1,720,978 research outputs found
Teoria di campo scalare planare in quattro dimensioni e problemi di Borel sommabilità
No full textWe review the issue of Borel summability in the framework of multiscale analysis and renormal- ization group, by discussing a proof of Borel summability of the φ4 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 φ4 planar theory as a toy model for future applications
Evolution of correlation functions in the hard sphere dynamics
No full textThe 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
On the stationary BBGKY hierarchy for equilibrium states
No full textWe consider infinite classical systems of particles interacting via a smooth, stable and regular two–body potential. We establish a new direct integration method to construct the solutions of the stationary BBGKY hierarchy, assuming the usual Gaussian distribution of momenta. We prove equivalence between the corresponding infinite hierarchy and the Kirkwood–Salsburg equa- tions. 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
Review: ``From Newton to Boltzmann: Hard Spheres and Short-range Potentials''
No full textReview of: From Newton to Boltzmann: Hard Spheres and Short-range Potentials, by Isabelle Gal- lagher, Laure Saint-Raymond and Benjamin Texier, Zurich Lectures in Advanced Mathematics, vol. 18, European Mathematical Society, Zu ̈rich, 2014, xi+135 pp., ISBN 978-3- 03719-129-3
On the theory of Lorentz gases with long range interactions
No full textWe construct and study the stochastic force field generated by a Poisson distribu- tion of sources at finite density, x1, x2, · · · in R3 each of them yielding a long range potential QiΦ(x − xi) with possibly different charges Qi ∈ 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
Two-dimensional Lorentz process for magnetotransport: Boltzmann-Grad limit
No full textWe study a system of charged, noninteracting classical particles moving in a Poisson distribution of hard-disk scatterers in two dimensions, under the effect of a magnetic field perpendicular to the plane. We prove that, in the low-density (Boltzmann- Grad) limit, the particle distribution evolves according to a generalized linear Boltzmann equation, previously derived and solved by Bobylev et al. (Phys. Rev. Lett. 75 (1995) 2, J. Stat. Phys. 87 (1997) 1205-1228, J. Stat. Phys. 102 (2001) 1133-1150). In this model, Boltzmann's chaos fails, and the kinetic equation includes non-Markovian terms. The ideas of (Phys. Rev. 185 (1969) 308-322) can be however adapted to prove convergence of the process with memory
Microscopic solutions of the Boltzmann-Enskog equation in the series representation
No full textThe 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
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