Cylindrical-wave diffraction by a rational wedge

In this paper, new expressions for the field produced by the diffraction of a cylindrical wave by a wedge, whose angle can be expressed as a rational multiple of π are given. The solutions are expressed in terms of source terms and real integrals that represent the diffracted field. The general result obtained includes as special cases, Macdonald's solution for diffraction by a half plane, a solution for Carslaw's problem of diffraction by a wedge of open angle 2π\3, and a new representation for the solution of the problem of diffraction by a mixed soft-hard half plane

Narrow-escape times for diffusion in microdomains with a particle-surface affinity: Mean-field results

We analyze the mean time t_{app} that a randomly moving particle spends in a bounded domain (sphere) before it escapes through a small window in the domain's boundary. A particle is assumed to diffuse freely in the bulk until it approaches the surface of the domain where it becomes weakly adsorbed, and then wanders diffusively along the boundary for a random time until it desorbs back to the bulk, and etc. Using a mean-field approximation, we define t_{app} analytically as a function of the bulk and surface diffusion coefficients, the mean time it spends in the bulk between two consecutive arrivals to the surface and the mean time it wanders on the surface within a single round of the surface diffusion.Comment: 8 pages, 1 figure, submitted to JC

Single-Species Three-Particle Reactions in One Dimension

Renormalization group calculations for fluctuation-dominated reaction-diffusion systems are generally in agreement with simulations and exact solutions. However, simulations of the single-species reactions 3A->(0,A,2A) at their upper critical dimension d_c=1 have found asymptotic densities argued to be inconsistent with renormalization group predictions. We show that this discrepancy is resolved by inclusion of the leading corrections to scaling, which we derive explicitly and show to be universal, a property not shared by the A+A->(0,A) reactions. Finally, we demonstrate that two previous Smoluchowski approaches to this problem reduce, with various corrections, to a single theory which yields, surprisingly, the same asymptotic density as the renormalization group.Comment: 8 pages, 5 figs, minor correction

Designing arrays of Josephson junctions for specific static responses

We consider the inverse problem of designing an array of superconducting Josephson junctions that has a given maximum static current pattern as function of the applied magnetic field. Such devices are used for magnetometry and as Terahertz oscillators. The model is a 2D semilinear elliptic operator with Neuman boundary conditions so the direct problem is difficult to solve because of the multiplicity of solutions. For an array of small junctions in a passive region, the model can be reduced to a 1D linear partial differential equation with Dirac distribution sine nonlinearities. For small junctions and a symmetric device, the maximum current is the absolute value of a cosine Fourier series whose coefficients (resp. frequencies) are proportional to the areas (resp. the positions) of the junctions. The inverse problem is solved by inverse cosine Fourier transform after choosing the area of the central junction. We show several examples using combinations of simple three junction circuits. These new devices could then be tailored to meet specific applications.Comment: The article was submitted to Inverse Problem

Quantum optomechanics of a multimode system coupled via photothermal and radiation pressure force

We provide a full quantum description of the optomechanical system formed by a Fabry-Perot cavity with a movable micro-mechanical mirror whose center-of-mass and internal elastic modes are coupled to the driven cavity mode by both radiation pressure and photothermal force. Adopting a quantum Langevin description, we investigate simultaneous cooling of the micromirror elastic and center-of-mass modes, and also the entanglement properties of the optomechanical multipartite system in its steady state.Comment: 11 pages, 7 figure

Exact Green's Function of the reversible diffusion-influenced reaction for an isolated pair in 2D

We derive an exact Green's function of the diffusion equation for a pair of spherical interacting particles in 2D subject to a back-reaction boundary condition.Comment: 6 pages, 1 Figur

Opinion dynamics: rise and fall of political parties

We analyze the evolution of political organizations using a model in which agents change their opinions via two competing mechanisms. Two agents may interact and reach consensus, and additionally, individual agents may spontaneously change their opinions by a random, diffusive process. We find three distinct possibilities. For strong diffusion, the distribution of opinions is uniform and no political organizations (parties) are formed. For weak diffusion, parties do form and furthermore, the political landscape continually evolves as small parties merge into larger ones. Without diffusion, a pattern develops: parties have the same size and they possess equal niches. These phenomena are analyzed using pattern formation and scaling techniques.Comment: 5 pages, 5 figure

Integral Equations for Heat Kernel in Compound Media

By making use of the potentials of the heat conduction equation the integral equations are derived which determine the heat kernel for the Laplace operator $-a^2\Delta$ in the case of compound media. In each of the media the parameter $a^2$ acquires a certain constant value. At the interface of the media the conditions are imposed which demand the continuity of the `temperature' and the `heat flows'. The integration in the equations is spread out only over the interface of the media. As a result the dimension of the initial problem is reduced by 1. The perturbation series for the integral equations derived are nothing else as the multiple scattering expansions for the relevant heat kernels. Thus a rigorous derivation of these expansions is given. In the one dimensional case the integral equations at hand are solved explicitly (Abel equations) and the exact expressions for the regarding heat kernels are obtained for diverse matching conditions. Derivation of the asymptotic expansion of the integrated heat kernel for a compound media is considered by making use of the perturbation series for the integral equations obtained. The method proposed is also applicable to the configurations when the same medium is divided, by a smooth compact surface, into internal and external regions, or when only the region inside (or outside) this surface is considered with appropriate boundary conditions.Comment: 26 pages, no figures, no tables, REVTeX4; two items are added into the Reference List; a new section is added, a version that will be published in J. Math. Phy

Ordering of Random Walks: The Leader and the Laggard

We investigate two complementary problems related to maintaining the relative positions of N random walks on the line: (i) the leader problem, that is, the probability {\cal L}_N(t) that the leftmost particle remains the leftmost as a function of time and (ii) the laggard problem, the probability {\cal R}_N(t) that the rightmost particle never becomes the leftmost. We map these ordering problems onto an equivalent (N-1)-dimensional electrostatic problem. From this construction we obtain a very accurate estimate for {\cal L}_N(t) for N=4, the first case that is not exactly soluble: {\cal L}_4(t) ~ t^{-\beta_4}, with \beta_4=0.91342(8). The probability of being the laggard also decays algebraically, {\cal R}_N(t) ~ t^{-\gamma_N}; we derive \gamma_2=1/2, \gamma_3=3/8, and argue that \gamma_N--> ln N/N$ as N-->oo.Comment: 7 pages, 4 figures, 2-column revtex 4 forma

From the solutions of diffusion equation to the solutions of subdiffusive one

Starting with the Green's functions found for normal diffusion, we construct exact time-dependent Green's functions for subdiffusive equation (with fractional time derivatives), with the boundary conditions involving a linear combination of fluxes and concentrations. The method is particularly useful to calculate the concentration profiles in a multi-part system where different kind of transport occurs in each part of it. As an example, we find the solutions of subdiffusive equation for the system composed from two parts with normal diffusion and subdiffusion, respectively.Comment: 11 pages, 2 figure