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Sergio Simonella

Publications and source records attributed to Sergio Simonella.

32 records · Page 2Linked to original sources
arXiv2020

Fluctuation theory in the Boltzmann--Grad limit

We develop a rigorous theory of hard-sphere dynamics in the kinetic regime, away from thermal equilibrium. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed by the Boltzmann equation. Deviations from this behaviour are described by dynamical correlations, which can be fully characterized for short times. This provides both a fluctuating Boltzmann equation and large deviation asymptotics.

math.AP↗
arXiv2018

On the size of chaos in the mean field dynamics

We consider the error arising from the approximation of an N-particle dynamics with its description in terms of a one-particle kinetic equation. We estimate the distance between the j-marginal of the system and the factorized state, obtained in a mean field limit as N $\rightarrow$ $\infty$. Our analysis relies on the evolution equation for the "correlation error" rather than on the usual BBGKY hierarchy. The rate of convergence is shown to be O(j 2 N) in any bounded interval of time (size of chaos), as expected from heuristic arguments. Our formalism applies to an abstract hierarchical mean field model with bounded collision operator and a large class of initial data, covering (a) stochastic jump processes converging to the homogeneous Boltzmann and the Povzner equation and (b) quantum systems giving rise to the Hartree equation.

math.AP↗
arXiv2018

Microscopic solutions of the Boltzmann-Enskog equation in the series representation

The Boltzmann-Enskog equation for a hard sphere gas is known to have so called microscopic solutions, i.e., solutions of the form of time-evolving empirical measures of a finite number of hard spheres. However, the precise mathematical meaning of these solutions should be discussed, since the formal substitution of empirical measures into the equation is not well-defined. Here we give a rigorous mathematical meaning to the microscopic solutions to the Boltzmann-Enskog equation by means of a suitable series representation.

math-ph↗
arXiv2017

On the theory of Lorentz gases with long range interactions

We construct and study the stochastic force field generated by a Poisson distribution of sources at finite density, $x_1,x_2,\cdots$ in $\mathbb{R}^3$ each of them yielding a long range potential $Q_iΦ(x-x_i)$ with possibly different charges $Q_i \in \mathbb{R}$. The potential $Φ$ is assumed to behave typically as $|x|^{-s}$ for large $|x|$, with $s > 1/2$. We will denote the resulting random field as "generalized Holtsmark field". We then consider the dynamics of one tagged particle in such random force fields, in several scaling limits where the mean free path is much larger than the average distance between the scatterers. We estimate the diffusive time scale and identify conditions for the vanishing of correlations. These results are used to obtain appropriate kinetic descriptions in terms of a linear Boltzmann or Landau evolution equation depending on the specific choices of the interaction potential.

math-ph↗
arXiv2016

One-sided convergence in the Boltzmann-Grad limit

We review various contributions on the fundamental work of Lanford deriving the Boltzmann equation from hard-sphere dynamics in the low density limit. We focus especially on the assumptions made on the initial data and on how they encode irreversibility. The impossibility to reverse time in the Boltzmann equation (expressed for instance by Boltzmann's H-theorem) is related to the lack of convergence of higher order marginals on some singular sets. Explicit counterexamples single out the microscopic sets where the initial data should converge in order to produce the Boltzmann dynamics.

math.PR↗
arXiv2016

Kinetic Theory of Cluster Dynamics

In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, defined as finite groups of particles having an influence on each other's trajectory during a given interval of time. For an ideal gas with short-range intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simplified context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in finite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard.

math-ph↗
arXiv2016

The Boltzmann-Grad Limit of a Hard Sphere System: Analysis of the Correlation Error

We present a quantitative analysis of the Boltzmann-Grad (low-density) limit of a hard sphere system. We introduce and study a set of functions (correlation errors) measuring the deviations in time from the statistical independence of particles (propagation of chaos). In the context of the BBGKY hierarchy, a correlation error of order $k$ measures the event where $k$ particles are connected by a chain of interactions preventing the factorization. We show that, provided $k < \varepsilon^{-α}$, such an error flows to zero with the average density $\varepsilon$, for short times, as $\varepsilon^{γk}$, for some positive $α,γ\in (0,1)$. This provides an information on the size of chaos, namely, $j$ different particles behave as dictated by the Boltzmann equation even when $j$ diverges as a negative power of $\varepsilon$. The result requires a rearrangement of Lanford perturbative series into a cumulant type expansion, and an analysis of many-recollision events.

math-ph↗
arXiv2015

On the evolution of the empirical measure for the Hard-Sphere dynamics

We prove that the evolution of marginals associated to the empirical measure of a finite system of hard spheres is driven by the BBGKY hierarchical expansion. The usual hierarchy of equations for $L^1$ measures is obtained as a corollary. We discuss the ambiguities arising in the corresponding notion of microscopic series solution to the Boltzmann-Enskog equation.

math-ph↗
arXiv2014

On the stationary BBGKY hierarchy for equilibrium states

A new direct integration method is established to construct the solutions of the stationary BBGKY hierarchy, assuming the usual form of the equilibrium correlation functions, for infinite classical systems of particles interacting via a smooth, stable and regular two body potential. The equivalence between the corresponding infinite hierarchy and the Kirkwood--Salsburg equations is proved. A problem of existence and uniqueness of the solutions of the hierarchy with appropriate boundary conditions is thus solved for low densities. The result is extended in a milder sense to systems with a hard core interaction. Comparisons are provided with different integration techniques.

math-ph↗
arXiv2014

Evolution of correlation functions in the hard sphere dynamics

The series expansion for the evolution of the correlation functions of a finite system of hard spheres is derived from direct integration of the solution of the Liouville equation, with minimal regularity assumptions on the density of the initial measure. The usual BBGKY hierarchy of equations is then recovered. A graphical language based on the notion of collision history originally introduced by Spohn is developed, as a useful tool for the description of the expansion and of the elimination of degrees of freedom.

math-ph↗
arXiv2014

On the validity of the Boltzmann equation for short range potentials

We consider a classical system of point particles interacting by means of a short range potential. We prove that, in the low--density (Boltzmann--Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann equation. This is a revisitation and an extension of the thesis of King (unpublished), appeared after the well known result of Lanford for hard spheres, and of a recent paper by Gallagher et al (arXiv: 1208.5753v1). Our analysis applies to any stable and smooth potential. In the case of repulsive potentials (with no attractive parts), we estimate explicitly the rate of convergence.

math-ph↗
arXiv2010

Borel summability of $ϕ^{4}_{4}$ planar theory via multiscale analysis

We review the issue of Borel summability in the framework of multiscale analysis and renormalization group, by discussing a proof of Borel summability of the $ϕ^{4}_4$ massive euclidean planar theory; this result is not new, since it was obtained by Rivasseau and 't Hooft. However, the techniques that we use have already been proved effective in the analysis of various models of consended matter and field theory; therefore, we take the $ϕ^{4}_4$ planar theory as a toy model for future applications.

math-ph↗