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.     

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.     

湍流路径积分参量与湍流大气中光的传播效应

根据光波在湍流大气中传播的理论分析了弱起伏条件下决定光波传播效应的折射率结构常数Ｃ＾２ｎ的各种路径积分参量之间的关系，导出了由光波的强度起伏所确定的等效折射率结构常数对计算大气相干长度与光波的到达角起伏，光束的扩展及漂移的相对偏差的表达式。针对折射率结构常数Ｃ＾２ｎ具有周期性，递减，递增和随     

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

Distinct Clusterings and Characteristic Path Lengths in Dynamic Small-World Networks with Identical Limit Degree Distribution

Many real-world networks belong to a particular class of structures, known as small-world networks, that display short distance between pair of nodes. In this paper, we introduce a simple family of growing small-world networks where both addition and deletion of edges are possible. By tuning the deletion probability q _t , the model undergoes a transition from large worlds to small worlds. By making use of analytical or numerical means we determine the degree distribution, clustering coefficient and average path length of our networks. Surprisingly, we find that two similar evolving mechanisms, which provide identical degree distribution under a reciprocal scaling as t goes to infinity, can lead to quite different clustering behaviors and characteristic path lengths. It is also worth noting that Farey graphs constitute the extreme case q _t ??0 of our random construction.     

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     

部分相干光在大气湍流中斜程传输路径上的展宽与漂移

基于Andrews和Philips经典漂移方差模型,利用部分相干高斯-谢尔光束在大气湍流中斜程传输的光束扩展半径,推导出考虑外尺度情况时部分相干高斯-谢尔光束斜程情况下的漂移方差表达式,应用随高度变化的大气结构常量模型进行数值计算,对比分析了部分相干光和完全相干光在大气湍流中的展宽和漂移特性.结果表明:相同的传输条件下,部分相干光比完全相干光的光束扩展更迅速,受湍流的影响也更小;初始半径越大,接收机高度越高,光束的扩展效应越小;随着传输距离的增大,光束的质心漂移方差随光束初始半径的增大而减小,不同相干性的光束漂移方差变化很小;完全相干光的光束漂移受波长的影响较小,而部分相干光的波长越长,漂移越明显.     

大气湍流对斜程传输准单色高斯-谢尔光束时间相干性的影响

由第一类零阶贝塞尔函数的级数展开推导出波结构函数在任意湍流条件下的近似表达式。由广义惠更斯-菲涅耳原理、随高度变化的Hufnagel-Valley湍流廓线模型以及波结构函数在任意湍流条件下的近似表达式,导出了斜程传输时准单色高斯-谢尔光束互相干函数的解析式。然后,利用表征光束时间相干性的纵向相干长度(可由互相干函数导出),研究了斜程传输时大气湍流对准单色高斯-谢尔光束时间相干性的影响。研究结果表明,准单色高斯-谢尔光束的时间相干性在整个斜程传输过程中保持不变。最后,对该结果在物理上给予了定性解释。     

Study on the Scintillation for Optical Wave Propagation in the Slant Path Through the Atmospheric Turbulence

Based on the theory of optical wave propagation in the slant path and the ITU-R turbulence structure constant model which is altitude dependent, the modified Rytov method, which is applicable to the optical wave propagation in the horizontal path, is extended to the propagation in the slant path. According to the spectrum model with non-zero inner scale, the scintillation index as a function of the Rytov variance is obtained from weak to strong fluctuation regions with plane wave and sphere wave incidence, respectively. Finally, the scintillation index with different zenith angles is analyzed, and comparison between the results of the modified method and experimental measurements is given and discussed in detail.     

Lengths on rotating platforms

The paper treats the issue of the length of a rotating circumference as seen from on board the moving disk and from an inertial reference frame. It is shown that, properly defining a measuring process, the result is in both cases 2πR thus dissolving the Ehrenfest paradox. The same holds good when considering that, for the rotating observer, the perceived radius coincides with the curvature radius of a space-time helix and a complete round trip corresponds to an angle which differs from the one seen by the inertial observer. The apparent contradiction with the Lorentz contraction is discussed.     

高斯波束在湍流大气斜程传输中的闪烁问题研究

根据光波斜程传输理论以及随高度变化的ITU-R湍流大气结构常数模型,将水平传输的修正里托夫方法推广到了斜程传输问题中。导出了高斯波束入射时,在零内尺度湍流谱模型及考虑湍流内尺度效应和外尺度效应条件下,从弱起伏湍流区到强起伏湍流区闪烁指数随斜程里托夫方差变化的计算公式。讨论了不同传播高度、不同湍流内、外尺度对高斯波束斜程传输闪烁指数的影响。最后将有关数值计算结果与实验测量结果进行了比较和验证。     

Gibbsian Hypothesis in Turbulence

We show that Kolmogorov multipliers in turbulence cannot be statistically independent of others at adjacent scales (or even a finite range apart) by numerical simulation of a shell model and by theory. As the simplest generalization of independent distributions, we suppose that the steady-state statistics of multipliers in the shell model are given by a translation-invariant Gibbs measure with a short-range potential, when expressed in terms of suitable spin variables: real-valued spins that are logarithms of multipliers and XY-spins defined by local dynamical phases. Numerical evidence is presented in favor of the hypothesis for the shell model, in particular novel scaling laws and derivative relations predicted by the existence of a thermodynamic limit. The Gibbs measure appears to be in a high-temperature, unique-phase regime with paramagnetic spin order.     

