Dipartimento di Matematica
Guido Castelnuovo

Alessandro Teta (Università di Roma La Sapienza)
La nascita della Meccanica Quantistica negli articoli di Heisenberg (1925) e di Schroedinger (1926)

La Meccanica Quantistica si afferma negli anni 1925/26 grazie soprattutto ai lavori di Heisenberg e di Schroedinger. Nel seminario, dopo una breve premessa sul contesto storico, si discuteranno le idee fondamentali di questi lavori, limitando al minimo gli aspetti tecnici. Si cercherà in particolare di mettere in evidenza le differenze dei due approcci, basati su modi differenti di concepire la descrizione del mondo fisico. Il seminario sarà in italiano.

[Seminario congiunto di Storia della Matematica e Fisica Matematica]

Rossana Marra (INFN, Università di Roma Tor Vergata)
Ghost effect equations from the Boltzmann theory

The diffusive hydrodynamic limit of the Boltzmann equation in the low Mach number regime is usually described by the incompressible Navier-Stokes-Fourier equations. When the density and temperature at initial time and/or the temperature on the boundary have gradients of order 1 the limiting equations (called "ghost effect equations") are different and cannot be predicted by the classical fluid theory. Proving the hydrodynamic limit under these conditions has been an open and challenging problem. In this talk I will discuss the rigorous proof of this non standard hydrodynamic behaviour for the stationary Boltzmann equation in a bounded domain with diffuse reflection boundary condition. Work in collaboration with R. Esposito, Y. Guo and Lei Wu.

Massimo Moscolari (Politecnico di Milano)
General bulk-edge correspondence at positive temperature

By extending the gauge covariant magnetic perturbation theory to operators defined on half planes, we prove that for general 2d random ergodic magnetic Schrödinger operators the celebrated bulk-edge correspondence is just a particular case of a much more general paradigm, which also includes the theory of diamagnetic currents and of Landau diamagnetism. Our main result is encapsulated in a formula, which states that the derivative of a large class of bulk partition functions with respect to the external constant magnetic field, equals the expectation of a corresponding edge distribution function of the velocity component which is parallel to the edge. Neither spectral gaps, nor mobility gaps, nor topological arguments are required. The equality between the bulk and edge indices, as stated by the conventional bulk-edge correspondence, is obtained as a corollary of our purely analytical arguments by imposing a gap condition and by taking a "zero temperature" limit. The talk is based on a joint work with H. Cornean and S. Teufel.

Lorenzo Bertini (Sapienza, Roma)
Current fluctuations in stochastic lattice gases: Donsker-Varadhan meets Freidlin-Wentzell

We discuss the large deviations asymptotic of the time-averaged empirical current in stochastic lattice gases in the limit in which both the number of particles and the time window diverges. For some models it has been shown that dynamical phase transitions occur: the optimal density profile to realize such deviations is given by travelling waves rather than by homogeneous profiles. We shall prove a variational representation, proposed by Varadhan, for the corresponding rate function that is obtained by projecting the large deviations at the level of the empirical process.

Angeliki Menegaki (IHES, Paris)
Quantitative framework for hydrodynamic limits

We will present a new quantitative approach to the problem of proving hydrodynamic limits from microscopic stochastic particle systems, namely the zero-range and the Ginzburg-Landau process with Kawasaki dynamics, to macroscopic partial differential equations. Our method combines a modulated Wasserstein-distance estimate comparing the law of the stochastic process to the local Gibbs measure, together with stability estimates a la Kruzhkov in weak distance and consistency estimates exploiting the regularity of the limit solution. It is simplified as it avoids the use of the block estimates. This is a joint work with Daniel Marahrens and Clément Mouhot (University of Cambridge).

Corentin Le Bihan (ENS Lyon)
Dynamical correlation of the Gibbs measure of a gas in a low density scaling

We consider a gas of N particles in a box of dimension 3, interacting pairwise with a potential α V(r/ε). We want to understand the behavior of the system in the limit N → ∞, with a suitable scaling for α and ε. If we choose ε = 1, α = 1/N, it is the mean field limit: particles interact weakly at long distance. We are interested in the low density limit ε = N ^-1/2, α = 1. Then the distance crossed by a particle is constant. This limit is well understood since the work of Lanford: the empirical law of the system converges to a solution of the Boltzmann equation. It is a kind of Law of Large Numbers. Sadly this convergence occurs only for a short time. In order to go to longer times, we study the fluctuations around the equilibrium, which follow a linearized version of the Boltzmann equation. The talk will present the idea of the proof of Bodineau, Gallagher, Saint Raymond and Simonella for hard sphere potentials and the idea of an improvement in case of realistic interaction potentials.

