Wim van Saarloos

Front propagation into unstable states

Abstract This paper is an introductory review of the problem of front propagation into unstable states. Our presentation is centered around the concept of the asymptotic linear spreading velocity v ∗ , the asymptotic rate with which initially localized perturbations spread into an unstable state according to the linear dynamical equations obtained by linearizing the fully nonlinear equations about the unstable state. This allows us to give a precise definition of pulled fronts, nonlinear fronts whose asymptotic propagation speed equals v ∗ , and pushed fronts, nonlinear fronts whose asymptotic speed v † is larger than v ∗ . In addition, this approach allows us to clarify many aspects of the front selection problem, the question whether for a given dynamical equation the front is pulled or pushed. It also is the basis for the universal expressions for the power law rate of approach of the transient velocity v(t) of a pulled front as it converges toward its asymptotic value v ∗ . Almost half of the paper is devoted to reviewing many experimental and theoretical examples of front propagation into unstable states from this unified perspective. The paper also includes short sections on the derivation of the universal power law relaxation behavior of v(t), on the absence of a moving boundary approximation for pulled fronts, on the relation between so-called global modes and front propagation, and on stochastic fronts.

The diffusion coefficient of propagating fronts with multiplicative noise

Recent studies have shown that in the presence of noise, both fronts propagating into a metastable state and so-called pushed fronts propagating into an unstable state, exhibit diffusive wandering about the average position. In this paper, we derive an expression for the effective diffusion coefficient of such fronts, which was motivated before on the basis of a multiple scale ansatz. Our systematic derivation is based on the decomposition of the fluctuating front into a suitably positioned average profile plus fluctuating eigenmodes of the stability operator. While the fluctuations of the front position in this particular decomposition are a Wiener process on all time scales, the fluctuations about the time-averaged front profile relax exponentially.

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes: pulled fronts and pushed fronts. The term “pulled front” expresses that these fronts are “pulled along” by the spreading of linear perturbations about the unstable state. Accordingly, their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state. The central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv. We show that when such fronts evolve from “sufficiently steep” initial conditions, which initially decay faster than e x forx !1 ,t hey have auniversal relaxation behavioras timet !1 : the velocity of a pulled front always relaxes algebraically like v.t/ D v 3=.2t/C 3 p D =.D 2 t/ 3=2 C O.1=t 2 /. The parametersv, , and D are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state. This front velocity is independent of the precise value of the front amplitude, which one tracks to measure the front position. The interior of the front is essentially slaved to the leading edge, and develops universally as .x;t/ D8v.t/.x Rt dt 0 v.t 0 // C O.1=t 2 /, where8v.x vt/ is a uniformly translating front solution with velocity v<v . Our result, which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well-known nonlinear diffusion equation of type @t D @ 2 x C 3 , where the invaded unstable state is D 0. Even for this simple case, the subdominant t 3=2 term extends an earlier result of Bramson. Our analysis is then generalized to more general (sets of) partial differential equations with higher spatial or temporal derivatives, to PDEs with memory kernels, and also to difference equations such as those that occur in numerical finite difference codes. Our universal result for pulled fronts thus implies independence (i) of the level curve which is used to track the front position, (ii) of the precise nonlinearities, (iii) of the precise form of the linear operators in the dynamical equation, and (iv) of the precise initial conditions, as long as they are sufficiently steep. The only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v, , and D. As our simulations confirm all our analytical predictions in every detail, it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts, if they are uniformly translating for t !1 . An immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions. In addition to our main result for pulled fronts, we also discuss the propagation and convergence of fronts emerging from

Singular or non-Fermi liquids

Abstract An introductory survey of the theoretical ideas and calculations and the experimental results which depart from Landau Fermi liquids is presented. The common themes and possible routes to the singularities leading to the breakdown of Landau Fermi liquids are categorized following an elementary discussion of the theory. Soluble examples of singular or non-Fermi liquids include models of impurities in metals with special symmetries and one-dimensional interacting fermions. A review of these is followed by a discussion of singular Fermi liquids in a wide variety of experimental situations and theoretical models. These include the effects of low-energy collective fluctuations, gauge fields due either to symmetries in the Hamiltonian or possible dynamically generated symmetries, fluctuations around quantum critical points, the normal state of high-temperature superconductors and the two-dimensional metallic state. For the last three systems, the principal experimental results are summarized and the outstanding theoretical issues are highlighted.

Large-Scale Finite-Wavelength Modulation within Turbulent Shear Flows

Investigations of counter-rotating Taylor-Couette flow (TCF) in the narrow gap limit are conducted in a very large aspect ratio apparatus. The phase diagram is presented and compared to that obtained by Andereck et al. The spiral turbulence regime is studied by varying both internal and external Reynolds numbers. Spiral turbulence is shown to emerge from the fully turbulent regime via a continuous transition appearing first as a modulated turbulent state, which eventually relaxes locally to the laminar flow. The connection with the intermittent regimes of the plane Couette flow (pCf) is discussed.

Critical scaling in linear response of frictionless granular packings near jamming.

