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Dario Benedetto

Publications and source records attributed to Dario Benedetto.

At least 19 recordsLinked to original sources
arXiv2026

Entropy dissipation inequality for general binary collision models

We introduce a ``two-particle factorization'' condition which allows us to formulate the homogeneous Boltzmann equation for non-reversible collision kernels in terms of an entropy inequality. This formulation yields an H-Theorem. We provide some examples of non-reversible binary collision models with a concentration/dispersion mechanism, as in opinion dynamics, which satisfy this condition. As a preliminary step, we also provide an analogous variational formulation of non-reversible continuous time Markov chains, expressed in terms of an entropy dissipation inequality.

math-ph↗
arXiv2026

Variational derivation of the homogeneous Boltzmann equation

We introduce a variational formulation of the homogeneous Boltzmann equation, with hard-sphere cross section, which selects the unique energy conserving solution. We prove that this solution arises from the microscopic dynamics, namely Kac's walk, and we establish the propagation in time of entropic chaoticity, under the minimal assumption that the initial distribution is entropically chaotic.

math-ph↗
arXiv2024

Large time analysis of the rate function associated to the Boltzmann equation: dynamical phase transitions

We analyse the large time behaviour of the rate function that describes the probability of large fluctuations of an underlying microscopic model associated to the homogeneous Boltzmann equation, such as the Kac walk. We consider in particular the asymptotic of the number of collisions, per particle and per unit of time, and show it exhibits a phase transition in the joint limit in which the number of particles N and the time interval [0,T] diverge. More precisely, due to the existence of Lu-Wennberg solutions, the corresponding limiting rate function vanishes for subtypical values of the number of collisions. We also analyse the second order large deviations showing that the probability of subtypical fluctuations is exponentially small in N, independently on T. As a key point, we establish the controllability of the homogeneous Boltzmann equation.

math.PR↗
arXiv2024

Propagation of chaos and hydrodynamic description for topological models

In this work, we study the deterministic Cucker-Smale model with topological interaction. Focusing on the solutions of the corresponding Liouville equation, we show that propagation of chaos holds. Moreover, considering monokinetic solutions, we also obtain a rigorous derivation of the hydrodynamic description given by a pressureless Euler-type system.

math.AP↗
arXiv2024

Scattering problem for Vlasov-type equations on the $d$-dimensional torus with Gevrey data

In this article, we consider Vlasov-type equations describing the evolution of single-species type plasmas, such as those composed of electrons (Vlasov-Poisson) or ions (screened Vlasov-Poisson/Vlasov-Poisson with massless electrons). We solve the final data problem on the torus $\mathbb{T}^d$, $d \geq 1$, by considering asymptotic states of regularity Gevrey-$\frac{1}γ$ with $γ>\frac13$, small perturbations of homogeneous equilibria satisfying the Penrose stability condition. This extends to the Gevrey perturbative case, and to higher dimension, the scattering result in analytic regularity obtained by E. Caglioti and C. Maffei in [14], and answers an open question raised by J. Bedrossian in arXiv:2211.13707.

math.AP↗
arXiv2023

Microcanonical phase transitions for the vortex system

We consider the Microcanonical Variational Principle for the vortex system in a bounded domain. In particular we are interested in the thermodynamic properties of the system in domains of second kind, i.e. for which the equivalence of ensembles does not hold. For connected domains close to the union of disconnected disks (dumbbell domains), we show that the system may exhibit an arbitrary number of fist-order phase transitions, while the entropy is convex for large energy.

math-ph↗
arXiv2022

Asymptotic probability of energy increasing solutions to the homogeneous Boltzmann equation

Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collision (Kac's model) and show that these solutions are atypical. More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate. This result is obtained by improving the established large deviation estimates in the canonical setting. Key ingredients are the extension of Sanov's theorem to the microcanonical ensemble and large deviations for the Kac's model in the microcanonical setting.

math-ph↗
arXiv2021

Comparison between the Cauchy problem and the scattering problem for the Landau damping in the Vlasov-HMF equation

We analyze the analytic Landau damping problem for the Vlasov-HMF equation, by fixing the asymptotic behavior of the solution. We use a new method for this "scattering problem", closer to the one used for the Cauchy problem. In this way we are able to compare the two results, emphasizing the different influence of the plasma echoes in the two approaches. In particular, we prove a non-perturbative result for the scattering problem.

math.AP↗
arXiv2021

Mean-field limit for particle systems with topological interactions

The mean-field limit for systems of self-propelled agents with topological interaction cannot be obtained by means of the usual Dobrushin approach. We get a result on this direction by adapting to the multidimensional case the techniques developed by Trocheris in 1986 to treat the Vlasov-Poisson equation in one dimension.

