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Continuous-time random walks and travelling fronts
No full textWe present a geometric approach to the problem of propagating fronts into an unstable state, valid for an arbitrary continuous-time random walk with a Fisher–Kolmogorov-Petrovski-Piskunov growth/reaction rate. We derive an integral Hamilton-Jacobi type equation for the action functional determining the position of reaction front and its speed. Our method does not rely on the explicit derivation of a differential equation for the density of particles. In particular, we obtain an explicit formula for the propagation speed for the case of anomalous transport involving non-Markovian random processes
Non-markovian random processes and travelling fronts in a reaction-transport system with memory and long-range interactions
No full textThe problem of finding the propagation rate for traveling waves in reaction-transport systems with memory and long-range interactions has been considered. Our approach makes use of the generalized master equation with logistic growth, hyperbolic scaling, and Hamilton-Jacobi theory. We consider the case when the waiting-time distribution for the underlying microscopic random walk is modeled by the family of gamma distributions, which in turn leads to non-Markovian random processes and corresponding memory effects on mesoscopic scales. We derive formulas that enable us to determine the front propagation rate and understand how the memory and long-range interactions influence the propagation rate for traveling fronts. Several examples involving the Gaussian and discrete distributions for jump densities are presented
Oscillatory flame edge propagation, isolated flame tubes and stability in a non-premixed counterflow
No full textAn investigation is carried out into the ranges of Damk¨ohler number and Lewis
number, less than unity, in which different forms of combustion phenomena
arise within a non-premixed counterflow; cases that are symmetrical across the
counterflow are chosen for study. These link oscillatory and steady propagation
of flame edges, zero propagation speeds, isolated flame tubes, quenching and
marginal stability of planar diffusion flames. Over a range of Lewis numbers,
the transition between steady and oscillatory propagation of a flame edge has
characteristics that are consistent with a subcritical Hopf bifurcation. Isolated
flame tubes are found to persist up to a Lewis number that is very close to unity
An arithmetic group associated with a pisot unit, and its symbolic-dynamical representation
No full textRegularity of invariant graphs over hyperbolic systems
No full textWe consider cocycles with negative Lyapunov exponents defined over a hyperbolic dynamical system. It is well known that such systems possess invariant graphs and that under spectral assumptions these graphs have some degree of Hölder regularity. When the invariant graph has a slightly higher Hölder exponent than the a priori lower bound on an open set (even on just a set of positive measure for certain systems), we show that the graph must be Lipschitz or (in the Anosov case) as smooth as the cocycle
The steady propagation of a semi-infinite bubble into a tube of elliptical or rectangular cross-section
No full textThis paper investigates the propagation of an air finger into a fluid-filled, axially uniform tube of elliptical or rectangular cross-section with transverse length scale a and aspect ratio [alpha]. Gravity is assumed to act parallel to the tube's axis. The problem is studied numerically by a finite-element-based direct solution of the free-surface Stokes equations.
In rectangular tubes, our results for the pressure drop across the bubble tip, [Delta]p, are in good agreement with the asymptotic predictions of Wong et al. (1995b) at low values of the capillary number, Ca (ratio of viscous to surface-tension forces). At larger Ca, Wong et al.'s (1995b) predictions are found to underestimate [Delta]p. In both elliptical and rectangular tubes, the ratio [Delta]p([alpha])/[Delta]p([alpha] = 1) is approximately independent of Ca and thus equal to the ratio of the static meniscus curvatures.
In non-axisymmetric tubes, the air-liquid interface develops a noticeable asymmetry near the bubble tip at all values of the capillary number. The tip asymmetry decays with increasing distance from the bubble tip, but the decay rate becomes very small as Ca increases. For example, in a rectangular tube with [alpha] = 1.5, when Ca = 10, the maximum and minimum finger radii still differ by more than 10% at a distance 100a behind the finger tip. At large Ca the air finger ultimately becomes axisymmetric with radius r[infty infinity]. In this regime, we find that r[infty infinity] in elliptical and rectangular tubes is related to r[infty infinity] in circular and square tubes, respectively, by a simple, empirical scaling law. The scaling has the physical interpretation that for rectangular and elliptical tubes of a given cross-sectional area, the propagation speed of an air finger, which is driven by the injection of air at a constant volumetric rate, is independent of the tube's aspect ratio.
For smaller Ca (Ca [alpha]^ dry spots will develop on the tube walls at all values of Ca
Estimating Gramians of large-scale time-varying systems.
Get PDFIn this paper we present a Smith-like updating technique to estimate a low rank approximation of the Gramians of a time-varying system. We obtain error estimates of our approximation and also explain how to use this for model reduction of time-varying systems
Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems
No full textAn important class of generalized eigenvalue problems Ax=λBx is those in which A and B are Hermitian and some real linear combination of them is definite. For the quadratic eigenvalue problem (QEP) with Hermitian A, B and C and positive definite A, particular interest focuses on problems in which (x*Bx)2−4(x*Ax)(x*Cx) is one-signed for all non-zero x—for the positive sign these problems are called hyperbolic and for the negative sign elliptic. The important class of overdamped problems arising in mechanics is a sub-class of the hyperbolic problems. For each of these classes of generalized and quadratic eigenvalue problems we show how to check that a putative member has the required properties and we derive the distance to the nearest problem outside the class. For definite pairs (A,B) the distance is the Crawford number, and we derive bisection and level set algorithms both for testing its positivity and for computing it. Testing hyperbolicity of a QEP is shown to reduce to testing a related pair for definiteness. The distance to the nearest non-hyperbolic or non-elliptic n×n QEP is shown to be the solution of a global minimization problem with n−1 degrees of freedom. Numerical results are given to illustrate the theory and algorithms
Interrupted coarsening in a driven kinetically constrained ising chain
No full textWe introduce a driven version of the one-dimensional kinetically constrained spin chain [J. Jackle and S. Eisinger, Zeitschr. Phys. B 84, 115 (1991)]. In its original undriven version, this model shows anomalous coarsening following a quench to a low temperature, with an equilibration time that diverges as ∼exp(1/T2) for T→0. We show that driving of constant rate γ̇ interrupts coarsening and stabilizes the chain in a state analogous to that of a coarsening chain of age 1/γ̇. We present an analytical theory for this steady state, and demonstrate it to be in excellent agreement with our simulation results
Fluctuation-dissipation relations and effective temperatures in simple non-mean field systems
No full textWe give a brief review of violations of the fluctuation-dissipation theorem (FDT) in out-of-equilibrium systems; in mean field scenarios the corresponding fluctuation-dissipation (FD) plots can, in the limit of long times, be used to define a effective temperature Teff that shares many properties of the true thermodynamic temperature T. We discuss carefully how correlation and response functions need to be represented to obtain meaningful limiting FD plots in non-mean field systems. A minimum requirement on the resulting effective temperatures is that they should be independent of the observable whose correlator and response are being considered; we show for two simple models with glassy dynamics (Bouchaud's trap model and the Glauber-Ising chain at zero temperature) that this is generically not the case. Consequences for the wider applicability of effective temperatures derived from FD relations are discussed; one intriguing possibility is that at least the limit of the FDT violation factor for well separated times may generically be observable-independent and so could yield a meaningful Teff