On the Distribution of Free Path Lengths for the Periodic Lorentz Gas III

 For r(0,1), let Z _r ={xR ² |dist(x,Z ²)>r/2} and define τ _r (x,v)=inf{t>0|x+tv∂Z _r }. Let Φ _r (t) be the probability that τ _r (x,v)≥t for x and v uniformly distributed in Z _r and §¹ respectively. We prove in this paper that

Effect of Correlations on the Exponents for the Power-Law Distributions in Self-Organized Criticality

The origin of power-law distributions in self-organized criticality is investigated by treating the variation of the number of active sites in the system as a stochastic process. An avalanche is mapped to a first-return random-walk process in a one-dimensional lattice. In order to understand the reason of variant exponents for the power-law distributions in different self-organized critical systems, we introduce the correlations among evolution steps. Power-law distributions of the lifetime and spatial size are found when the random walk is unbiased with equal probability to move in opposite directions. It is found that the longer the correlation length, the smaller values of the exponents for the power-law distributions.     

Current in Periodic Lorentz Gases with Twists

We study electrical current in two-dimensional periodic Lorentz gas in the presence of a twist force on the scatterers. In this deterministic system, billiard orbits are still geodesics between collisions, but do not reflect elastically when reaching the boundary. When the horizon is finite, i.e. the free flights between collisions are bounded, the resulting current J is proportional to the strength of the twist force measured by ε. We also prove the existence of a unique SRB measure, for which the Pesin entropy formula and Young’s expression for the fractal dimension are valid.     

The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatterer Limit

We study the free path length and the geometric free path length in the model of the periodic two-dimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the small-scatterer limit and explicitly compute them. As a corollary one gets a complete proof for the existence of the constant term in the asymptotic formula of the KS entropy of the billiard map in this model, as conjectured by P. Dahlqvist.In memory of Walter Philipp     

On Lorentz Invariance and the Minimum Length

JETP Letters - It was shown by Kirzhnits and Chechen, following an earlier paper by Mead, that the minimum length scale l is constrained by the Mössbauer effect, which leads to the...     

Some Estimates for 2-Dimensional Infinite and Bounded Dilute Random Lorentz Gases

We present a (mostly) rigorous approach to unbounded and bounded (open) dilute random Lorentz gases. Relying on previous rigorous results on the dilute (Boltzmann–Grad) limit we compute the asymptotics of the Lyapunov exponent in the unbounded case. For the bounded open case in a circular region we give here an incomplete rigorous analysis which gives the asymptotics for large radius of the escape rate and of the rescaled quasi-invariant (q.i., or quasi-stationary) measure. We finally give a complete proof on existence and asymptotic properties of the q.i. measure in a one-dimensional caricature.     

Free Path Lengths in Quasicrystals

Previous studies of kinetic transport in the Lorentz gas have been limited to cases where the scatterers are distributed at random (e.g., at the points of a spatial Poisson process) or at the vertices of a Euclidean lattice. In the present paper we investigate quasicrystalline scatterer configurations, which are non-periodic, yet strongly correlated. A famous example is the vertex set of a Penrose tiling. Our main result proves the existence of a limit distribution for the free path length, which answers a question of Wennberg. The limit distribution is characterised by a certain random variable on the space of higher dimensional lattices, and is distinctly different from the exponential distribution observed for random scatterer configurations. The key ingredients in the proofs are equidistribution theorems on homogeneous spaces, which follow from Ratner’s measure classification.     

A Formula of Average Path Length for Unweighted Networks

In this paper, based on the adjacency matrix of the network and its powers, the formulas are derived for the shortest path and the average path length, and an effective algorithm is presented. Furthermore, an example is provided to demonstrate the proposed method.     

Lorentz Covariant Canonical Formalism for Free Massive, Massless and Tachyonic Particles

We construct a consistent Lorentz-covariant canonical formalism for a free massive, massless and tachyonic particle in the framework of the absolute synchronization scheme of clocks. In the case of a massive particle our approach is canonically equivalent to the standard formulation which is not manifestly covariant. The absolute synchronization scheme seems to be the only one we can apply in the case of massless and tachyonic particles.     

