and the second law (Clausiuss statement). The ideal gas with the micro-canonical ensemble. The internal energy and the equation of state. Mixing entropy and the Gibbs paradox. Maxwell distribution of velocity and heat capacities. Two-level systems. Classical and quantum harmonic oscillators. The canonical ensemble: physical definition. Mathematical definition and equivalence with the micro-canonical ensemble in the thermodynamical limit. Canonical examples: ideal gas, two-level systems. Non-interacting spins in a external magnetic field: free energy, entropy, energy, magnetization and susceptibility.
• PART IV - The one-dimensional, first neighbors, Ising model in presence of an external magnetic field: transfer matrices method. Low and high temperature limits: free energy, entropy, energy, magnetization and suscepti- bility. The mean field (fully connected) Ising model or Curie-Weiss model. The spontaneous magnetization in absence of external magnetic field and the phase transition. The critical exponents near critical temperature and their scaling. Behavior of energy and heat capacity close to the critical temperature and to the zero temperature.
Discontinuities of thermodynamical functions and the order of a phase transition.
• PART V - The grand canonical ensemble: the chemical potential and the grand potential. The perfect gas and a system of non-interacting spins in a external magnetic field. Basic facts of quantum mechanics. Fermi and Bose statistics: the role of spin. Simple model o non interacting particles with a single state of zero energy and (thermodynamically) many states of identical positive energy: the fermionic case (Fermi energy) and the bosonic case (Bose-Einstein condensation). Ideal quantum gases (Fermi and Bose): average occupation number, pressure, density and energy. High temperature limit and the classical ideal gas. Fermi gas: Fermi energy and Fermi pressure. Bose gas: transition and Bose-Einstein condensation. Bose-Einstein condensation for a toy model o non interacting particles with degenerate energy levels equally spaced.
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Textbooks
M. Kardar, Statistical Physics of Particles, Cambridge University Press, 2007.
M. Falcioni and A. Vulpiani, Meccanica Statistica Elementare, Springer-Verlag Italia, 2015.
E. Presutti, Lezioni di Meccanica Statistica, Aracne, 1995.
R. J. Baxter. Exactly Solved Models in Statistical Mechanics, Academic Press, 1989.
Entry requirements:
Probability theory, classical mechanics, thermodynamics.
Learning outcomes:
This course aims to enable students to make a connection between mechanics and thermodynamics Teaching methods:
Lectures and exercises Assessment methods:
Written and oral examination
