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December 7 |
Michela Ottobre (Heriot Watt University, Edinburgh, UK) |
Diffusion limits for Markov Chain Monte Carlo algorithms out of stationarity
Markov Chain Monte Carlo (MCMC) algorithms are powerful sampling tools to extract information from a given probability measure (commonly referred to as the target measure). The basic prescription behind MCMC is the following: construct a Markov Chain which admits the target measure as unique invariant measure. If, additionally, such a Markov Chain enjoys good ergodic properties, then, running the chain "long enough" produces samples from the desired measure. By the point of view of computational efficiency, the construction of such a chain comes with two requirements: i) the chain should converge to equilibrium as fast as possible and ii) once equilibrium is reached, it should explore the state space quickly and thoroughly. The study of MCMC-generated chains has been a compelling motivation for several interesting developments in the theory of Markov Processes. In this talk we will first introduce the MCMC framework and present some applications and then move on to present recent results on diffusion limits for certain MCMC processes. In particular we will explain how obtaining diffusion limits for Markov Chains (produced through the MCMC paradigm) can help quantifying the "efficiency" of the associated algorithm. |
November 30 |
Francesco Salvarani (Università di Pavia) |
Gas di Knudsen in domini tempo-dipendenti
Il seminario illustrerà alcuni problemi relativi allo studio di dinamiche gassose definite in domini dipendenti dal tempo. Nella prima parte, di taglio più teorico, verrà sviluppata la teoria di esistenza di soluzioni per il gas di Knudsen in domini evolutivi, basata sulla nozione di tempo di uscita retrogrado. La seconda parte sar àconsacrata allo studio numerico del problema e alla descrizione dei risultati delle simulazioni. |
November 23 |
Marco Falconi (Università degli Studi Roma Tre) |
External Potentials Generated by the Interaction with a Semiclassical Boson Field
In physical experiments of condensed matter, quantum particles (atoms) are immersed in controllable external potentials, such as harmonic traps and optical lattices. These potentials are obtained using finely tuned lasers, or magnetic fields, acting on the atoms. In this talk I will present some results, obtained in collaboration with Michele Correggi, that shed light on the procedure above. We consider a coupled system consisting of N quantum particles in (linear) interaction with a scalar boson field. Using tools of infinite dimensional semiclassical analysis, a partially classical limit on the scalar field is studied. We prove the convergence of the partial trace of the full Hamiltonian to an effective reduced Hamiltonian on the particles alone (in the strong resolvent sense). The effective Hamiltonian contains an induced external potential, that has an explicit form that depends only on the state of the field. We also prove the convergence of the ground state energy of the full quantum system to the infimum on all possible classical field configurations of the ground state energy of the effective Hamiltonian. |
November 22–24 |
Marco Falconi (Università degli Studi Roma Tre) |
An introduction to semiclassical analysis in infinite dimensions, and its applications to mean and quantum field theories (mini-course)
In this short course we introduce the basic tools of semiclassical analysis, for systems with an infinite dimensional phase space. In order to do so, we will touch upon many interesting topics of modern analysis, from operator algebras to optimal transportation. Some physical applications will also be discussed, in particular Bohr's correspondence principle for bosonic quantum field theories, and the mean field evolution of general quantum states. |
November 15 |
Nicolò Catapano (Università degli Studi di Roma "La Sapienza") |
The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit
We consider a system of N classical particles interacting by a smooth short-range potential, in an intermediate scaling between the low-density and the weak-coupling . We adapt to our context some recent tecniques to prove the rigorous derivation of the linear Boltzmann equation. Then, following the ideas of Landau, we prove the rigorous derivation of the linear Landau equation from the linear Boltzmann equation. From this two results we obtain the rigorous derivation of the Linear Landau equation from this particle system. |
November 2 |
Marius Mantoiu (University of Chile, Santiago de Chile, Chile) |
Algebras of Pseudodifferential Quantum Observables on Groups
We present a short overview of quantization by pseudodifferential operators. Besides the well-known setting, we will also describe new pseudodifferential calculi associated to locally compact and Lie groups. |
October 11 |
Serena Cenatiempo (GSSI, L'Aquila) |
Quantum many-body fluctuations around nonlinear Schrödinger dynamics
We consider the many body quantum dynamics of a system of a large number N of bosons interacting through a two-body potential N^{3\beta -1} V(N^\beta x). For $0<\beta<1$, we obtain a norm approximation of the evolution of an appropriate class of data on the Fock space. To this end, we need to correct the evolution of the condensate described by the one-particle nonlinear Schrödinger equation by means of a fluctuation dynamics, governed by a quadratic generator. Obstructions to the extension of this result to the Gross-Pitaevskii scaling limit $\beta=1$ will be also discussed. Joint work with C. Boccato and B. Schlein. |