We study the origin of the scaling behavior in frictionless granular media above the jamming transition by analyzing their linear response. The response to local forcing is non-self-averaging and fluctuates over a length scale that diverges at the jamming transition. The response to global forcing becomes increasingly nonaffine near the jamming transition. This is due to the proximity of floppy modes, the influence of which we characterize by the local linear response. We show that the local response also governs the anomalous scaling of elastic constants and contact number.

Sources, sinks and wavenumber selection in coupled CGL equations and experimental implications for counter-propagating wave systems

We study the coupled complex Ginzburg–Landau (CGL) equations for traveling wave systems, and show that sources and sinks are the important coherent structures that organize much of the dynamical properties of traveling wave systems. We focus on the regime in which sources and sinks separate patches of left and right-traveling waves, i.e., the case that these modes suppress each other. We present in detail the framework to analyze these coherent structures, and show that the theory predicts a number of general properties which can be tested directly in experiments. Our counting arguments for the multiplicities of these structures show that independently of the precise values of the coefficients in the equations, there generally exists a symmetric stationary source solution, which sends out waves with a unique frequency and wave number. Sinks, on the other hand, occur in two-parameter families, and play an essentially passive role, being sandwiched between the sources. These simple but general results imply that sources are important in organizing the dynamics of the coupled CGL equations. Simulations show that the consequences of the wavenumber selection by the sources is reminiscent of a similar selection by spirals in the 2D complex Ginzburg–Landau equations; sources can send out stable waves, convectively unstable waves, or absolutely unstable waves. We show that there exists an additional dynamical regime where both single- and bimodal states are unstable; the ensuing chaotic states have no counterpart in single amplitude equations. A third dynamical mechanism is associated with the fact that the width of the sources does not show simple scaling with the growth rate e. This is related to the fact that the standard coupled CGL equations are not uniform in e. In particular, when the group velocity term dominates over the linear growth term, no stationary source can exist; however, sources displaying nontrivial dynamics can often survive here. Our results for the existence, multiplicity, wavelength selection, dynamics and scaling of sources and sinks and the patterns they generate are easily accessible by experiments. We therefore advocate a study of the sources and sinks as a means to probe traveling wave systems and compare theory and experiment. In addition, they bring up a large number of new research issues and open problems, which are listed explicitly in the concluding section.

Elastic wave propagation in confined granular systems

We present numerical simulations of acoustic wave propagation in confined granular systems consisting of particles interacting with the three-dimensional Hertz-Mindlin force law. The response to a short mechanical excitation on one side of the system is found to be a propagating coherent wave front followed by random oscillations made of multiply scattered waves. We find that the coherent wave front is insensitive to details of the packing: force chains do not play an important role in determining this wave front. The coherent wave propagates linearly in time, and its amplitude and width depend as a power law on distance, while its velocity is roughly compatible with the predictions of macroscopic elasticity. As there is at present no theory for the broadening and decay of the coherent wave, we numerically and analytically study pulse propagation in a one-dimensional chain of identical elastic balls. The results for the broadening and decay exponents of this system differ significantly from those of the random packings. In all our simulations, the speed of the coherent wave front scales with pressure as p1/6; we compare this result with experimental data on various granular systems where deviations from the p1/6 behavior are seen. We briefly discuss the eigenmodes of the system and effects of damping are investigated as well.

Soft glassy rheology of supercooled molecular liquids

We probe the mechanical response of two supercooled liquids, glycerol and ortho-terphenyl, by conducting rheological experiments at very weak stresses. We find a complex fluid behavior suggesting the gradual emergence of an extended, delicate solid-like network in both materials in the supercooled state—i.e., above the glass transition. This network stiffens as it ages, and very early in this process it already extends over macroscopic distances, conferring all well known features of soft glassy rheology (yield-stress, shear thinning, aging) to the supercooled liquids. Such viscoelastic behavior of supercooled molecular glass formers is difficult to observe because the large stresses in conventional rheology can easily shear-melt the solid-like structure. The work presented here, combined with evidence for long-lived heterogeneity from previous single-molecule studies [Zondervan R, Kulzer F, Berkhout GCG, Orrit M (2007) Local viscosity of supercooled glycerol near Tg probed by rotational diffusion of ensembles and single dye molecules. Proc Natl Acad Sci USA 104:12628–12633], has a profound impact on the understanding of the glass transition because it casts doubt on the widely accepted assumption of the preservation of ergodicity in the supercooled state.

Critical and noncritical jamming of frictional grains

We probe the nature of the jamming transition of frictional granular media by studying their vibrational properties as a function of the applied pressure p and friction coefficient mu. The density of vibrational states exhibits a crossover from a plateau at frequencies omega > or similar to omega*(p,mu) to a linear growth for omega < or similar to omega*(p,mu). We show that omega* is proportional to Deltaz, the excess number of contacts per grain relative to the minimally allowed, isostatic value. For zero and infinitely large friction, typical packings at the jamming threshold have Deltaz-->0, and then exhibit critical scaling. We study the nature of the soft modes in these two limits, and find that the ratio of elastic moduli is governed by the distance from isostaticity.