Seminario di Fisica Matematica
A partire da Aprile 2023, i Seminari di Fisica Matematica si svolgono all'interno delle attività del Progetto Eccellenza (CUP B83C23001390001).
L'Area di Ricerca "Mathematical Models in Statistical Physics" del Dipartimento di Eccellenza organizza in particolare un periodo tematico su "Statistical Mechanics and Non Equilibrium Processes" che si svolgerà fra febbraio e giugno 2025. Il periodo verrà lanciato dalla scuola invernale "Statistical Mechanics, Nonequilibrium Processes and Probability" che si terrà presso il Dipartimento dal 17 al 28 febbraio 2025. Maggiori informazioni sono disponibili alla pagina web dedicata.
SEMINARI 2024 | |
Martedì, 15/10/2024 h15:00 Aula E |
Suren Poghosyan (Institute of Mathematics, Armenian National Academy of Sciences) We consider classical continuous system of interacting particles in Euclidean space (classical gas). Our approach to the limit theorems for the particle number is based on the method of cluster expansions which is well known for nite Gibbs processes. In case of the limiting Gibbs process with empty boundary conditions, we use our result on the cluster representation of such processes which goes back to Malyshev and Minlos. We prove integral and local central limit theorems for a large class of stable and regular pair potentials (which include physically relevant interactions) if the activity is small. In case of local central limit theorem we obtain also an estimate of convergence rate. |
9/10/2024 |
Giuliano Angelone (Università di Roma La Sapienza) In this seminar I will present some of my research activities related to the rigorous analysis of a quantum particle confined in a bounded region of space. In the first part of the seminar I will focus on a free quantum particle, living in the one-dimensional segment, considering all the possible self-adjoint extensions of the Hamiltonian operator. The latter are described in terms of suitable boundary conditions. In particular, I will discuss both the emergence of a non-trivial unitary equivalence between different self-adjoint extensions, that is connected with the problem of "hearing" a given extension, and how some quantum boundary conditions persist when we perform the classical limit in the phase-space. In the second part of the seminar I will then move to a simple model of the quantum Hall effect, described by the magnetic Laplacian acting in an infinitely long two-dimensional strip. For this system I will describe how the quantization of the Hall conductivity is affected by finite-size effects and by non-conventional boundary conditions, and also how the latter are associated with a new kind of quantum states having no classical counterpart. [Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022AKRC5P "Interacting Quantum Systems: Topological Phenomena and Effective Theories" finanziato dall’Unione europea – Next Generation EU.] |
25/09/2024 |
Heide Langhammer (Weierstrass Institute for Applied Analysis and Stochastics, Berlin) In this talk we consider a spatial version of the Marcus-Lushnikov process, which models the evolution of particles that merge pairwise in a series of coagulation events. The particles are equipped with a spatial location and an integer mass. In a coagulation event a pair of particles is merged and replaced by a new particle. The mass of the newly formed particle is the sum of the former two and its location is sampled according to a given distribution. The exponential rate of a coagulation event depends on the locations and masses of the particle pair via a coagulation kernel. For the non-spatial version of this model it is known that certain kernels exhibit a gelation phase transition, which describes the formation of one or several large particles after a finite time. In this talk we introduce a new approach for studying the process. We will provide a formula that connects its distribution to a Poisson point process and show how this can be used to derive a large deviation principle. From this we gain new insights into the limiting behavior of the process and obtain criteria for the gelation phase transition. [Il seminario si svolgerà all'interno delle attività del progetto PRIN 202277WX43 "Emergence of condensation-like phenomena in interacting particle systems: kinetic and lattice models" finanziato dall’Unione europea – Next Generation EU.] |
22/05/2024 |
Abdallah Maichine (Mohammed V University in Rabat) For a uniform magnetic field, we provide a decomposition of operators commuting with the associated magnetic translations. As an application, we rewrite the energy per unit surface for an electronic system with 2d symmetry as a function of the magnetic field intensity. We also investigate the ground state of such a system. Besides, we also derive the Thomas-Fermi-like energy density in 2d and 3d. [Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022AKRC5P "Interacting Quantum Systems: Topological Phenomena and Effective Theories" finanziato dall’Unione europea – Next Generation EU.] |
