Site percolation and random walks on d-dimensional Kagome lattices

The site percolation problem is studied on d-dimensional generalisations of the Kagome' lattice. These lattices are isotropic and have the same coordination number q as the hyper-cubic lattices in d dimensions, namely q=2d. The site percolation thresholds are calculated numerically for d= 3, 4, 5, and 6. The scaling of these thresholds as a function of dimension d, or alternatively q, is different than for hypercubic lattices: p_c ~ 2/q instead of p_c ~ 1/(q-1). The latter is the Bethe approximation, which is usually assumed to hold for all lattices in high dimensions. A series expansion is calculated, in order to understand the different behaviour of the Kagome' lattice. The return probability of a random walker on these lattices is also shown to scale as 2/q. For bond percolation on d-dimensional diamond lattices these results imply p_c ~ 1/(q-1).Comment: 11 pages, LaTeX, 8 figures (EPS format), submitted to J. Phys.

Complex-Temperature Singularities in the d=2d=2 Ising Model. III. Honeycomb Lattice

We study complex-temperature properties of the uniform and staggered susceptibilities $\chi$ and $\chi^{(a)}$ of the Ising model on the honeycomb lattice. From an analysis of low-temperature series expansions, we find evidence that $\chi$ and $\chi^{(a)}$ both have divergent singularities at the point $z=-1 \equiv z_{\ell}$ (where $z=e^{-2K}$), with exponents $\gamma_{\ell}'= \gamma_{\ell,a}'=5/2$. The critical amplitudes at this singularity are calculated. Using exact results, we extract the behaviour of the magnetisation $M$ and specific heat $C$ at complex-temperature singularities. We find that, in addition to its zero at the physical critical point, $M$ diverges at $z=-1$ with exponent $\beta_{\ell}=-1/4$, vanishes continuously at $z=\pm i$ with exponent $\beta_s=3/8$, and vanishes discontinuously elsewhere along the boundary of the complex-temperature ferromagnetic phase. $C$ diverges at $z=-1$ with exponent $\alpha_{\ell}'=2$ and at $v=\pm i/\sqrt{3}$ (where $v = \tanh K$) with exponent $\alpha_e=1$, and diverges logarithmically at $z=\pm i$. We find that the exponent relation $\alpha'+2\beta+\gamma'=2$ is violated at $z=-1$; the right-hand side is 4 rather than 2. The connections of these results with complex-temperature properties of the Ising model on the triangular lattice are discussed.Comment: 22 pages, latex, figures appended after the end of the text as a compressed, uuencoded postscript fil

Advanced materials for space

The principal thrust of the LSST program is to develop the materials technology required for confident design of large space systems such as antennas and platforms. Areas of research in the FY-79 program include evaluation of polysulfones, measurement of the coefficient of thermal expansion of low expansion composite laminates, thermal cycling effects, and cable technology. The development of new long thermal control coatings and adhesives for use in space is discussed. The determination of radiation damage mechanisms of resin matrix composites and the formulation of new polymer matrices that are inherently more stable in the space environment are examined

Universal Formulae for Percolation Thresholds

A power law is postulated for both site and bond percolation thresholds. The formula writes $p_c=p_0[(d-1)(q-1)]^{-a}d^{\ b}$, where $d$ is the space dimension and $q$ the coordination number. All thresholds up to $d\rightarrow \infty$ are found to belong to only three universality classes. For first two classes $b=0$ for site dilution while $b=a$ for bond dilution. The last one associated to high dimensions is characterized by $b=2a-1$ for both sites and bonds. Classes are defined by a set of value for $\{p_0; \ a\}$. Deviations from available numerical estimates at $d \leq 7$ are within $\pm 0.008$ and $\pm 0.0004$ for high dimensional hypercubic expansions at $d \geq 8$. The formula is found to be also valid for Ising critical temperatures.Comment: 11 pages, latex, 3 figures not include