Turbulence in nuclear reactions

An explanation of the emergence of turbulence in nuclear collisions is proposed on the basis of the assumption that, in an excited nucleus, there arise nonequilibrium steady-state distributions n(?) of occupation numbers.     

Turbulence noise

We show that the large-eddy motions in turbulent fluid flow obey a modified hydrodynamic equation with a stochastic turbulent stress whose distribution is a causal functional of the large-scale velocity field itself. We do so by means of an exact procedure of statistical filtering of the Navier-Stokes equations, which formally solves the closure problem, and we discuss the relation of our analysis with the decimation theory of Kraichnan. We show that the statistical filtering procedure can be formulated using field-theoretic path-integral methods within the Martin-Siggia-Rose (MSR) formalism for classical statistical dynamics. We also establish within the MSR formalism a least-effective-action principle for mean turbulent velocity profiles, which generalizes Onsager's principle of least dissipation. This minimum principle is a consequence of a simple realizability inequality and therefore holds also in any realizable closure. Symanzik's theorem in field theory—which characterizes the static effective action as the minimum expected value of the quantum Hamiltonian over all state vectors with prescribed expectations of fields—is extended to MSR theory with non-Hermitian Hamiltonian. This allows stationary mean velocity profiles and other turbulence statistics to be calculated variationally by a Rayleigh-Ritz procedure. Finally, we develop approximations of the exact Langevin equations for large eddies, e.g., a random-coupling DIA model, which yield new stochastic LES models. These are compared with stochastic subgrid modeling schemes proposed by Rose, Chasnov, Leith, and others, and various applications are discussed.     

Turbulence spectra

The scaling invariance of the Navier-Stokes equations in the limit of infinite Reynolds number is used to derive laws for the inertial range of the turbulence spectrum. Whether the flow is homogeneous or not, the spectrum is chosen to be that given by a well-chosen biorthogonal decomposition. If the flow is hoogeneous, this spectrum coincides with the classical Fourier (energy) spectrum which exhibits Kolmogorov's k^–5/3 power law if the scaling exponent is assumed to be 1/3. In the more general case where the homogeneity assumption is relaxed, the spectrum is discrete and decays exponentially fast under the assumption that the flow is invariant (in a deterministic or statistical sense) under only one subgroup of the scaling coefficient of one scaling group of the equations (corresponding to one value of the scaling exponent). If the flow is invariant under two subgroups of scaling coefficients and, the spectrum becomes maximal, equal toR ₊. Finally, when a full symmetry, namely an invariance under a whole group, is assumed and the spectrum becomes continuous, the decaying law for the spectral density is derived and found to be independent of the specific value ofh These ideas are then applied to locally self-similar flows with multiple dilation centers (localized in space and time) and multiple scaling exponents, extending the concept of multifractals to space and time.     

Predicting Optimal Lengths of Random Knots

In a thermally fluctuating long linear polymeric chain in a solution, the ends, from time to time, approach each other. At such an instance, the chain can be regarded as closed and thus will form a knot or rather a virtual knot. Several earlier studies of random knotting demonstrated that simpler knots show a higher occurrence for shorter random walks than do more complex knots. However, up to now there have been no rules that could be used to predict the optimal length of a random walk, i.e. the length for which a given knot reaches its highest occurrence. Using numerical simulations, we show here that a power law accurately describes the relation between the optimal lengths of random walks leading to the formation of different knots and the previously characterized lengths of ideal knots of a corresponding type.     

Turbulence in a gas laser

We test a recent assertion [A. Muriel, Physica A 388 (4) (2009) 311] that a gas consisting of excited molecules is turbulent, in contrast to the laminar state of a gas of ground state molecules. Since a lasing gas is made up of excited molecules, we examine if a lasing gas system is indeed turbulent. Surprisingly, from a literature search, it appears that turbulence in a lasing gas medium has never been addressed. To test for turbulence, we use a recently proposed criterion for the existence of turbulence, the presence of multivalued steady-state velocity fields [P. Getreur, A. Albano, A. Muriel, Phys. Lett. A 366 (2007) 101]. To study this subject, we improve an old model of a gas of two-level atoms in a one-dimensional model [A. Muriel, M. Dresden, Physica D 94 (1996) 103] by including the effect of a radiation field with the use of Einstein A and B coefficients. A set of coupled equations for the velocity fields in one dimension are derived. The zeroth order implementation of an iterative solution establishes that the steady-state velocity fields are multivalued, given by the Lambert function. We obtain signature characteristics of turbulence such as velocity reversals, infinite gradients, and stagnation points.     

Turbulence in nonabelian gauge theory

Kolmogorov wave turbulence plays an important role for the thermalization process following plasma instabilities in nonabelian gauge theories. We show that classical-statistical simulations in SU(2)$\mathrm{SU}(2)$ gauge theory indicate a Kolmogorov scaling exponent known from scalar models. In the range of validity of resummed perturbation theory this result is shown to agree with analytical estimates. We study the effect of classical-statistical versus quantum corrections and demonstrate that the latter lead to the absence of turbulence in the far ultraviolet.