Diffusion, super-diffusion and coalescence from single step

From the exact single step evolution equation of the two-point correlation function of a particle distribution subjected to a stochastic displacement field \bu(\bx), we derive different dynamical regimes when \bu(\bx) is iterated to build a velocity field. First we show that spatially uncorrelated fields \bu(\bx) lead to both standard and anomalous diffusion equation. When the field \bu(\bx) is spatially correlated each particle performs a simple free Brownian motion, but the trajectories of different particles result to be mutually correlated. The two-point statistical properties of the field \bu(\bx) induce two-point spatial correlations in the particle distribution satisfying a simple but non-trivial diffusion-like equation. These displacement-displacement correlations lead the system to three possible regimes: coalescence, simple clustering and a combination of the two. The existence of these different regimes, in the one-dimensional system, is shown through computer simulations and a simple theoretical argument.Comment: RevTeX (iopstyle) 19 pages, 5 eps-figure

Antisymmetric PT-photonic structures with balanced positive and negative index materials

We propose a new class of synthetic optical materials in which the refractive index satisfies n(-\bx)=-n^*(\bx). We term such systems antisymmetric parity-time (APT) structures. Unlike PT-symmetric systems which require balanced gain and loss, i.e. n(-\bx)=n^*(\bx), APT systems consist of balanced positive and negative index materials. Despite the seemingly PT-symmetric optical potential V(\bx)\equiv n(\bx)^2\omega^2/c^2, APT systems are not invariant under combined PT operations due to the discontinuity of the spatial derivative of the wavefunction. We show that APT systems can display intriguing properties such as spontaneous phase transition of the scattering matrix, bidirectional invisibility, and a continuous lasing spectrum.Comment: 5 pages, 4 figure

Nuclear effects and their interplay in nuclear DVCS amplitudes

In this paper we analyze nuclear medium effects on DVCS amplitudes in the \Bx range of $0.1-0.0001$ for a large range of $Q^2$ and four different nuclei. We use our nucleon GPD model capable of describing all currently available DVCS data on the proton and extend it to the nuclear case using two competing parameterizations of nuclear effects. The two parameterizations, though giving different absolute numbers, yield the same type and magnitude of effects for the imaginary and real part of the nuclear DVCS amplitude. The imaginary part shows stronger nuclear shadowing effects compared to the inclusive case i.e. $F^N_2$, whereas in the real part nuclear shadowing at small \Bx and anti-shadowing at large \Bx combine through evolution to yield an even greater suppression than in the imaginary part up to large values of \Bx. This is the first time that such a combination of nuclear effects has been observed in a hadronic amplitude. The experimental implications will be discussed in a subsequent publication.Comment: 8 pages, 5 figures, uses RevTex4, final version to appear in PHys. Rev.

Elliptic operators with honeycomb symmetry: Dirac points, Edge States and Applications to Photonic Graphene

Consider electromagnetic waves in two-dimensional {\it honeycomb structured media}. The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operator \LA=-\nabla_\bx\cdot A(\bx) \nabla_\bx, where A(\bx) is $\Lambda_h-$ periodic ($\Lambda_h$ denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, A(\bx) is $\mathcal{P}\mathcal{C}-$ invariant (A(\bx)=\overline{A(-\bx)}) and $120^\circ$ rotationally invariant (A(R^*\bx)=R^*A(\bx)R, where $R$ is a $120^\circ$ rotation in the plane). We first obtain results on the existence, stability and instability of Dirac points, conical intersections between two adjacent Floquet-Bloch dispersion surfaces. We then show that the introduction through small and slow variations of a {\it domain wall} across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) {\it edge states}. These are time-harmonic solutions of Maxwell's equations which are propagating parallel to the line-defect and spatially localized transverse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term. These results imply the existence of {\it uni-directional} propagating edge states for two classes of time-reversal invariant media in which $\mathcal{C}$ symmetry is broken: magneto-optic media and bi-anisotropic media. Our analysis applies and extends the tools previously developed in the context of honeycomb Schr\"odinger operators.Comment: 65 pages, 8 figures, To appear in Archive for Rational Mechanics and Analysi

Continuous horizontally rigid functions of two variables are affine

Cain, Clark and Rose defined a function f\colon \RR^n \to \RR to be \emph{vertically rigid} if \graph(cf) is isometric to \graph (f) for every $c \neq 0$. It is \emph{horizontally rigid} if \graph(f(c \vec{x})) is isometric to \graph (f) for every $c \neq 0$ (see \cite{CCR}). In an earlier paper the authors of the present paper settled Jankovi\'c's conjecture by showing that a continuous function of one variable is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ (a,b,k \in \RR). Later they proved that a continuous function of two variables is vertically rigid if and only if after a suitable rotation around the z-axis it is of the form $a + bx + dy$, $a + s(y)e^{kx}$ or $a + be^{kx} + dy$ (a,b,d,k \in \RR, $k \neq 0$, s : \RR \to \RR continuous). The problem remained open in higher dimensions. The characterization in the case of horizontal rigidity is surprisingly simpler. C. Richter proved that a continuous function of one variable is horizontally rigid if and only if it is of the form $a+bx$ (a,b\in \RR). The goal of the present paper is to prove that a continuous function of two variables is horizontally rigid if and only if it is of the form $a + bx + dy$ (a,b,d \in \RR). This problem also remains open in higher dimensions. The main new ingredient of the present paper is the use of functional equations