Asymptotic statistics of the n-sided planar Voronoi cell: II. Heuristics

We develop a set of heuristic arguments to explain several results on planar Poisson-Voronoi tessellations that were derived earlier at the cost of considerable mathematical effort. The results concern Voronoi cells having a large number n of sides. The arguments start from an entropy balance applied to the arrangement of n neighbors around a central cell. It is followed by a simplified evaluation of the phase space integral for the probability p_n that an arbitrary cell be n-sided. The limitations of the arguments are indicated. As a new application we calculate the expected number of Gabriel (or full) neighbors of an n-sided cell in the large-n limit.Comment: 22 pages, 10 figure

Two interacting Ising chains in relative motion

We consider two parallel cyclic Ising chains counter-rotating at a relative velocity v, the motion actually being a succession of discrete steps. There is an in-chain interaction between nearest-neighbor spins and a cross-chain interaction between instantaneously opposite spins. For velocities v>0 the system, subject to a suitable markovian dynamics at a temperature T, can reach only a nonequilibrium steady state (NESS). This system was introduced by Hucht et al., who showed that for v=\infty it undergoes a para- to ferromagnetic transition, essentially due to the fact that each chain exerts an effective field on the other one. The present study of the v=\infty case determines the consequences of the fluctuations of this effective field when the system size N is finite. We show that whereas to leading order the system obeys detailed balancing with respect to an effective time-independent Hamiltonian, the higher order finite-size corrections violate detailed balancing. Expressions are given to various orders in 1/N for the interaction free energy between the chains, the spontaneous magnetization, the in-chain and cross-chain spin-spin correlations, and the spontaneus magnetization. It is shown how finite-size scaling functions may be derived explicitly. This study was motivated by recent work on a two-lane traffic problem in which a similar phase transition was found.Comment: 30 pages, 1 figur

New Monte Carlo method for planar Poisson-Voronoi cells

By a new Monte Carlo algorithm we evaluate the sidedness probability p_n of a planar Poisson-Voronoi cell in the range 3 \leq n \leq 1600. The algorithm is developed on the basis of earlier theoretical work; it exploits, in particular, the known asymptotic behavior of p_n as n\to\infty. Our p_n values all have between four and six significant digits. Accurate n dependent averages, second moments, and variances are obtained for the cell area and the cell perimeter. The numerical large n behavior of these quantities is analyzed in terms of asymptotic power series in 1/n. Snapshots are shown of typical occurrences of extremely rare events implicating cells of up to n=1600 sides embedded in an ordinary Poisson-Voronoi diagram. We reveal and discuss the characteristic features of such many-sided cells and their immediate environment. Their relevance for observable properties is stressed.Comment: 35 pages including 10 figures and 4 table

Sylvester's question and the Random Acceleration Process

Let n points be chosen randomly and independently in the unit disk. "Sylvester's question" concerns the probability p_n that they are the vertices of a convex n-sided polygon. Here we establish the link with another problem. We show that for large n this polygon, when suitably parametrized by a function r(phi) of the polar angle phi, satisfies the equation of the random acceleration process (RAP), d^2 r/d phi^2 = f(phi), where f is Gaussian noise. On the basis of this relation we derive the asymptotic expansion log p_n = -2n log n + n log(2 pi^2 e^2) - c_0 n^{1/5} + ..., of which the first two terms agree with a rigorous result due to Barany. The nonanalyticity in n of the third term is a new result. The value 1/5 of the exponent follows from recent work on the RAP due to Gyorgyi et al. [Phys. Rev. E 75, 021123 (2007)]. We show that the n-sided polygon is effectively contained in an annulus of width \sim n^{-4/5} along the edge of the disk. The distance delta_n of closest approach to the edge is exponentially distributed with average 1/(2n).Comment: 29 pages, 4 figures; references added and minor change

Large-n conditional facedness m_n of 3D Poisson-Voronoi cells

We consider the three-dimensional Poisson-Voronoi tessellation and study the average facedness m_n of a cell known to neighbor an n-faced cell. Whereas Aboav's law states that m_n=A+B/n, theoretical arguments indicate an asymptotic expansion m_n = 8 + k_1 n^{-1/6} +.... Recent new Monte Carlo data due to Lazar et al., based on a very large data set, now clearly rule out Aboav's law. In this work we determine the numerical value of k_1 and compare the expansion to the Monte Carlo data. The calculation of k_1 involves an auxiliary planar cellular structure composed of circular arcs, that we will call the Poisson-Moebius diagram. It is a special case of more general Moebius diagrams (or multiplicatively weighted power diagrams) and is of interest for its own sake. We obtain exact results for the total edge length per unit area, which is a prerequisite for the coefficient k_1, and a few other quantities in this diagram.Comment: 18 pages, 5 figure

A Supervisor for Control of Mode-switch Process

Many processes operate only around a limited number of operation points. In order to have adequate control around each operation point, and adaptive controller could be used. When the operation point changes often, a large number of parameters would have to be adapted over and over again. This makes application of conventional adaptive control unattractive, which is more suited for processes with slowly changing parameters. Furthermore, continuous adaptation is not always needed or desired. An extension of adaptive control is presented, in which for each operation point the process behaviour can be stored in a memory, retrieved from it and evaluated. These functions are co-ordinated by a ¿supervisor¿. This concept is referred to as a supervisor for control of mode-switch processes. It leads to an adaptive control structure which quickly adjusts the controller parameters based on retrieval of old information, without the need to fully relearn each time. This approach has been tested on experimental set-ups of a flexible beam and of a flexible two-link robot arm, but it is directly applicable to other processes, for instance, in the (petro) chemical industry