Playing with sandpiles

The Bak–Tang–Wiesenfeld sandpile model provides a simple and elegant system with which to demonstrate self-organized criticality. This model has rather remarkable mathematical properties first elucidated by Dhar. I demonstrate some of these properties graphically with a simple computer simulation.     

Diffusion in stochastic sandpiles

We study diffusion of particles in large-scale simulations of one-dimensional stochastic sandpiles, in both the restricted and unrestricted versions. The results indicate that the diffusion constant scales in the same manner as the activity density, so that it represents an alternative definition of an order parameter. The critical behavior of the unrestricted sandpile is very similar to that of its restricted counterpart, including the fact that a data collapse of the order parameter as a function of the particle density is possible, but with a narrow scaling region. We also develop a series expansion, in inverse powers of the density, for the collective diffusion coefficient in a variant of the stochastic sandpile in which the toppling rate at a site with n particles is n(n-1), and compare the theoretical prediction with simulation results.     

Bethe lattice representation for sandpiles

Avalanches in sandpiles are represented by a process of percolation in a Bethe lattice with a feedback mechanism. The results indicate that the frequency spectrum and probability distribution of avalanches provide a better resemblance to the experimental results than other models using cellular automata simulations. Apparent discrepancies between experiments performed by different authors are reconciled. Critical behavior is expressed here by the critical properties of percolation phenomena.     

颗粒堆的体积分数与制备流量关系的实验研究

通过实验研究发现,颗粒堆的体积分数随制备颗粒堆的颗粒流流量指数衰减,减小或增加流量到一定程度时,体积分数都达到饱和;出料口直径与颗粒粒径的比值小到一临界值时,随着流量的减小体积分数增加急剧变缓;颗粒粒径小到一临界值时,随着流量的增加,体积分数的减小急剧变缓.结合颗粒物质的强耗散性、空气作用、瓶颈效应和碰撞理论解释了实验现象,从连续性原理出发推出的颗粒堆体积分数随制备流量变化的函数与实验数据的拟合公式相同.     

Are avalanches in sandpiles a chaotic phenomenon?

We show that deterministic systems with strong nonlinearities seem to be more appropriate to model sandpiles than stochastic systems or deterministic systems in which discontinuities are the only nonlinearity. In particular, we are able to reproduce the breakdown of self-organized criticality found in two well known experiments, that is, a centrally fueled pile [Phys. Rev. Lett. 65 (1990) 1120] and sand in a rotating tray [Phys. Rev. Lett. 69 (1992) 2431]. By varying the parameters of the model we recover self-organized criticality, in agreement with other experiments and other models. We show that chaos plays a fundamental role in the dynamics of the system.     

Kinematic segregation of granular mixtures in sandpiles

We study the segregation of granular mixtures in two-dimensional silos using a recently proposed set of coupled equations for surface flows of grains. We study the thick flow regime, where the grains are segregated in the rolling phase. We incorporate this dynamical segregation process, called kinematic sieving, free-surface segregation or percolation, into the theoretical formalism and calculate the profiles of the rolling species and the concentration of grains in the bulk in the steady state. Our solution shows the segregation of the mixture with the large grains being found at the bottom of the pile in qualitative agreement with experiments. Received: 6 July 1998 / Revised and Accepted: 13 August 1998     

Large scale structures, symmetry, and universality in sandpiles

We introduce a sandpile model where, at each unstable site, all grains are transferred randomly to downstream neighbors. The model is local and conservative, but not Abelian. This does not appear to change the universality class for the avalanches in the self-organized critical state. It does, however, introduce long-range spatial correlations within the metastable states. For the transverse direction d(perpendicular)>0, we find a fractal network of occupied sites, whose density vanishes as a power law with distance into the sandpile.     

Landslides on sandpiles: Some moment relations in one dimension

Several relations between the structure of stable recurrent states and the statistics of avalanches in a one-dimensional sandpile automaton are derived and numerically verified. In particular, it is shown that the average avalanche size is determined by the second rather than the first moment of the distribution of trough distances. The two moments scale differently with system size, which implies multiscaling for the distribution. Moreover, the scaling of edge events (avalanches which fall off the pile) is shown to differ from that of bulk events (avalanches which remain on the pile).     

Precise toppling balance, quenched disorder, and universality for sandpiles

A single sandpile model with quenched random toppling matrices captures the crucial features of different models of self-organized criticality. With symmetric matrices avalanche statistics falls in the multiscaling Bak-Tang-Wiesenfeld universality class. In the asymmetric case the simple scaling of the Manna model is observed. The presence or absence of a precise toppling balance between the amount of sand released by a toppling site and the total quantity the same site receives when all its neighbors topple once determines the appropriate universality class.     

