On the spectrum of the Laplace operator of metric graphs attached at a vertex -- Spectral determinant approach

We consider a metric graph $\mathcal{G}$ made of two graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ attached at one point. We derive a formula relating the spectral determinant of the Laplace operator $S_\mathcal{G}(\gamma)=\det(\gamma-\Delta)$ in terms of the spectral determinants of the two subgraphs. The result is generalized to describe the attachment of $n$ graphs. The formulae are also valid for the spectral determinant of the Schr\"odinger operator $\det(\gamma-\Delta+V(x))$.Comment: LaTeX, 8 pages, 7 eps figures, v2: new appendix, v3: discussions and ref adde

Ordered spectral statistics in 1D disordered supersymmetric quantum mechanics and Sinai diffusion with dilute absorbers

Some results on the ordered statistics of eigenvalues for one-dimensional random Schr\"odinger Hamiltonians are reviewed. In the case of supersymmetric quantum mechanics with disorder, the existence of low energy delocalized states induces eigenvalue correlations and makes the ordered statistics problem nontrivial. The resulting distributions are used to analyze the problem of classical diffusion in a random force field (Sinai problem) in the presence of weakly concentrated absorbers. It is shown that the slowly decaying averaged return probability of the Sinai problem, \mean{P(x,t|x,0)}\sim \ln^{-2}t, is converted into a power law decay, \mean{P(x,t|x,0)}\sim t^{-\sqrt{2\rho/g}}, where $g$ is the strength of the random force field and $\rho$ the density of absorbers.Comment: 10 pages ; LaTeX ; 4 pdf figures ; Proceedings of the meeting "Fundations and Applications of non-equilibrium statistical mechanics", Nordita, Stockholm, october 2011 ; v2: appendix added ; v3: figure 2.left adde

Spectral determinants and zeta functions of Schr\"odinger operators on metric graphs

A derivation of the spectral determinant of the Schr\"odinger operator on a metric graph is presented where the local matching conditions at the vertices are of the general form classified according to the scheme of Kostrykin and Schrader. To formulate the spectral determinant we first derive the spectral zeta function of the Schr\"odinger operator using an appropriate secular equation. The result obtained for the spectral determinant is along the lines of the recent conjecture.Comment: 16 pages, 2 figure

Understanding the friction mechanisms between the human finger and flat contacting surfaces in moist conditions

Human hands sweat in different circumstances and the presence of sweat can alter the friction between the hand and contacting surface. It is, therefore, important to understand how hand moisture varies between people, during different activities and the effect of this on friction. In this study, a survey of fingertip moisture was done. Friction tests were then carried out to investigate the effect of moisture. Moisture was added to the surface of the finger, the finger was soaked in water, and water was added to the counter-surface; the friction of the contact was then measured. It was found that the friction increased, up until a certain level of moisture and then decreased. The increase in friction has previously been explained by viscous shearing, water absorption and capillary adhesion. The results from the experiments enabled the mechanisms to be investigated analytically. This study found that water absorption is the principle mechanism responsible for the increase in friction, followed by capillary adhesion, although it was not conclusively proved that this contributes significantly. Both these mechanisms increase friction by increasing the area of contact and therefore adhesion. Viscous shearing in the liquid bridges has negligible effect. There are, however, many limitations in the modelling that need further exploration

Derivation of the Zakharov equations

This paper continues the study of the validity of the Zakharov model describing Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for well-prepared initial data. We apply this result to the Euler-Maxwell equations describing laser-plasma interactions, to obtain, in a high-frequency limit, an asymptotic estimate that describes solutions of the Euler-Maxwell equations in terms of WKB approximate solutions which leading terms are solutions of the Zakharov equations. Because of transparency properties of the Euler-Maxwell equations, this study is led in a supercritical (highly nonlinear) regime. In such a regime, resonances between plasma waves, electromagnetric waves and acoustic waves could create instabilities in small time. The key of this work is the control of these resonances. The proof involves the techniques of geometric optics of Joly, M\'etivier and Rauch, recent results of Lannes on norms of pseudodifferential operators, and a semiclassical, paradifferential calculus

Dephasing due to electron-electron interaction in a diffusive ring

We study the effect of the electron-electron interaction on the weak localization correction of a ring pierced by a magnetic flux. We compute exactly the path integral giving the magnetoconductivity for an isolated ring. The results are interpreted in a time representation. This allows to characterize the nature of the phase coherence relaxation in the ring. The nature of the relaxation depends on the time regime (diffusive or ergodic) but also on the harmonics $n$ of the magnetoconductivity. Whereas phase coherence relaxation is non exponential for the harmonic $n=0$, it is always exponential for harmonics $n\neq0$. Then we consider the case of a ring connected to reservoirs and discuss the effect of connecting wires. We recover the behaviour of the harmonics predicted recently by Ludwig & Mirlin for a large perimeter (compared to the Nyquist length). We also predict a new behaviour when the Nyquist length exceeds the perimeter.Comment: 21 pages, RevTeX4, 8 eps figures; version of 10/2006 : eqs.(100-102) of section V.C correcte

Quantum oscillations in mesoscopic rings and anomalous diffusion

We consider the weak localization correction to the conductance of a ring connected to a network. We analyze the harmonics content of the Al'tshuler-Aronov-Spivak (AAS) oscillations and we show that the presence of wires connected to the ring is responsible for a behaviour different from the one predicted by AAS. The physical origin of this behaviour is the anomalous diffusion of Brownian trajectories around the ring, due to the diffusion in the wires. We show that this problem is related to the anomalous diffusion along the skeleton of a comb. We study in detail the winding properties of Brownian curves around a ring connected to an arbitrary network. Our analysis is based on the spectral determinant and on the introduction of an effective perimeter probing the different time scales. A general expression of this length is derived for arbitrary networks. More specifically we consider the case of a ring connected to wires, to a square network, and to a Bethe lattice.Comment: 17 pages, 7 eps figure

Conditional stability of unstable viscous shock waves in compressible gas dynamics and MHD

Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic--parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\cap H^3$ perturbations, converging time-asymptotically to a translate of the unperturbed wave. That is, for a shock with $p$ unstable eigenvalues, we establish conditional stability on a codimension-$p$ manifold of initial data, with sharp rates of decay in all $L^p$. For $p=0$, we recover the result of unconditional stability obtained by Mascia and Zumbrun. The main new difficulty in the hyperbolic--parabolic case is to construct an invariant manifold in the absence of parabolic smoothing.Comment: 32p

Thermal noise and dephasing due to electron interactions in non-trivial geometries

We study Johnson-Nyquist noise in macroscopically inhomogeneous disordered metals and give a microscopic derivation of the correlation function of the scalar electric potentials in real space. Starting from the interacting Hamiltonian for electrons in a metal and the random phase approximation, we find a relation between the correlation function of the electric potentials and the density fluctuations which is valid for arbitrary geometry and dimensionality. We show that the potential fluctuations are proportional to the solution of the diffusion equation, taken at zero frequency. As an example, we consider networks of quasi-1D disordered wires and give an explicit expression for the correlation function in a ring attached via arms to absorbing leads. We use this result in order to develop a theory of dephasing by electronic noise in multiply-connected systems.Comment: 9 pages, 6 figures (version submitted to PRB