15/05/2024 h16:15 |
Raffaele Scandone (Università degli Studi di Napoli "Federico II") I will discuss the existence of turbulent solutions to a quantum hydrodynamic (QHD) system, with periodic boundary conditions. A suitable nonlinear change of variables (the Madelung transform) formally connects the QHD system to a non-linear Schrödinger (NLS) equation, for which we can construct smooth solutions displaying arbitrarily large growth of Sobolev norms above the energy regularity level. This amounts to a cascade in time of the energy to higher Fourier modes. In addition, these solutions can be designed to be small amplitude perturbations of stationary states, which implies in particular absence of quantum vortices. This allows to exploit an equivalence between high regularity QHD- and NLS- norms, which eventually yields the existence of smooth, turbulent solutions to the quantum hydrodynamic system. [Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022CHELC7 "Singular Interactions and Effective Models in Mathematical Physics" finanziato dall’Unione europea – Next Generation EU.] |
24/04/2024 h16:45 |
Morris Brooks (Universität Zürich) In this talk we will review some recent results on the Fröhlich polaron, which is a model for a charged particle coupled to a polarizable medium. We will especially focus on a conjecture due to Landau and Pekar from 1948, claiming that the mass of the charged particle is effectively increased due to its interaction with the environment according to the asymptotic formula m_eff=\alpha^4 m_LP, where \alpha is the coupling strength between particle and medium and m_LP is an explicit constant. Notably the recent progress is due to two rather distinct approaches, one working with functional integration (probabilistic/commutative) and the other one working with functional analytic methods (operator theory/non-commutative), which are connected by the celebrated Feynman-Kac formula. |
20/03/2024 |
Carlo Presilla (Università di Roma La Sapienza) The mere existence and the spectroscopic behavior of pyramidal molecules like ammonia have remained unsolved problems since the early developments of quantum mechanics. A possible solution can be found on the idea of spontaneous symmetry breaking. I discuss a model proposed some years ago with G. Jona-Lasinio and C. Toninelli and its more recent developments. Despite the simplicity of the model, results are obtained which are in quantitative agreement with experiments and provide predictions for the behavior of general chiral gases and their mixtures in fields from cold atoms to astronomical data. However, many open questions remain including whether this model can be considered a step to understanding homochirality. |
13/03/2024 |
Jacob Shapiro (Princeton University) In this talk I will describe some of the mathematical aspects of disordered topological insulators. These are novel materials which insulate in their bulk but (may) conduct along their edge; the quintessential example is that of the integer quantum Hall effect. What characterizes these materials is the existence of a topological index, experimentally measurable and macroscopically quantized. Mathematically this is explained by applying algebraic topology to the space of appropriate quantum mechanical Hamiltonians; I will survey some recent results mainly concentrating on the classification problem in one dimension, where the problem reduces to studying spaces of unitaries (resp. orthogonal projections) which essentially-commute with a fixed projection. Joint with Jui-Hui Chung. [Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022AKRC5P "Interacting Quantum Systems: Topological Phenomena and Effective Theories" finanziato dall’Unione europea – Next Generation EU.] |
6/03/2024 |
Giulia Basti (GSSI, L'Aquila) In the last decades, since the first experimental realizations of Bose- Einstein condensates, the study of large bosonic systems has been a very active field of research both in physics and in mathematics. In experiments Bose gases are often very dilute and can be well described in the Gross-Pitaevskii limit, i.e. as quantum systems of N confined particles, interacting through a potential with scattering length of order 1/N where N tends to infinity. In this talk we present a recent result on a hard-sphere Bose gas in this regime. Namely, we prove a second order upper bound on the ground state energy matching the known expression of the energy for integrable potentials. We also discuss a new upper bound for hard-spheres in the thermodynamic limit, where the number of particles and the size of the box are sent to infinity keeping the density fixed. Our result resolves the ground state energy up to an error of the order of the so-called Lee-Huang-Yang correction. Based on joint works with S. Cenatiempo, A. Giuliani, A. Olgiati, G. Pasqualetti, B. Schlein. [Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022AKRC5P "Interacting Quantum Systems: Topological Phenomena and Effective Theories" finanziato dall’Unione europea – Next Generation EU.] |
