A Kac model for kinetic annihilation

In this paper we consider the stochastic dynamics of a finite system of particles in a finite volume (Kac-like particle system) which annihilate with probability $\alpha \in (0,1)$ or collide elastically with probability $1-\alpha$. We first establish the well-posedness of the particle system which exhibits no conserved quantities. We rigorously prove that, in some thermodynamic limit, a suitable hierarchy of kinetic equations is recovered for which tensorized solution to the homogenous Boltzmann with annihilation is a solution. For bounded collision kernels, this shows in particular that propagation of chaos holds true. Furthermore, we make conjectures about the limit behaviour of the particle system when hard-sphere interactions are taken into account.Comment: 40 page

Further Analysis of the Zipf Law: Does the Rank-Size Rule Really Exist?

The widely-used Zipf law has two striking regularities: excellent fit and close-to-one exponent. When the exponent equals to one, the Zipf law collapses into the rank-size rule. This paper further analyzes the Zipf exponent. By changing the sample size, the truncation point, and the mix of cities in the sample, we found that the exponent is close to one only for some selected sub-samples. Using the values of estimated exponent from the rolling sample method, we obtained an elasticity of the exponent with respect to sample size.Zipf law; Rank-size rule; Rolling sample method

Further Analysis of the Zipf's Law: Does the Rank-Size Rule Really Exist?

The widely-used Zipf’s law has two striking regularities. One is its excellent fit; the other is its close-to-one exponent. When the exponent equals to one, the Zipf’s law collapses into the rank-size rule. This paper further analyzes the Zipf exponent. By changing the sample size, the truncation point, and the mix of cities in the sample, we found that the exponent is close to one only for some selected sub-samples. Small samples of large cities alone provide higher value of the exponent whereas small cities introduce high variance and lower the value of the exponent. Using the values of estimated exponent from the rolling sample method, we obtained an elasticity of the exponent with respect to sample size. We concluded that the rank-size rule is not an economic regularity but a statistical phenomenon.Zipf's law; Rank-size rule; Rolling sample method

A diffusion limit for a test particle in a random distribution of scatterers

We consider a point particle moving in a random distribution of obstacles described by a potential barrier. We show that, in a weak-coupling regime, under a diffusion limit suggested by the potential itself, the probability distribution of the particle converges to the solution of the heat equation. The diffusion coefficient is given by the Green-Kubo formula associated to the generator of the diffusion process dictated by the linear Landau equation

Pre-Main sequence Turn-On as a chronometer for young clusters: NGC346 as a benchmark

We present a novel approach to derive the age of very young star clusters, by using the Turn-On (TOn). The TOn is the point in the color-magnitude diagram (CMD) where the pre-main sequence (PMS) joins the main sequence (MS). In the MS luminosity function (LF) of the cluster, the TOn is identified as a peak followed by a dip. We propose that by combining the CMD analysis with the monitoring of the spatial distribution of MS stars it is possible to reliably identify the TOn in extragalactic star forming regions. Compared to alternative methods, this technique is complementary to the turn-off dating and avoids the systematic biases affecting the PMS phase. We describe the method and its uncertainties, and apply it to the star forming region NGC346, which has been extensively imaged with the Hubble Space Telescope (HST). This study extends the LF approach in crowded extragalactic regions and opens the way for future studies with HST/WFC3, JWST and from the ground with adaptive optics.Comment: 6 pages, 4 figures. Accepted for publication in ApJ Letter