Tanaka Theorem for Inelastic Maxwell Models

We show that the Euclidean Wasserstein distance is contractive for inelastic homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its associated Kac-like caricature. This property is as a generalization of the Tanaka theorem to inelastic interactions. Consequences are drawn on the asymptotic behavior of solutions in terms only of the Euclidean Wasserstein distance

Collisional rates for the inelastic Maxwell model: application to the divergence of anisotropic high-order velocity moments in the homogeneous cooling state

The collisional rates associated with the isotropic velocity moments $$and the anisotropic moments$$ and $$ are exactly derived in the case of the inelastic Maxwell model as functions of the exponent $r$, the coefficient of restitution $\alpha$, and the dimensionality $d$. The results are applied to the evolution of the moments in the homogeneous free cooling state. It is found that, at a given value of $\alpha$, not only the isotropic moments of a degree higher than a certain value diverge but also the anisotropic moments do. This implies that, while the scaled distribution function has been proven in the literature to converge to the isotropic self-similar solution in well-defined mathematical terms, nonzero initial anisotropic moments do not decay with time. On the other hand, our results show that the ratio between an anisotropic moment and the isotropic moment of the same degree tends to zero.Comment: 7 pages, 2 figures; v2: clarification of some mathematical statements and addition of 7 new references; v3: Published in "Special Issue: Isaac Goldhirsch - A Pioneer of Granular Matter Theory

Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model

This paper deals with a one--dimensional model for granular materials, which boils down to an inelastic version of the Kac kinetic equation, with inelasticity parameter $p>0$. In particular, the paper provides bounds for certain distances -- such as specific weighted $\chi$--distances and the Kolmogorov distance -- between the solution of that equation and the limit. It is assumed that the even part of the initial datum (which determines the asymptotic properties of the solution) belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent \a=2/(1+p). With such initial data, it turns out that the limit exists and is just the aforementioned stable distribution. A necessary condition for the relaxation to equilibrium is also proved. Some bounds are obtained without introducing any extra--condition. Sharper bounds, of an exponential type, are exhibited in the presence of additional assumptions concerning either the behaviour, near to the origin, of the initial characteristic function, or the behaviour, at infinity, of the initial probability distribution function

The dissipative linear Boltzmann equation for hard spheres

We prove the existence and uniqueness of an equilibrium state with unit mass to the dissipative linear Boltzmann equation with hard--spheres collision kernel describing inelastic interactions of a gas particles with a fixed background. The equilibrium state is a universal Maxwellian distribution function with the same velocity as field particles and with a non--zero temperature lower than the background one, which depends on the details of the binary collision. Thanks to the H--theorem we then prove strong convergence of the solution to the Boltzmann equation towards the equilibrium.Comment: 17 pages, submitted to Journal of Statistical Physic

A new approach to the creation and propagation of exponential moments in the Boltzmann equation

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous $d$-dimensional Boltzmann equation. In particular, when the collision kernel is of the form $|v-v_*|^\beta b(\cos(\theta))$ for $\beta \in (0,2]$ with $\cos(\theta)= |v-v_*|^{-1}(v-v_*)\cdot \sigma$ and $\sigma \in \mathbb{S}^{d-1}$, and assuming the classical cut-off condition $b(\cos(\theta))$ integrable in $\mathbb{S}^{d-1}$, we prove that there exists $a > 0$ such that moments with weight $\exp(a \min{t,1} |v|^\beta)$ are finite for $t>0$, where $a$ only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.Comment: 14 pages. Many typos corrected in this revised versio

Exact steady state solution of the Boltzmann equation: A driven 1-D inelastic Maxwell gas

The exact nonequilibrium steady state solution of the nonlinear Boltzmann equation for a driven inelastic Maxwell model was obtained by Ben-Naim and Krapivsky [Phys. Rev. E 61, R5 (2000)] in the form of an infinite product for the Fourier transform of the distribution function $f(c)$. In this paper we have inverted the Fourier transform to express $f(c)$ in the form of an infinite series of exponentially decaying terms. The dominant high energy tail is exponential, $f(c)\simeq A_0\exp(-a|c|)$, where $a\equiv 2/\sqrt{1-\alpha^2}$ and the amplitude $A_0$ is given in terms of a converging sum. This is explicitly shown in the totally inelastic limit ($\alpha\to 0$) and in the quasi-elastic limit ($\alpha\to 1$). In the latter case, the distribution is dominated by a Maxwellian for a very wide range of velocities, but a crossover from a Maxwellian to an exponential high energy tail exists for velocities $|c-c_0|\sim 1/\sqrt{q}$ around a crossover velocity $c_0\simeq \ln q^{-1}/\sqrt{q}$, where $q\equiv (1-\alpha)/2\ll 1$. In this crossover region the distribution function is extremely small, $\ln f(c_0)\simeq q^{-1}\ln q$.Comment: 11 pages, 4 figures; a table and a few references added; to be published in PR

Stable Equilibrium Based on L\'evy Statistics: Stochastic Collision Models Approach

We investigate equilibrium properties of two very different stochastic collision models: (i) the Rayleigh particle and (ii) the driven Maxwell gas. For both models the equilibrium velocity distribution is a L\'evy distribution, the Maxwell distribution being a special case. We show how these models are related to fractional kinetic equations. Our work demonstrates that a stable power-law equilibrium, which is independent of details of the underlying models, is a natural generalization of Maxwell's velocity distribution.Comment: PRE Rapid Communication (in press

Kinetic models with randomly perturbed binary collisions

We introduce a class of Kac-like kinetic equations on the real line, with general random collisional rules, which include as particular cases models for wealth redistribution in an agent-based market or models for granular gases with a background heat bath. Conditions on these collisional rules which guarantee both the existence and uniqueness of equilibrium profiles and their main properties are found. We show that the characterization of these stationary solutions is of independent interest, since the same profiles are shown to be solutions of different evolution problems, both in the econophysics context and in the kinetic theory of rarefied gases

Anomalous Negative Magnetoresistance Caused by Non-Markovian Effects

A theory of recently discovered anomalous low-field magnetoresistance is developed for the system of two-dimensional electrons scattered by hard disks of radius $a,$ randomly distributed with concentration $n.$ For small magnetic fields the magentoresistance is found to be parabolic and inversely proportional to the gas parameter, $\delta \rho_{xx}/\rho \sim - (\omega_c \tau)^2 / n a^2.$ With increasing field the magnetoresistance becomes linear $\delta \rho_{xx}/\rho \sim - \omega_c \tau$ in a good agreement with the experiment and numerical simulations.Comment: 4 pages RevTeX, 5 figure