Organizing Committee: M. Correggi , D. Fermi , A. Lucia , M. Moscolari , A. Olgiati

Poster

Mattia Cantoni

Politecnico di Milano

title: An overview of the Boltzmann-Nordheim equation 

abstract: The development of Quantum Mechanics at the beginning of the last century led Nordheim to propose a correction of the Boltzmann equation in order to describe the behaviour of particles obeying Bose-Einstein or Fermi-Dirac statistics. In this talk, I will introduce the Boltzmann-Nordheim equation for a gas of bosons and its main properties: conservation laws, existence and uniqueness of the equilibrium distribution and H-theorem. Moreover, I will discuss its application to the description of the emergence of a Bose-Einstein condensate when a gas of bosons is cooled below a critical temperature. This process can be divided into three stages: the first and the third ones are kinetic stages, which occur before and after the nucleation of the condensate. They can be treated with the Boltzmann-Nordheim equation, which describes restructuring of the distribution function at the first stage and growth of the condensate at the third one. In the end, I will present some open problems, including a rigorous derivation of the Boltzmann-Nordheim equation from the fundamental laws of Quantum Mechanics and the analytical derivation of self-similar solutions.

Filippo Perani

Politecnico di Milano

title: The quantum Heisenberg model: a semiclassical approach

abstract: A significant open problem in theoretical and mathematical physics is the proof of the long-range order for the three-dimensional ferromagnetic Quantum Heisenberg Model (QHM) at low temperature (T), unlike the antiferromagnetic QHM and the Classical Heisenberg Model, whose long-range order is proved by reflection positivity methods. At a heuristic level, the low-temperature thermodynamics of ferromagnetic QHM can be deduced from spin wave theory. This model is an approximation that neglects both the kinematical and dynamical interactions that appear by writing the Heisenberg Hamiltonian in the Holstein-Primakoff bosonic representation. By studying the free energy of the QHM, rigorous results have been proved as S → ∞ and for the low temperature behaviour, in the thermodynamic limit. In this talk, I will present results obtained by further pursuing the analysis of the high-spin limit of the QHM free energy (finite volume). In particular, we studied the regime used by Lieb (T ∼ S2), which is known to be a classical one in the high-spin limit. By using the Holstein-Primakoff representation of the model, we were able to prove how the kinematic interaction behaves in this limit. Also, we characterized the Holstein-Primakoff representation by noticing a geometrical link between the bosonic and the spin classical phase spaces.

Tommaso Pistillo

Politecnico di Milano

title: Ground states for the Hartree energy functional in the limit case

abstract: It is well known that Hartree type equation arise as mean field limits of systems of interacting bosons. Results on existence of ground states and global dynamics have been proved from the '80s using Lions' concentration-compactness method. We extend previous results to a larger (and almost maximal) class of convolutional potentials, in the case where no external potential is present. To do so, we rely only on variational methods, i.e. the aforementioned concentration-compactness principle and the direct method of calculus of variations.