813 research outputs found
A Boltzmann model for rod alignment and schooling fish
We consider a Boltzmann model introduced by Bertin, Droz and Greegoire as a
binary interaction model of the Vicsek alignment interaction. This model
considers particles lying on the circle. Pairs of particles interact by trying
to reach their mid-point (on the circle) up to some noise. We study the
equilibria of this Boltzmann model and we rigorously show the existence of a
pitchfork bifurcation when a parameter measuring the inverse of the noise
intensity crosses a critical threshold. The analysis is carried over rigorously
when there are only finitely many non-zero Fourier modes of the noise
distribution. In this case, we can show that the critical exponent of the
bifurcation is exactly 1/2. In the case of an infinite number of non-zero
Fourier modes, a similar behavior can be formally obtained thanks to a method
relying on integer partitions first proposed by Ben-Naim and Krapivsky.Comment: 22 pages, 3 figure
Celebrating Cercignani's conjecture for the Boltzmann equation
Cercignani's conjecture assumes a linear inequality between the entropy and
entropy production functionals for Boltzmann's nonlinear integral operator in
rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities
and spectral gap inequalities, this issue has been at the core of the renewal
of the mathematical theory of convergence to thermodynamical equilibrium for
rarefied gases over the past decade. In this review paper, we survey the
various positive and negative results which were obtained since the conjecture
was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani,
powerful mind and great scientist, one of the founders of the modern theory
of the Boltzmann equation. 24 pages. V2: correction of some typos and one
ref. adde
On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials
The paper concerns - convergence to equilibrium for weak solutions of
the spatially homogeneous Boltzmann Equation for soft potentials (-4\le
\gm<0), with and without angular cutoff. We prove the time-averaged
-convergence to equilibrium for all weak solutions whose initial data have
finite entropy and finite moments up to order greater than 2+|\gm|. For the
usual -convergence we prove that the convergence rate can be controlled
from below by the initial energy tails, and hence, for initial data with long
energy tails, the convergence can be arbitrarily slow. We also show that under
the integrable angular cutoff on the collision kernel with -1\le \gm<0, there
are algebraic upper and lower bounds on the rate of -convergence to
equilibrium. Our methods of proof are based on entropy inequalities and moment
estimates.Comment: This version contains a strengthened theorem 3, on rate of
convergence, considerably relaxing the hypotheses on the initial data, and
introducing a new method for avoiding use of poitwise lower bounds in
applications of entropy production to convergence problem
Strong Convergence towards homogeneous cooling states for dissipative Maxwell models
We show the propagation of regularity, uniformly in time, for the scaled
solutions of the inelastic Maxwell model for small inelasticity. This result
together with the weak convergence towards the homogenous cooling state present
in the literature implies the strong convergence in Sobolev norms and in the
norm towards it depending on the regularity of the initial data. The
strategy of the proof is based on a precise control of the growth of the Fisher
information for the inelastic Boltzmann equation. Moreover, as an application
we obtain a bound in the distance between the homogeneous cooling state
and the corresponding Maxwellian distribution vanishing as the inelasticity
goes to zero.Comment: 2 figure
Relative entropy of entanglement for certain multipartite mixed states
We prove conjectures on the relative entropy of entanglement (REE) for two
families of multipartite qubit states. Thus, analytic expressions of REE for
these families of states can be given. The first family of states are composed
of mixture of some permutation-invariant multi-qubit states. The results
generalized to multi-qudit states are also shown to hold. The second family of
states contain D\"ur's bound entangled states. Along the way, we have discussed
the relation of REE to two other measures: robustness of entanglement and
geometric measure of entanglement, slightly extending previous results.Comment: Single column, 22 pages, 9 figures, comments welcom
Complete characterization of convergence to equilibrium for an inelastic Kac model
Pulvirenti and Toscani introduced an equation which extends the Kac
caricature of a Maxwellian gas to inelastic particles. We show that the
probability distribution, solution of the relative Cauchy problem, converges
weakly to a probability distribution if and only if the symmetrized initial
distribution belongs to the standard domain of attraction of a symmetric stable
law, whose index is determined by the so-called degree of
inelasticity, , of the particles: . This result is
then used: (1) To state that the class of all stationary solutions coincides
with that of all symmetric stable laws with index . (2) To determine
the solution of a well-known stochastic functional equation in the absence of
extra-conditions usually adopted
- …