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

Derivation of the linear Landau equation and linear Boltzmann equation from the Lorentz model with magnetic field

We consider a test particle moving in a random distribution of obstacles in the plane, under the action of a uniform magnetic field, orthogonal to the plane. We show that, in a weak coupling limit, the particle distribution behaves according to the linear Landau equation with a magnetic transport term. Moreover, we show that, in a low density regime, when each obstacle generates an inverse power law potential, the particle distribution behaves according to the linear Boltzmann equation with a magnetic transport term. We provide an explicit control of the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. We compare these results with those ones obtained for a system of hard disks in \cite{BMHH}, which show instead that the memory effects are not negligible in the Boltzmann-Grad limit.Comment: 22 pages, 4 figures in Journal of Statistical Physics 201

Effect of low-stiffness closeout overwrap on rocket thrust-chamber life

Three rocket thrust chambers with copper liners and a thrust level of 20.9 kN were cyclically test fired to failure. Two of the liners were made from oxygen free, high conductivity (OFHC) copper and from annealed Amzirc. The milled coolant channels were closed out with a thin copper closeout over which a fiberglass composite was wrapped to provide hoop strength only. Experimental data are presented, along with the results of a preliminary analysis that was performed before fabrication to evaluate the life extending potential of a thin copper closeout with a fiberglass overwrap