math.AP↗
arXiv2021

Random assignment problems on ${2d}$ manifolds

We consider the assignment problem between two sets of $N$ random points on a smooth, two-dimensional manifold $Ω$ of unit area. It is known that the average cost scales as $E_Ω(N)\sim\frac{1}{2π}\ln N$ with a correction that is at most of order $\sqrt{\ln N\ln\ln N}$. In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $Ω$-dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace--Beltrami operator on $Ω$. We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.

math-ph↗
arXiv2021

Large deviations for Kac-like walks

We introduce a Kac's type walk whose rate of binary collisions preserves the total momentum but not the kinetic energy. In the limit of large number of particles we describe the dynamics in terms of empirical measure and flow, proving the corresponding large deviation principle. The associated rate function has an explicit expression. As a byproduct of this analysis, we provide a gradient flow formulation of the Boltzmann-Kac equation.

math.PR↗
arXiv2020

Euclidean random matching in 2D for non-constant densities

We consider the 2-dimensional random matching problem in $\mathbb{R}^2.$ In a challenging paper, Caracciolo et. al. arXiv:1402.6993 on the basis of a subtle linearization of the Monge Ampere equation, conjectured that the expected value of the square of the Wasserstein distance, with exponent $2,$ between two samples of $N$ uniformly distributed points in the unit square is $\log N/2πN$ plus corrections, while the expected value of the square of the Wasserstein distance between one sample of $N$ uniformly distributed points and the uniform measure on the square is $\log N/4πN$. These conjectures has been proved by Ambrosio et al. arXiv:1611.04960. Here we consider the case in which the points are sampled from a non uniform density. For first we give formal arguments leading to the conjecture that if the density is regular and positive in a regular, bounded and connected domain $Λ$ in the plane, then the leading term of the expected values of the Wasserstein distances are exactly the same as in the case of uniform density, but for the multiplicative factor equal to the measure of $Λ$. We do not prove these results but, in the case in which the domain is a square, we prove estimates from above that coincides with the conjectured result.

math-ph↗
arXiv2019

Some aspects of the inertial spin model for flocks and related kinetic equations

In this paper we study the macroscopic behavior of the inertial spin (IS) model. This model has been recently proposed to describe the collective dynamics of flocks of birds, and its main feature is the presence of an auxiliary dynamical variable, a sort of internal spin, which conveys the interaction among the birds with the effect of better describing the turning of flocks. After discussing the geometrical and mechanical properties of the IS model, we show that, in the case of constant interaction among the birds, its mean-field limit is described by a non-linear Fokker-Planck equation, whose equilibria are fully characterized. Finally, in the case of non-constant interactions, we derive the kinetic equation for the mean-field limit of the model in the absence of thermal noise, and explore its macroscopic behavior by analyzing the mono-kinetic solutions.

math-ph↗
arXiv2014

Exponential dephasing of oscillators in the Kinetic Kuramoto Model

We study the kinetic Kuramoto model for coupled oscillators with coupling constant below the synchronization threshold. We manage to prove that, for any analytic initial datum, if the interaction is small enough, the order parameter of the model vanishes exponentially fast, and the solution is asymptotically described by a free flow. This behavior is similar to the phenomenon of Landau damping in plasma physics. In the proof we use a combination of techniques from Landau damping and from abstract Cauchy-Kowalewskaya theorem.

math-ph↗
arXiv2014

Dephasing of Kuramoto oscillators in kinetic regime towards a fixed asymptotically free state

We study the kinetic Kuramoto model for coupled oscillators. We prove that for any regular asymptotically free state, if the interaction is small enough, it exists a solution which is asymptotically close to it. For this class of solution the order parameter vanishes to zero, showing a behavior similar to the phenomenon of Landau damping in plasma physics. We obtain an exponential decay of the order parameter in the case on analytical regularity of the asymptotic state, and a polynomial decay in the case of Sobolev regularity.

math-ph↗
arXiv2014

An invariant region for the collisional dynamics of two bodies on Keplerian orbits

We study the dynamics of two bodies moving on elliptic Keplerian orbits around a fixed center of attraction and interacting only by means of elastic or inelastic collisions. We show that there exists a bounded invariant region: for suitable values of the total energy and the total angular momentum (explicitly computable) the orbits of the bodies remain elliptic, whatever are the number and the details of the collisions. We show that there exists a bounded invariant region even in the case of two bodies interacting by short range potential.

math.DS↗