28/02/2024 Aula Picone |
Horia Cornean (Università di Aalborg) We provide sufficient conditions such that the time evolution of a mesoscopic tight-binding open system with a local Hartree–Fock non-linearity converges to a self-consistent non-equilibrium steady state, which is independent of the initial condition from the "small sample". We also show that the steady charge current intensities are given by Landauer–Büttiker-like formulas, and make the connection with the case of weakly self-interacting many-body systems. This is joint work with Giovanna Marcelli (https://arxiv.org/abs/2309.01564). [Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022AKRC5P "Interacting Quantum Systems: Topological Phenomena and Effective Theories" finanziato dall’Unione europea – Next Generation EU.] |
27/02/2024 h16:30 |
Alessandro Teta (Università di Roma La Sapienza) Una particella alfa, emessa come un’onda sferica in una camera a nebbia, produce una traccia visibile che si interpreta in modo naturale come traiettoria della particella. La spiegazione teorica di questo fatto nell’ambito della Meccanica Quantistica è un problema non banale e fu uno dei problemi rilevanti discussi subito dopo la formulazione della teoria. Nel seminario si richiameranno il contesto storico e i primi contributi all’analisi del problema. Quindi verrà descritto in dettaglio l’articolo di N.F. Mott (1929), rilevante non solo nell’ambito del dibattito sui fondamenti della Meccanica Quantistica ma anche come contributo anticipatore della moderna teoria della decoerenza. [Seminario congiunto di Fisica Matematica e Storia della Matematica] |
14/02/2024 |
Giovanni Gallavotti (Università di Roma La Sapienza e Rutgers University) Le implicazioni fisiche del vuoto torricelliano conducono Pascal, 1647, a una disputa con il gesuita E. Noël sulla natura del vuoto, fra la Fisica aristotelica e la Fisica moderna, e ad esprimere lusinghieri riconoscimenti a Torricelli. Dopo un decennio, in relazione alla natura dell'infinito matematico e alle applicazioni del metodo degli indivisibili, Pascal, da estimatore di Torricelli, è ormai (1659) suo accusatore di plagio verso il matematico P. Roberval. Una breve storia che coinvolge anche altri, tra cui Fermat, Descartes e Wallis. [Seminario congiunto di Fisica Matematica e Storia della Matematica] |
31/01/2024 |
Luigi Barletti (Università degli Studi di Firenze) In recent times the hydrodynamic behaviour of electrons in graphene has attracted much interest from both the theoretical and experimental viewpoints. The usual approach to graphene hydrodynamics is "semiclassical", to the extent that the macroscopic equations are derived from a classical Boltzmann equation with conical momentum-energy relation. However, it would be interesting to obtain quantum corrections, similarly to what is done for traditional semiconductors, which can be achieved by using the quantum maximum entropy principle. However, due to the conical singular point, such procedure proves to be difficult in graphene and, so far, quantum corrections have been computed only for a regularized energy band. In the present work we show that singularities can actually be avoided, at least in the case of inviscid equations. [Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022CHELC7 "Singular Interactions and Effective Models in Mathematical Physics" finanziato dall’Unione europea – Next Generation EU.] |
24/01/2024 |
Stefano Olla (Università di Paris-Dauphine e GSSI) We derive the heat equation for the thermal energy under diffusive space-time scaling from a purely deterministic microscopic dynamics satisfying Newton equations perturbed by an external chaotic force acting like a magnetic field. Joint work with Giovanni Canestrari and Carlangelo Liverani. |
10/01/2024 |
Minh-Binh Tran (Texas A&M University) In this talk, we describe the kinetic equation for the Bogoliubov excitations of the Bose-Einstein Condensate. We find three collisional processes: One of them describes the 1↔2 interactions between the condensate and the excited atoms. The other two describe the 2↔2 and 1↔3 interactions between the excited atoms themselves. |
8/01/2024 h14:30 |
