Random Magnetic Impurities and the Landau Problem

The 2-dimensional density of states of an electron is studied for a Poissonian random distribution of point vortices carrying $\alpha$ flux in unit of the quantum of flux. It is shown that, for any given density of impurities, there is a transition, when $\alpha\simeq 0.3-0.4$, from an "almost free" density of state -with only a depletion of states at the bottom of the spectrum characterized by a Lifschitz tail- to a Landau density of state with sharp Landau level oscillations. Several evidences and arguments for this transition -numerical and analytical- are presented.Comment: 22 pages, latex, 4 figures upon reques

Arithmetic area for m planar Brownian paths

We pursue the analysis made in [1] on the arithmetic area enclosed by m closed Brownian paths. We pay a particular attention to the random variable S{n1,n2, ...,n} (m) which is the arithmetic area of the set of points, also called winding sectors, enclosed n1 times by path 1, n2 times by path 2, ...,nm times by path m. Various results are obtained in the asymptotic limit m->infinity. A key observation is that, since the paths are independent, one can use in the m paths case the SLE information, valid in the 1-path case, on the 0-winding sectors arithmetic area.Comment: 12 pages, 2 figure

MicroMegascope

Atomic Force Microscopy (AFM) allows to reconstruct the topography of surface with a resolution in the nanometer range. The exceptional resolution attainable with the AFM makes this instrument a key tool in nanoscience and technology. The core of the set-up relies on the detection of the mechanical properties of a micro-oscillator when approached to a sample to image. Despite the fact that AFM is nowadays a very common instrument for research and development applications, thanks to the exceptional performances and the relative simplicity to use it, the fabrication of the micrometric scale mechanical oscillator is still a very complicated and expensive task requiring a dedicated platform. Being able to perform atomic force microscopy with a macroscopic oscillator would make the instrument more versatile and accessible for an even larger spectrum of applications and audiences. We present for the first time atomic force imaging with a centimetric oscillator. We show how it is possible to perform topographical images with nanometric resolution with a grams tuning fork. The images presented here are obtained with an aluminum tuning fork of centimeter size as sensor on which an accelerometer is glued on one prong to measure the oscillation of the resonator. In addition to the stunning sensitivity, by imaging both in air and in liquid, we show the high versatility of such oscillator. The set up proposed here can be extended to numerous experiments where the probe needs to be heavy and/or very complex as well as the environment

Quantifying and mapping covalent bond scission during elastomer fracture

Many new soft but tough rubbery materials have been recently discovered and new applications such as flexible prosthetics, stretchable electrodes or soft robotics continuously emerge. Yet, a credible multi-scale quantitative picture of damage and fracture of these materials has still not emerged, due to our fundamental inability to disentangle the irreversible scission of chemical bonds along the fracture path from dissipation by internal molecular friction. Here, by coupling new fluorogenic mechanochemistry with quantitative confocal microscopy mapping, we uncover how many and where covalent bonds are broken as an elastomer fractures. Our measurements reveal that bond scission near the crack plane can be delocalized over up to hundreds of micrometers and increase by a factor of 100 depending on temperature and stretch rate, pointing to an intricated coupling between strain rate dependent viscous dissipation and strain dependent irreversible network scission. These findings paint an entirely novel picture of fracture in soft materials, where energy dissipated by covalent bond scission accounts for a much larger fraction of the total fracture energy than previously believed. Our results pioneer the sensitive, quantitative and spatially-resolved detection of bond scission to assess material damage in a variety of soft materials and their applications

Quantifying Rate-and Temperature-Dependent Molecular Damage in Elastomer Fracture

Elastomers are highly valued soft materials finding many applications in the engineering and biomedical fields for their ability to stretch reversibly to large deformations. Yet their maximum extensibility is limited by the occurrence of fracture, which is currently still poorly understood. Because of a lack of experimental evidence, current physical models of elastomer fracture describe the rate and temperature dependence of the fracture energy as being solely due to viscoelastic friction, with chemical bond scission at the crack tip assumed to remain constant. Here, by coupling new fluorogenic mechanochemistry with quantitative confocal microscopy mapping, we are able to quantitatively detect, with high spatial resolution and sensitivity, the scission of covalent bonds as ordinary elastomers fracture at different strain rates and temperatures. Our measurements reveal that, in simple networks, bond scission, far from being restricted to a constant level near the crack plane, can both be delocalized over up to hundreds of micrometers and increase by a factor of 100, depending on the temperature and stretch rate. These observations, permitted by the high fluorescence and stability of the mechanophore, point to an intricate coupling between strain-rate-dependent viscous dissipation and strain-dependent irreversible network scission. These findings paint an entirely novel picture of fracture in soft materials, where energy dissipated by covalent bond scission accounts for a much larger fraction of the total fracture energy than previously believed. Our results pioneer the sensitive, quantitative, and spatially resolved detection of bond scission to assess material damage in a variety of soft materials and their applications. © 2020 authors. Published by the American Physical Society

The Local Time Distribution of a Particle Diffusing on a Graph

We study the local time distribution of a Brownian particle diffusing along the links on a graph. In particular, we derive an analytic expression of its Laplace transform in terms of the Green's function on the graph. We show that the asymptotic behavior of this distribution has non-Gaussian tails characterized by a nontrivial large deviation function.Comment: 8 pages, two figures (included

Numerical studies of planar closed random walks

Lattice numerical simulations for planar closed random walks and their winding sectors are presented. The frontiers of the random walks and of their winding sectors have a Hausdorff dimension $d_H=4/3$. However, when properly defined by taking into account the inner 0-winding sectors, the frontiers of the random walks have a Hausdorff dimension $d_H\approx 1.77$.Comment: 15 pages, 15 figure

Brownian Motion in wedges, last passage time and the second arc-sine law

We consider a planar Brownian motion starting from $O$ at time $t=0$ and stopped at $t=1$ and a set $F= \{OI_i ; i=1,2,..., n\}$ of $n$ semi-infinite straight lines emanating from $O$. Denoting by $g$ the last time when $F$ is reached by the Brownian motion, we compute the probability law of $g$. In particular, we show that, for a symmetric $F$ and even $n$ values, this law can be expressed as a sum of $\arcsin$ or $(\arcsin)^2$ functions. The original result of Levy is recovered as the special case $n=2$. A relation with the problem of reaction-diffusion of a set of three particles in one dimension is discussed

Spectral determinant on quantum graphs

We study the spectral determinant of the Laplacian on finite graphs characterized by their number of vertices V and of bonds B. We present a path integral derivation which leads to two equivalent expressions of the spectral determinant of the Laplacian either in terms of a V x V vertex matrix or a 2B x 2B link matrix that couples the arcs (oriented bonds) together. This latter expression allows us to rewrite the spectral determinant as an infinite product of contributions of periodic orbits on the graph. We also present a diagrammatic method that permits us to write the spectral determinant in terms of a finite number of periodic orbit contributions. These results are generalized to the case of graphs in a magnetic field. Several examples illustrating this formalism are presented and its application to the thermodynamic and transport properties of weakly disordered and coherent mesoscopic networks is discussed.Comment: 33 pages, submitted to Ann. Phy