Infinitesimal Lorentz Transformations and the States of Polarization of Free Photons

Applying the tangent dynamics technique as generated by infinitesimal Lorentz transformations, it is shown that the components of an electromagnetic field correspond to the Lie-Cartan parameters of the group SO(3,1). Thus the maximum number of non-null components of the magnetic field (i.e., the number of the states of polarization) is not decided by the number of group generators.     

Free Path Lengths in Quasi Crystals

The Lorentz gas is a model for a cloud of point particles (electrons) in a distribution of scatterers in space. The scatterers are often assumed to be spherical with a fixed diameter d, and the point particles move with constant velocity between the scatterers, and are specularly reflected when hitting a scatterer. There is no interaction between point particles. An interesting question concerns the distribution of free path lengths, i.e. the distance a point particle moves between the scattering events, and how this distribution scales with scatterer diameter, scatterer density and the distribution of the scatterers. It is by now well known that in the so-called Boltzmann–Grad limit, a Poisson distribution of scatterers leads to an exponential distribution of free path lengths, whereas if the scatterer distribution is periodic, the free path length distribution asymptotically behaves as a power law.     

自由空间中超短啁啾洛仑兹脉冲光束的空间奇异性

给出了超短啁啾洛仑兹脉冲光束在自由空间中的传输方程，研究了啁啾对洛仑兹脉冲光束空间奇异性的影响．理论及数值模拟的结果表明，由于啁啾的作用，对于脉冲宽度大于一个光波振荡周期的洛仑兹脉冲光束同样存在空间奇异性．在初始位置，啁啾洛仑兹脉冲光束的空间奇异性出现在脉冲的前沿或后沿．当洛仑兹脉冲光束传输以后，啁啾洛仑兹脉冲光束的空间奇异性将更加强烈．     

激光陀螺新型光路程长控制镜的研制

爪盘结构程长控制镜由于其稳定性好等优点,已越来越广泛地应用于激光陀螺的腔长控制,但这种只有程长控制能力的控制镜,对陀螺腔体变形等原因引起的谐振光路变化却是无能为力。通过分析该类腔长控制镜的特点,提出了一种新颖的具有角度控制功能的光路程长控制镜。在原有的控制镜上增加一组角度控制部件,使其不仅有程长控制功能同时还具备了光路控制功能的方法,设计了新的槽片及角度控制元件,研制出了高性能的光路程长控制镜。系统地对控制镜的性能进行了测试。结果表明,它不仅稳定性好(陀螺使用温度范围内最大歪扭变化量小于0.3″),而且灵敏度高(大于0.1″/V)。     

Power-Law Tail in the Chinese Wealth Distribution

We analyse the data from the recently published lists of the richest Chinese from the year 2003 to 2005. The results confirm that in these years the wealth is distributed according to a power law with exponents between 1.758 and 2.285 in the high end. The power distribution is found to be quite robust although the persons in the list change drastically and the wealth increases rapidly. The relation between the wealth and the absolute change of wealth rejects the notion that the wealth evolution is a multiplicative stochastic process.     

Hamilton's Turns for the Lorentz Group

Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by “turns,” which are equivalence classes of directed great circle arcs on the unit sphere S ², in such a manner that the rule for composition of group elements takes the form of the familiar parallelogram law for the Euclidean translation group. It is only recently that this construction has been generalized to the simplest noncompact group SU(1, 1)=Sp(2, R)=SL(2, R), the double cover of SO(2, 1). The present work develops a theory of turns for SL(2, C), the double and universal cover of SO(3, 1) and SO(3, C), rendering a geometric representation in the spirit of Hamilton available for all low dimensional semisimple Lie groups of interest in physics. The geometric construction is illustrated through application to polar decomposition, and to the composition of Lorentz boosts and the resulting Wigner or Thomas rotation. PACS numbers: 02.20.-a     

An Estimation for the Power-Law Distribution Parameter Based on Entropy

A new estimation based on the Shannon entropy for the power-law distribution parameter is presented, The maximum likelihood estimation （MLE） of the parameter is discussed and the relation between the MLE and the moment estimation of the parameter is given out. It is shown that the minimum Shannon entropy estimation is equivalent to the MLE giving the log expectation.