Mikaela Iacobelli (ETH Zurich) In this talk, we will present two kinetic models that are used to describe the evolution of charged particles in plasmas: the Vlasov-Poisson system and the Vlasov-Poisson system with massless electrons. These systems model respectively the evolution of electrons, and ions in a plasma. We will discuss the well-posedness of these systems, the stability of solutions, and their behavior under singular limits. Finally, we will introduce a new class of Wasserstein-type distances specifically designed to tackle stability questions for kinetic equations. [Seminario congiunto di Analisi e Fisica Matematica] |
SEMINARI 2023 | |
22/11/2023 |
Diego Alberici (Università degli Studi dell'Aquila) In this talk we will consider a SDE characterised by different time scales, different diffusion coefficients, and a confining potential V that couples all the degrees of freedom. The rates of convergence toward an explicit non-equilibrium measure can be controlled for long time and widely separated time scales, by using logarithmic Sobolev inequalities. The connection of the limit measure with the Parisi functional and Guerra's interpolation method will be discussed. |
02/11/2023 h16:15 |
Chiara Boccato (Università degli Studi di Milano) The interacting Bose gas is a system in quantum statistical mechanics where a collective behavior emerges from the underlying many-body theory, posing interesting challenges to its rigorous mathematical description. While at temperature close to zero we have precise information on the ground state energy and the low-lying spectrum of excitations (at least in certain scaling limits), much less is known close to the critical point. In this talk I will discuss how thermal excitations can be described by Bogoliubov theory, allowing us to estimate the free energy of the Bose gas in the Gross-Pitaevskii regime. This is joint work with A. Deuchert and D. Stocker. |
25/10/2023 h17:00, Aula B |
Federico Laudisa (Università di Trento) Niels Bohr è certamente uno dei grandi protagonisti della fisica del XX secolo. La tradizione ha definito in suo onore quella che sarebbe diventata per molti decenni l’interpretazione canonica della teoria, fortemente improntata a una visione strumentalistica e puramente operativa della meccanica quantistica, come interpretazione di Copenhagen. Questa circostanza ha finito per appiattire la complessa figura del fisico danese, relegato in una visione che ancora negli anni Sessanta e Settanta era considerata rinunciataria o persino oscurantistica. In anni recenti si è assistito a una rilettura attenta e a una profonda rivalutazione della figura di Bohr: con la fine degli anni Venti, Bohr si applica con passione e determinazione al tentativo di costruire una visione organica del mondo fisico, capace di accogliere le lezioni scientifiche della meccanica quantistica in una vera e propria ‘filosofia naturale’, in un dibattito serrato con i protagonisti delle rivoluzioni della fisica contemporanea, primo fra tutti Albert Einstein, e attento ad alcuni dei grandi temi filosofici collegati alla giustificazione della conoscenza scientifica. [Seminario congiunto di Storia della Matematica e Fisica Matematica] |
19/10/2023 h16:15 |
Kasper Studsgaard Sørensen (Università di Roma La Sapienza) We consider two-dimensional unbounded magnetic Dirac operators, either defined on the whole plane, or with infinite mass boundary conditions on a half-plane. Our main results use techniques from elliptic PDEs and integral operators, while their topological consequences are presented as corollaries of some more general identities involving magnetic derivatives of local traces of fast decaying functions of the bulk and edge operators. One of these corollaries leads to the so-called Středa formula: if the bulk operator has an isolated compact spectral island, then the integrated density of states of the corresponding bulk spectral projection varies linearly with the magnetic field as long as the gaps between the spectral island and the rest of the spectrum are not closed, and the slope of this variation is given by the Chern character of the projection. The same bulk Chern character is related to the number of edge states which appear in the gaps of the bulk operator. |
12/09/2023 h14:00 |
Armand Bernou (Università di Roma La Sapienza) Harris theorems are powerful tools for studying the asymptotic behavior of Markovian models. Originating from the probabilistic literature, they were recently adapted to the PDE framework by Hairer-Mattingly and Cañizo-Mischler. I will present some recent works in which this method furnishes an alternative strategy for treating kinetic models with boundary conditions. I will discuss in particular the long-time behavior of the free-transport kinetic equation and of the linear Boltzmann models when the gas is enclosed in a domain with diffuse or Cercignani-Lampis boundary condition and variable temperature at the wall, which, in general, can not be handled with usual hypocoercivity methods. [Seminario congiunto di Probabilità e Fisica Matematica] |