Daniele Ferretti (Sapienza, Roma)
Contact interactions for many-particle quantum systems in dimension three

We discuss a class of regularized zero-range Hamiltonians for three different problems satisfying a bosonic symmetry in dimension three. Following the standard approach in defining such Hamiltonians in three dimensions, one comes up with the so-called Ter-Martirosyan Skornyakov Hamiltonian that turns out to be unbounded from below (Thomas collapse occurs in case of usual two-body point interactions since zero-range interactions become too singular when three or more particle get close). In order to avoid this energetical instability, we consider a many-body repulsion meant to weaken the strength of the interaction as more than two particles coincide. More precisely, developing a suggestion made in the early '60s by Minlos and Faddeev, we introduce an effective scattering length depending on the positions of the particles. In case of a three-boson problem (or a Bose gas of non-interacting particles interacting only with an impurity) such a function vanishes as a third particle gets closer to the couple of interacting particles. Similarly, dealing with an interacting Bose gas, we also take into account a four-body repulsion in order to handle the ultraviolet singularity associated with the collapse of two distinct couples of interacting particles. We show that the Hamiltonians corresponding to these regularizations are self adjoint and bounded from below, provided that the strength of the many-body force is large enough. Moreover, we compare our results with the ones obtained in the early '80s by Albeverio et al, which exploits an alternative method based on Dirichlet forms, providing the construction of a one-parameter family of many-body regularized zero-range Hamiltonians. In particular, we prove that such a class of regularized Hamiltonians is a special case of what can be obtained with our approach.

Chiara Saffirio (University of Basel)
Uniqueness criteria for the Vlasov-Poisson system and applications to mean-field and semiclassical problems

The Vlasov-Poisson system is a non-linear PDE describing the mean-field time-evolution of particles forming a plasma. In this talk I will present uniqueness criteria for the Vlasov-Poisson equation, emerging as corollaries of stability estimates in strong (L^p) topologies or in weak topologies (induced by Wasserstein distances), and show how they serve as a guideline to solve mean-field and semiclassical problems. Different topologies will allow us to treat different classes of quantum states.

Thierry Paul (LYSM, Rome)
From microscopic to macroscopic: Vlasov, Euler and graph-limit description of large number classical dynamics of multi-agent, particles and cells possibly interacting with a chemical background

We will present several results concerned with the collective motion involving N agents (distinguishable) or N cells/particles (indistinguishable) possibly involving chemotaxis as N becomes large. We will describe the different steps forming the way along which genuine non-linear partial differential equations posed on the one particle-space (Vlasov, Euler, graph-limit) can be rigorously derived out of the ordinary differential equations driving the microscopic dynamics. Recent numerical simulations will be presented, showing how the striking effects, e.g. of alignment dynamics, remain (somehow mysteriously) visible when passing from the microscopic to the macroscopic scale. Finally we will show that "any" quasi-linear PDE can be seen as deriving from a multi-agent system on the limit of large numbers of agents.

Matteo Gallone (SISSA, Trieste)
Prethermalization in classical and quantum physics

Understanding the route to thermalization of a physical system is a fundamental problem in statistical mechanics. When a system is initialized far from thermodynamical equilibrium, the situation is much complex as many interesting phenomena may arise. Historically, the first discovery in this direction is the so-called "Fermi-Pasta-Ulam paradox" that is the fact that, when excited with low energy, a classical chain of anharmonic oscillators has a quasi-periodic dynamics for very long-time scales. Since then, a lot of effort has been spent in trying to understand the mechanism behind the long thermalization time observed in numerical experiments and some rigorous results appeared in the last 20 years explaining the phenomenon with integrable normal forms of the equations of motion. Problem of prethermalization has become topical in the last years after the observation of the FPU-like recurrence in Bose-Einstein condensates, optical fibers and graphene resonators. Many challenging questions are still open and deserve investigations in the next future. In this talk I will present recent results that we obtained in the analysis of classical lattice systems and quantum spin chains.

Marco Coco (Università Politecnica delle Marche)
Monte Carlo study of charge transport in graphene: computation and theoretical perspectives

The Ensemble Monte Carlo (EMC) method has become a standard tool for the study of transport problems in electronic devices. When the Pauli principle is no longer negligible, however, the EMC suffers from some drawbacks regarding the correct reconstruction of the carrier distribution. We will show a new Monte Carlo scheme which correctly takes into account the Pauli principle. Almost all of the previous approaches were based on some approximations in the description of the distribution function or of the scattering terms even if earlier a novel procedure was proposed for silicon, by adding the Pauli principle also at the end of the free flight. We address also the question of the correctness of representing the free flight in such a quantum view in place of the semiclassical one with the Liouville operator, in the case of a suspended monolayer graphene. Some theoretical perspectives arising from the presented numerical works will be discussed as well.