Cluster Percolation in O(n) Spin Models

The spontaneous symmetry breaking in the Ising model can be equivalently described in terms of percolation of Wolff clusters. In O(n) spin models similar clusters can be built in a general way, and they are currently used to update these systems in Monte Carlo simulations. We show that for 3-dimensional O(2), O(3) and O(4) such clusters are indeed the physical `islands' of the systems, i.e., they percolate at the physical threshold and the percolation exponents are in the universality class of the corresponding model. For O(2) and O(3) the result is proven analytically, for O(4) we derived it by numerical simulations.Comment: 11 pages, 8 figures, 2 tables, minor modification

The Gibbs-Thomson formula at small island sizes - corrections for high vapour densities

In this paper we report simulation studies of equilibrium features, namely circular islands on model surfaces, using Monte-Carlo methods. In particular, we are interested in studying the relationship between the density of vapour around a curved island and its curvature-the Gibbs-Thomson formula. Numerical simulations of a lattice gas model, performed for various sizes of islands, don't fit very well to the Gibbs-Thomson formula. We show how corrections to this form arise at high vapour densities, wherein a knowledge of the exact equation of state (as opposed to the ideal gas approximation) is necessary to predict this relationship. Exploiting a mapping of the lattice gas to the Ising model one can compute the corrections to the Gibbs-Thomson formula using high field series expansions. We also investigate finite size effects on the stability of the islands both theoretically and through simulations. Finally the simulations are used to study the microscopic origins of the Gibbs-Thomson formula. A heuristic argument is suggested in which it is partially attributed to geometric constraints on the island edge.Comment: 27 pages including 7 figures, tarred, gzipped and uuencoded. Prepared using revtex and espf.sty. To appear in Phys. Rev.

Percolation and cluster distribution. II. layers, variable-range interactions, and exciton cluster model

Monte Carlo simulations for the site percolation problem are presented for lattices up to 64 x 10 6 sites. We investigate for the square lattice the variable-range percolation problem, where distinct trends with bond-length are found for the critical concentrations and for the critical exponents β and γ . We also investigate the layer problem for stacks of square lattices added to approach a simple cubic lattice, yielding critical concentrations as a functional of layer number as well as the correlation length exponent ν . We also show that the exciton migration probability for a common type of ternary lattice system can be described by a cluster model and actually provides a cluster generating function.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45139/1/10955_2005_Article_BF01011724.pd

Moons Are Planets: Scientific Usefulness Versus Cultural Teleology in the Taxonomy of Planetary Science

We argue that taxonomical concept development is vital for planetary science as in all branches of science, but its importance has been obscured by unique historical developments. The literature shows that the concept of planet developed by scientists during the Copernican Revolution was theory-laden and pragmatic for science. It included both primaries and satellites as planets due to their common intrinsic, geological characteristics. About two centuries later the non-scientific public had just adopted heliocentrism and was motivated to preserve elements of geocentrism including teleology and the assumptions of astrology. This motivated development of a folk concept of planet that contradicted the scientific view. The folk taxonomy was based on what an object orbits, making satellites out to be non-planets and ignoring most asteroids. Astronomers continued to keep primaries and moons classed together as planets and continued teaching that taxonomy until the 1920s. The astronomical community lost interest in planets ca. 1910 to 1955 and during that period complacently accepted the folk concept. Enough time has now elapsed so that modern astronomers forgot this history and rewrote it to claim that the folk taxonomy is the one that was created by the Copernican scientists. Starting ca. 1960 when spacecraft missions were developed to send back detailed new data, there was an explosion of publishing about planets including the satellites, leading to revival of the Copernican planet concept. We present evidence that taxonomical alignment with geological complexity is the most useful scientific taxonomy for planets. It is this complexity of both primary and secondary planets that is a key part of the chain of origins for life in the cosmos.Comment: 68 pages, 16 figures. For supplemental data files, see https://www.philipmetzger.com/moons_are_planets