28/06/2023 |
Tai-Ping Liu (Stanford University) The standard Hadamard's well-posedness theory has been established for linear systems, or for smooth solutions of nonlinear systems. For weak solutions of nonlinear systems, the usual calculus and functional analytic methods do not yield well-posedness theory. The difficulty is due in large part to the lack of understanding of the structure of weak solutions. In fact, there are several striking ill-posed examples for incompressible Euler and Navier-Stokes equations, and also for compressible Euler equations. On the other hand, there is the well-known well-posedness theory for the system of hyperbolic conservation laws, starting with the existence theory of James Glimm. We will survey the development of these. We will also explain the recent well-posedness theory for the compressible Navier-Stokes equation done in collaboration with Shih-Hsien Yu. |
14/06/2023 |
Giovanna Marcelli (Aalborg Universitet) We consider finite-range, many-body fermionic lattice models and we study the evolution of their thermal equilibrium state after introducing a weak and slowly-varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. Under a spectral gap assumption on the unperturbed Hamiltonian, convergence is also uniform in temperature. As an application, our expansion allows to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. In particular, we recover the zero temperature many-body adiabatic theorem by first taking the thermodynamic limit and then the zero temperature limit. As a corollary, we also establish the validity of linear response. Our strategy is based on a rigorous version of the Wick rotation and fermionic cluster expansion. |
07/06/2023 (h16:00 & h17:15) |
Giulia Sebastiani (Goethe-Universität Frankfurt) The free energy of TAP-solutions for the SK-model of mean-field spin glasses can be expressed as a nonlinear functional of local terms: we exploit this feature in order to contrive abstract REM-like models which we then solve by a classical Sanov-type large deviations treatment. This allows to identify the origin of the physically unsettling quadratic (in the inverse of temperature) correction to the Parisi free energy for the SK-model. From a non-spin glass point of view, this work is the first in a series of refinements that addresses the stability of hierarchical structures in models of evolving populations. Based on joint work with N. Kistler and M.A. Schmidt.
Marco Olivieri (Aarhus University) Dilute Bose gases interacting through a positive, pairwise, radial potential have a common expression for the first terms of the energy density expansion given by the so-called Lee-Huang-Yang (LHY) formula. The LHY formula depends only on the density of the gas and the scattering length of the potential, showing a universal behavior of the bosonic system independent of the particular shape of the potential. We present a proof of the derivation of the LHY formula in 2D and 3D in thermodynamic regime based on a rigorous Bogoliubov theory and localization techniques. Based on a joint work with S. Fournais, T. Girardot, L. Junge and L. Morin. |
17/05/2023 |
Christian Hainzl (LMU Muenchen) The correlation energy of a high density fermionic Coulomb gas, called Jellium, is expected to be given by the Gell-Mann Brueckner formula. I will discuss an analogue of this formula for the mean-field regime. I present a rigorous upper bound established by variational methods in the case of Coulomb interaction. I will further review similar results for more regular potentials, which is meanwhile well understood for particles in a fixed box. The talk is based on joint work with Martin Christiansen and P. T. Nam. |
03/05/2023 (h16:00 & h17:15) |
Emanuela L. Giacomelli (LMU Muenchen) In recent decades, the study of many-body systems has been an active area of research in both physics and mathematics. In this talk, we will consider a system of N spin 1/2 interacting fermions confined in a box in the dilute regime, with a particular focus on the correlation energy which is defined as the difference between the ground state energy and that of the free Fermi gas. We will discuss some recent results about a first order asymptotics for the correlation energy in the thermodynamic limit where the number of particles and the size of the box are sent to infinity keeping the density fixed. In particular, we will present a new upper bound for the correlation energy, which is consistent with the well-known Huang-Yang formula from 1957.
