693 research outputs found
On the spectrum of the Laplace operator of metric graphs attached at a vertex -- Spectral determinant approach
We consider a metric graph made of two graphs
and attached at one point. We derive a formula relating the
spectral determinant of the Laplace operator
in terms of the spectral
determinants of the two subgraphs. The result is generalized to describe the
attachment of graphs. The formulae are also valid for the spectral
determinant of the Schr\"odinger operator .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 is the strength of the random force field and 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
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
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 perturbations, converging
time-asymptotically to a translate of the unperturbed wave. That is, for a
shock with unstable eigenvalues, we establish conditional stability on a
codimension- manifold of initial data, with sharp rates of decay in all
. For , 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
We investigate weak localization in metallic networks etched in a two
dimensional electron gas between mK and 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 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
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
A one-dimensional model of classical diffusion in a random force field with a
weak concentration 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 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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