October 6,7,10 |
Serena Cenatiempo (GSSI, L'Aquila) |
Gross-Pitaevskii Equation from Quantum Dynamics (mini-course)
We consider the many body dynamics of bosonic systems in the Gross-Pitaevskii regime, a special case of a dilute limit which is relevant for the description of initially trapped Bose condensates. In this regime the dynamics is known to be approximated by the Gross-Pitaevskii equation. In particular, for positive potentials, one can show that the reduced density matrices associated with the solution of the many body Schrödinger equation converge in trace norm towards orthogonal projections onto products of solutions of the Gross-Pitaevskii equation. In these lectures we present a rigorous derivation of the Gross-Pitaevskii equation based on a representation of the system on the bosonic Fock space, and on the study of the time evolution of a special class of initial data, known as coherent states. This approach, first introduced by Rodnianski-Schlein [1] in the mean field contest, and later extended to the Gross-Pitaevskii regime in Benedikter-De Oliveira-Schlein [2], also provides precise bounds on the rate of the convergence towards the limiting Gross-Pitaevskii dynamics. Moreover, in some limiting regimes it has been used to describe fluctuations around the effective dynamics. [1] I. Rodnianski, B. Schlein. Quantum fluctuations and rate of convergence towards mean-field dynamics, Commun. Math. Phys. 291 (2009) [2] N. Benedikter, G. De Oliveira and B. Schlein. Quantitative derivation of the Gross-Pitaevskii equation, Comm. Pure Appl. Math. 68 (2015) |
September 28 |
Giambattista Giacomin (LPMA, Université Paris 7, France) |
Dynamical models on random graphs: mean field versus long time behavior
I will address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and the corresponding mean field model, i.e. a model on the complete graph with a suitably renormalized interaction parameter. Examples include Erdos-Renyi graphs with edge probability p(n), n is the number of vertices, such that p(n)n tends to infinity as n does. I will present a general result establishing this proximity on finite time horizon, by exploiting the fact that both systems are accurately described by a Fokker-Planck PDE in the limit. I will then try to put this result into perspective by discussing the limitations of our approach. |
June 1 |
Antonio Moro (Northumbria University, Newcastle, UK) |
Quasitriviality and Universality of Integrable Conservation Laws and Statistical Mechanics
Conservation laws, intended as nonlinear (diffusive and/or dispersive) systems of PDEs, naturally arise in a large variety of physical and mathematical problems, from fluid dynamics to differential geometry. Quasitriviality is the property of such systems that can be reduced to a system of hydrodynamic type via a quasi-Miura transformation. The quasi-Miura transformation is a powerful tool to construct formal solutions to a PDEs given any solution to the associated hydrodynamic type equation. We discuss the quasitriviality of scalar conservation laws in relation with the classical method of transport equations with a focus on the integrable viscous conservation laws. The application of the quasi-miura transformation to the hodograph equation also provides an effective method to construct perturbed solutions in both the Hamiltonian and non-Hamiltonian case. We also provide evidence of their universal behaviour in the vicinity of the critical point of gradient catastrophe. We finally propose a statistical mechanical interpretation of fundamental objects from the theory of integrable conservation laws showing how this theory can be effectively deployed for the analysis and solution of statistical mechanical models in both finite size and thermodynamic regime. |
May 24 |
Joel Lebowitz (Rutgers University, NJ, US) |
On the Definition of (and Approach to) Thermal Equilibrium in an Isolated Macroscopic System: Quantum and Classical
This seminar will discuss the notion of thermal equilibrium for an individual closed macroscopic quantum system in a pure state, i.e., described by a wave function. The macroscopic properties in thermal equilibrium of such a system must be the same as those obtained from thermodynamics, e.g., spatial uniformity of temperature and chemical potential. When this is true we say that the system is in macroscopic thermal equilibrium (MATE). Such a system may however not be in microscopic thermal equilibrium (MITE). The latter requires that the reduced density matrices of small subsystems be close to those obtained from the microcanonical, equivalently the canonical, ensemble for the whole system. The distinction between MITE and MATE is particularly relevant for systems with many-body localization (MBL) for which the energy eigenfuctions fail to be in MITE while necessarily most of them, but not all, are in MATE. For a classical macroscopic system, MATE holds for most phase points on the energy surface, but MITE fails to hold for any phase point. The approach to equilibrium from an initial nonequilibrium state will also be discussed. |
May 18 |
Pierre Picco (CNRS & Université de Provence, Marseille, France) |
Recent Results on the Phase Separation Phenomenon for One-dimensional Long-Range Ising Model
In this talk we first give a Roma oriented review of the history of one-dimensional long-range Ising model. We present recent results done in collaboration with M. Cassandro and I. Merola on the phenomenon of separation of phase, this is the one-dimensional analogue of the Minlos & Sinai theory, the Dobrushin, Kotecky & Shlosman and Pfister theory. |