Semi-simple real subalgebras of non-compact semi-simple real Lie algebras IV

The investigation of the problem of embedding a semi-simple real Lie algebra $\text{L}$′ in a non-compact semi-simple real Lie algebra $\text{L}$ is extended to the case in which at least one of the real Lie algebras has a semi-simple complex extension, which consists of the direct sum of two simple complex Lie algebras. Detailed procedures are given, which together with those given previously, allow the construction of all embeddings of $\text{L}$′ in $\text{L}$ when their complex extensions are A₁, B₁, C₁, D₁ or a direct sum of any two of these. The procedures are illustrated by considering examples corresponding to complex Lie algebra embeddings A₁?(A₂⊕A₂), (A₁⊕A₁)?(A₂⊕A₂), (A₁⊕A₁)?A₃, (A₁⊕A₁)??(A₃⊕A₃) and (A₁⊕A₁)?(A₃⊕A₂). Because of its physical significanc embeddings of SL(2,C) in simple and semi-simple real Lie algebras are studied in detail.     

Semi-simple real subal gebras of non-compact semi-simple real lie algebras V

The investigation of the problem of embedding a semi-simple real Lie algebra $\text{L}$ in a non-compact semi-simple real Lie algebra $\text{L}$ is extended to the case when $\text{L}$ and/or $\text{L}$ is exceptional. Matrix representations for all the exceptional Lie algebras are calculated. Detailed procedures are given, which, together with those given in previous papers, allow the construction of all embeddings of $\text{L}$ in $\text{L}$, when their complex extensions are A₁, B₁, C₁, D₁, E₆, E₇, E₈, F₄, G₂ or a direct sum of any two of there. The procedures are illustrated by examples, including all real semi-simple Lie subalgebras of real forms of G₂ and sub-algebras of real forms of F₄ whose complex extensions are B₄ or A₁ (representation (16) + (9)). Because of its physical significance, all embeddings of SL (2, C) in real forms of F₄ and E₆ are given. Many of these are new results.     

Statistics of real eigenvalues in Ginibre's ensemble of random real matrices

The integrable structure of Ginibre's orthogonal ensemble of random matrices is looked at through the prism of the probability p(n,k) to find exactly k real eigenvalues in the spectrum of an n x n real asymmetric Gaussian random matrix. The exact solution for the probability function p(n,k) is presented, and its remarkable connection to the theory of symmetric functions is revealed. An extension of the Dyson integration theorem is a key ingredient of the theory presented.     

火焰原子吸收光谱法测定XF-210膦羧酸缓蚀阻垢剂中的钴

本文研究了XF-210膦羧酸缓蚀阻垢剂中微量金属元素钴的火焰原子吸收分光光度法的测定,方法简便,省时,具有良好的精密度和准确度.平均相对标准偏差为1.42%,平均回收率为95.7%.     

Isotherms of real gases

A refined kinetic equation is proposed for the adsorption of molecules on a solid surface, taking account of factors such as the interaction of the molecules and their migration over the surface. Detailed theoretical analysis of the process is complemented by experimental verification of the resulting formulas by pulsed photodesorption. The time of monolayer formation is found to be two orders of magnitude greater than the values calculated from the Langmuir isotherm. This indicates that the new formula for the adsorption isotherm satisfactorily describes the process in high vacuum. Kazan’ State Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 41–48, April, 1998.     

Apparent and real roughness

Some works have studied the light scattering from a rough surface immersed in a liquid, where the light scattered from the immersed rough surface was assumed to be equivalent to that scattered from a fictitious rough surface with a different texture. This work deals with this kind of problems analyzing the light scattered by a reflecting strong scatterer and the light scattered by a rough surface immersed in a transparent liquid. The general Kirchhoff solution for scattering from a rough surface has been used.It is shown that under certain conditions, the mean scattered intensity (MSI) from a surface immersed in a liquid can be quasi-indistinguishable from that scattered from a non-immersed surface with an “equivalent texture”. An expression relating the equivalent texture and the immersed surface texture was obtained.The results of this work allow to evaluate the characteristics of any surface immersed in a liquid with a known refractive index.     

Resonantly enhanced real hyperpolarizability

We show theoretically and experimentally how the first hyperpolarizability of a three-level molecule at a two-photon resonance can obtain a real value with strong resonant enhancement. This phenomenon arises from destructive interference of the usually dominant imaginary parts of the resonant hyperpolarizability.     

火焰原子吸收光谱测定XF—120膦羧酸缓蚀阻垢剂中的钴

本文研究了ＸＦ－２１０膦羧酸缓蚀阻垢剂中微量金属元素的火焰原子吸收光分光光度法的测定，方法简便，省时，具有良好的精密度和准确度，平均相对标准偏差为１．４２％，平均回收率为９５．７％。