Virtual dielectric waveguide mode description of a high-gain free-electron laser I: Theory

A set of mode-coupled excitation equations for the slowly-growing amplitudes of dielectric waveguide eigenmodes is derived as a description of the electromagnetic signal field of a high-gain free-electron laser, or FEL, including the effects of longitudinal space-charge. This approach of describing the field basis set has notable advantages for FEL analysis in providing an efficient characterization of eigenmodes, and in allowing a clear connection to free-space propagation of the input (seeding) and output radiation. The formulation describes the entire evolution of the radiation wave through the linear gain regime, prior to the onset of saturation, with arbitrary initial conditions. By virtue of the flexibility in the expansion basis, this technique can be used to find the direct coupling and amplification of a particular mode. A simple transformation converts the derived coupled differential excitation equations into a set of coupled algebraic equations and yields a matrix determinant equation for the FEL eigenmodes. A quadratic index medium is used as a model dielectric waveguide to obtain an expression for the predicted spot size of the dominant system eigenmode, in the approximation that it is a single gaussian mode.Comment: 14 page

Mean-field solution of the parity-conserving kinetic phase transition in one dimension

A two-offspring branching annihilating random walk model, with finite reaction rates, is studied in one-dimension. The model exhibits a transition from an active to an absorbing phase, expected to belong to the $DP2$ universality class embracing systems that possess two symmetric absorbing states, which in one-dimensional systems, is in many cases equivalent to parity conservation. The phase transition is studied analytically through a mean-field like modification of the so-called {\it parity interval method}. The original method of parity intervals allows for an exact analysis of the diffusion-controlled limit of infinite reaction rate, where there is no active phase and hence no phase transition. For finite rates, we obtain a surprisingly good description of the transition which compares favorably with the outcome of Monte Carlo simulations. This provides one of the first analytical attempts to deal with the broadly studied DP2 universality class.Comment: 4 Figures. 9 Pages. revtex4. Some comments have been improve

Cluster approximation solution of a two species annihilation model

A two species reaction-diffusion model, in which particles diffuse on a one-dimensional lattice and annihilate when meeting each other, has been investigated. Mean field equations for general choice of reaction rates have been solved exactly. Cluster mean field approximation of the model is also studied. It is shown that, the general form of large time behavior of one- and two-point functions of the number operators, are determined by the diffusion rates of the two type of species, and is independent of annihilation rates.Comment: 9 pages, 7 figure

Exactly solvable models through the empty interval method, for more-than-two-site interactions

Single-species reaction-diffusion systems on a one-dimensional lattice are considered, in them more than two neighboring sites interact. Constraints on the interaction rates are obtained, that guarantee the closedness of the time evolution equation for $E_n(t)$'s, the probability that $n$ consecutive sites are empty at time $t$. The general method of solving the time evolution equation is discussed. As an example, a system with next-nearest-neighbor interaction is studied.Comment: 19 pages, LaTeX2

Seeing the Forest for the Trees: Using the Gene Ontology to Restructure Hierarchical Clustering

Motivation: There is a growing interest in improving the cluster analysis of expression data by incorporating into it prior knowledge, such as the Gene Ontology (GO) annotations of genes, in order to improve the biological relevance of the clusters that are subjected to subsequent scrutiny. The structure of the GO is another source of background knowledge that can be exploited through the use of semantic similarity. Results: We propose here a novel algorithm that integrates semantic similarities (derived from the ontology structure) into the procedure of deriving clusters from the dendrogram constructed during expression-based hierarchical clustering. Our approach can handle the multiple annotations, from different levels of the GO hierarchy, which most genes have. Moreover, it treats annotated and unannotated genes in a uniform manner. Consequently, the clusters obtained by our algorithm are characterized by significantly enriched annotations. In both cross-validation tests and when using an external index such as protein–protein interactions, our algorithm performs better than previous approaches. When applied to human cancer expression data, our algorithm identifies, among others, clusters of genes related to immune response and glucose metabolism. These clusters are also supported by protein–protein interaction data. Contact: [email protected] Supplementary information: Supplementary data are available at Bioinformatics online.Lynne and William Frankel Center for Computer Science; Paul Ivanier center for robotics research and production; National Institutes of Health (R01 HG003367-01A1

Quantum phase transitions, frustration, and the Fermi surface in the Kondo lattice model

The quantum phase transition from a spin-Peierls phase with a small Fermi surface to a paramagnetic Luttinger-liquid phase with a large Fermi surface is studied in the framework of a one-dimensional Kondo-Heisenberg model that consists of an electron gas away from half filling, coupled to a spin-1/2 chain by Kondo interactions. The Kondo spins are further coupled to each other with isotropic nearest-neighbor and next-nearest-neighbor antiferromagnetic Heisenberg interactions which are tuned to the Majumdar-Ghosh point. Focusing on three-eighths filling and using the density-matrix renormalization-group (DMRG) method, we show that the zero-temperature transition between the phases with small and large Fermi momenta appears continuous, and involves a new intermediate phase where the Fermi surface is not well defined. The intermediate phase is spin gapped and has Kondo-spin correlations that show incommensurate modulations. Our results appear incompatible with the local picture for the quantum phase transition in heavy fermion compounds, which predicts an abrupt change in the size of the Fermi momentum.Comment: 9 pages, 8 figure

Deterministic reaction models with power-law forces

We study a one-dimensional particles system, in the overdamped limit, where nearest particles attract with a force inversely proportional to a power of their distance and coalesce upon encounter. The detailed shape of the distribution function for the gap between neighbouring particles serves to discriminate between different laws of attraction. We develop an exact Fokker-Planck approach for the infinite hierarchy of distribution functions for multiple adjacent gaps and solve it exactly, at the mean-field level, where correlations are ignored. The crucial role of correlations and their effect on the gap distribution function is explored both numerically and analytically. Finally, we analyse a random input of particles, which results in a stationary state where the effect of correlations is largely diminished

Inter-Particle Distribution Functions for One-Species Diffusion-Limited Annihilation, A+A->0

Diffusion-limited annihilation, $A+A\to 0$, and coalescence, $A+A\to A$, may both be exactly analyzed in one dimension. While the concentrations of $A$ particles in the two processes bear a simple relation, the inter-particle distribution functions (IPDF) exhibit remarkable differences. However, the IPDF is known exactly only for the coalescence process. We obtain the IPDF for the annihilation process, based on the Glauber spin approach and assuming that the IPDF's of nearest-particle pairs are statistically independent. This assumption is supported by computer simulations. Our analysis sheds further light on the relationship between the annihilation and the coalescence models.Comment: 15 pages, plain TeX, 3 figures - available upon request (snail mail

Diffusion-Limited Coalescence with Finite Reaction Rates in One Dimension

We study the diffusion-limited process $A+A\to A$ in one dimension, with finite reaction rates. We develop an approximation scheme based on the method of Inter-Particle Distribution Functions (IPDF), which was formerly used for the exact solution of the same process with infinite reaction rate. The approximation becomes exact in the very early time regime (or the reaction-controlled limit) and in the long time (diffusion-controlled) asymptotic limit. For the intermediate time regime, we obtain a simple interpolative behavior between these two limits. We also study the coalescence process (with finite reaction rates) with the back reaction $A\to A+A$, and in the presence of particle input. In each of these cases the system reaches a non-trivial steady state with a finite concentration of particles. Theoretical predictions for the concentration time dependence and for the IPDF are compared to computer simulations. P. A. C. S. Numbers: 82.20.Mj 02.50.+s 05.40.+j 05.70.LnComment: 13 pages (and 4 figures), plain TeX, SISSA-94-0