Eric A. Carlen (Rutgers University) We present a study of spectral gaps, entropy production and log Sobolev inequalities for some Lindblad equations modeling systems of N particles interacting pairwise. The bounds obtained, some of which are sharp, are uniform in N. This is joint work with Michael Loss. |
12/04/2023 |
Chiara Marullo (Università di Roma La Sapienza) Neural networks have become a powerful tool in various domains of scientific research and industrial applications. However, the fundamental working principles of neural architectures still lacks of a solid theoretical framework, which prevents a true understanding of their information processing features and the associated emergence of collective properties thus making the realization of optimized models and algorithm a challenging problem. This talk provides a mathematical perspective on the theory of neural networks. After a brief introduction of the fundamental concepts of statistical mechanics and mean-field models of spin-glasses, we will move to the analysis of the so-called Dense Associative Memories. We will show how to frame the statical-mechanics description of these models with the theory of partial differential equations in order to gain a deeper comprehension of the system's subtle mechanisms. |
05/04/2023 |
Fraydoun Rezakhanlou (University of Berkeley) In this talk I will discuss a family of Gibbsian measures on the set of Laguerre tessellations. These measures may be used to provide a systematic approach for constructing Gibbsian solutions to Hamilton-Jacobi PDEs by exploring the Eularian description of the shock dynamics. Such solutions depend on kernels satisfying kinetic-like equations reminiscent of the Smoluchowski model for coagulating and fragmenting particles. [Seminario congiunto di Probabilità e Fisica Matematica] |
22/03/2023 (h14:30, Aula B) |
Alessandro Teta (Università di Roma La Sapienza) 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] |
08/03/2023 |
Rossana Marra (INFN, Università di Roma Tor Vergata) 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. |
01/03/2023 (h17:00) |
Massimo Moscolari (Politecnico di Milano) 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. |
15/02/2023 |
Lorenzo Bertini (Sapienza, Roma) 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. |
01/02/2023 |
Angeliki Menegaki (IHES, Paris) 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). |
25/01/2023 |
Corentin Le Bihan (ENS Lyon) 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. |
11/01/2023 |
Daniele Ferretti (Sapienza, Roma) 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. |
SEMINARI 2022 | |
21/12/2022 |
Chiara Saffirio (University of Basel) 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) 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. |
14/12/2022 |
Matteo Gallone (SISSA, Trieste) 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. |
07/12/2022 |
Marco Coco (Università Politecnica delle Marche) 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. |
16/11/2022 |
Daniel Heydecker (Max Planck Institute, Leipzig) 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. |
19/10/2022 |
Elena Pulvirenti (TU Delft) 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. |
22/06/2022 |
Alberto Fachechi (Sapienza, Roma) 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. |
06/04/2022 |
Francesca Elisa Leonelli (Sapienza, Roma) |
16/03/2022 |
Raphael Winter (ENS Lyon) |
09/02/2022 |
Alessia Nota (Università degli Studi dell'Aquila) 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. |
19/01/2022 |
Serena Cenatiempo (GSSI, L'Aquila) 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. |
SEMINARI 2021 | |
24/11/2021 |
Lorenzo Panebianco (Sapienza, Roma) |
05/05/2021 |
Davide Fermi (Sapienza, Roma) |
17/03/2021 |
Biagio Cassano (Università di Bari) |
03/02/2023 |
Luca Oddis (Sapienza, Roma) |