693 research outputs found

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

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    We consider a metric graph G\mathcal{G} made of two graphs G1\mathcal{G}_1 and G2\mathcal{G}_2 attached at one point. We derive a formula relating the spectral determinant of the Laplace operator SG(γ)=det(γΔ)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 nn graphs. The formulae are also valid for the spectral determinant of the Schr\"odinger operator det(γΔ+V(x))\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

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    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 gg is the strength of the random force field and ρ\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

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

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    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

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    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

    Quantum oscillations in mesoscopic rings and anomalous diffusion

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    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

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    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 L1H3L^1\cap H^3 perturbations, converging time-asymptotically to a translate of the unperturbed wave. That is, for a shock with pp unstable eigenvalues, we establish conditional stability on a codimension-pp manifold of initial data, with sharp rates of decay in all LpL^p. For p=0p=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

    Geometrical dependence of decoherence by electronic interactions in a GaAs/GaAlAs square network

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    We investigate weak localization in metallic networks etched in a two dimensional electron gas between 2525\:mK and 750750\:mK when electron-electron (e-e) interaction is the dominant phase breaking mechanism. We show that, at the highest temperatures, the contributions arising from trajectories that wind around the rings and trajectories that do not are governed by two different length scales. This is achieved by analyzing separately the envelope and the oscillating part of the magnetoconductance. For T0.3T\gtrsim0.3\:K we find \Lphi^\mathrm{env}\propto{T}^{-1/3} for the envelope, and \Lphi^\mathrm{osc}\propto{T}^{-1/2} for the oscillations, in agreement with the prediction for a single ring \cite{LudMir04,TexMon05}. This is the first experimental confirmation of the geometry dependence of decoherence due to e-e interaction.Comment: LaTeX, 5 pages, 4 eps figure

    Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices

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    The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products of random matrices pioneered by Furstenberg. After a brief historical survey, we describe some recent work that exploits the close connections between these topics. We review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case. Finally, we point out some limitations of the Lyapunov exponent as a means of studying localisation properties.Comment: LaTeX, 23 pages, 3 pdf figures ; review for a special issue on "Lyapunov analysis" ; v2 : typo corrected in eq.(3) & minor change

    One-dimensional classical diffusion in a random force field with weakly concentrated absorbers

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    A one-dimensional model of classical diffusion in a random force field with a weak concentration ρ\rho of absorbers is studied. The force field is taken as a Gaussian white noise with \mean{\phi(x)}=0 and \mean{\phi(x)\phi(x')}=g \delta(x-x'). Our analysis relies on the relation between the Fokker-Planck operator and a quantum Hamiltonian in which absorption leads to breaking of supersymmetry. Using a Lifshits argument, it is shown that the average return probability is a power law \smean{P(x,t|x,0)}\sim{}t^{-\sqrt{2\rho/g}} (to be compared with the usual Lifshits exponential decay exp(ρ2t)1/3\exp{-(\rho^2t)^{1/3}} in the absence of the random force field). The localisation properties of the underlying quantum Hamiltonian are discussed as well.Comment: 6 pages, LaTeX, 5 eps figure
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