Daniel Heydecker (Max Planck Institute, Leipzig)
Large Deviations of Kac’s Random Walk & the H-Theorem

We consider the dynamical large deviations of Kac’s model for the Boltzmann Equation in the many-particle limit N → ∞. With the expected rate function, the large deviations lower bound is only true on a restricted class of paths, and we find counterexamples to a global lower bound related to Lu and Wennberg’s energy non-conserving solutions to the Boltzmann equation: we will give a direct proof by mimicking the argument of Lu and Wennberg. On the other hand, the class of paths where we have a matching upper and lower bounds is sufficiently rich to rederive the celebrated Boltzmann H-Theorem.

Elena Pulvirenti (TU Delft)
Metastability of Glauber dynamics with inhomogeneous coupling disorder

I will first introduce a general class of mean-field-like spin systems with random couplings that comprises both the Ising model on inhomogeneous dense random graphs and the randomly diluted Hopfield model. I will then present quantitative estimates of metastability in large volumes at fixed temperatures when these systems evolve according to a Glauber dynamics, i.e. where spins flip with Metropolis rates at inverse temperature β. The main result identifies conditions ensuring that with high probability the system behaves like the corresponding system where the random couplings are replaced by their averages. More precisely, we prove that the metastability of the former system is implied with high probability by the metastability of the latter. Moreover, we consider relevant metastable hitting times of the two systems and find the asymptotic tail behaviour and the moments of their ratio. This result provides an extension of the results known for the Ising model on the the Erdos–Renyi random graph. Our proofs use the potential-theoretic approach to metastability in combination with concentration inequalities. Based on a joint work in collaboration with Anton Bovier, Frank den Hollander, Saeda Marello and Martin Slowik.

Alberto Fachechi (Sapienza, Roma)
PDE/Statistical Mechanics relation: from p-spin ferromagnets to dense associative memories

Spin-glass models have always been a challenge both for physicists and mathematicians, as they exhibit - also in their simplest version - complex, intriguing emergent behaviors whose rigorous description is particularly tricky. Among the techniques developed to deal with the thermodynamics of these systems, an important example is constituted by Guerra's interpolating techniques, which turn out to be connected with PDE systems. In this talk, we will present some recent results concerning the relation between Guerra’s interpolating partition functions of p-spin models and the Burgers hierarchy. Finally, we also discuss a generalization of these findings in some AI-relevant models, the so-called dense associative memories.

Francesca Elisa Leonelli (Sapienza, Roma)
Addressing climate variability and extreme events in the ocean with spectral time series analysis and neural network models

Raphael Winter (ENS Lyon)
Collective effects in scaling limits of interacting particle systems

Alessia Nota (Università degli Studi dell'Aquila)
Homoenergetic solutions of the Boltzmann equation

In this talk I will consider a particular class of solutions of the Boltzmann equation, known as homoenergetic solutions, which were introduced by Galkin and Truesdell in the 1960s. These are a particular type of non-equilibrium solutions of the Boltzmann equation and they are useful to describe the dynamics of Boltzmann gases under shear, expansion or compression. Due to the fact that these solutions describe far-from-equilibrium phenomena their long-time asymptotics cannot always be described by Maxwellian distributions. For several collision kernels the asymptotics of homoenergetic solutions is given by particle distributions which do not satisfy the detailed balance condition. I will discuss different possible long-time asymptotics of homoenergetic solutions of the Boltzmann equation, as well as some conjectures and open problems in this direction. These are joint works with A. V. Bobylev, R. D. James and J. J. L. Velàzquez.

Serena Cenatiempo (GSSI, L'Aquila)
A second order upper bound for confined hard sphere bosons

In this talk we present a second order upper bound for the ground state energy of a gas of N bosons confined in the three dimensional unit torus and interacting through a hard sphere potential with radius of order 1/N (Gross-Pitaevskii regime). Our result matches the known expression for the energy in the case of integrable potentials, and represents the first example where an upper bound for a hard sphere Bose gas capturing the second order term is obtained. The proof is based on a suitable modification of the trial state used by Dyson in his pioneering '57 paper. Joint work with G. Basti, A. Olgiati, G. Pasqualetti and B. Schlein.

Lorenzo Panebianco (Sapienza, Roma)
Relative entropy distributions in local QFT

Davide Fermi (Sapienza, Roma)
Semiclassical limit with zero-range potentials in one dimension

Biagio Cassano (Università di Bari)
General δ-shell interactions for the two-dimensional Dirac operator

Luca Oddis (Sapienza, Roma)
Two-anyon Schrödinger operators