Showing papers by "David A. Kessler published in 1998"

Front propagation: Precursors, cutoffs, and structural stability

Abstract: We discuss the problem of fronts propagating into metastable and unstable states. We examine the time development of the leading edge, discovering a precursor which in the metastable case propagates out ahead of the front at a velocity more than double that of the front and establishes the characteristic exponential behavior of the steady-state leading edge. We also study the dependence of the velocity on the imposition of a cutoff in the reaction term. These studies shed light on the problem of velocity selection in the case of propagation into an unstable state. We also examine how discreteness in a particle simulation acts as an effective cutoff in this case.

Fluctuation-induced diffusive instabilities

Abstract: The formation of complex patterns in many non-equilibrium systems, ranging from solidifying alloys to multiphase flow1, nonlinear chemical reactions2 and the growth of bacterial colonies3,4, involves the propagation of an interface that is unstable to diffusive motion. Most existing theoretical treatments of diffusive instabilities are based on mean-field approaches, such as the use of reaction–diffusion equations, that neglect the role of fluctuations. Here we show that finite fluctuations in particle number can be essential for such an instability to occur. We study, both analytically and with computer simulations, the planar interface separating different species in the simple two-component reaction A+ B → 2A (which can also serve as a simple model of bacterial growth in the presence of a nutrient). The interface displays markedly different dynamics within the reaction–diffusion treatment from that when fluctuations are taken into account. Our findings suggest that fluctuations can provide a new and general pattern-forming mechanism in non-equilibrium growth.

Mutator Dynamics on a Smooth Evolutionary Landscape

Abstract: We investigate a model of evolutionary dynamics on a smooth landscape which features a ``mutator'' allele which increases the mutation rate. We show that when the fitness is far from its equilibrium value the expected proportion of mutators approaches a value governed solely by the transition rates into and out of the mutator state, resulting in a much faster fitness increase than would be the case without the mutator allele. Near the fitness equilibrium, the mutators are severely suppressed, due to the detrimental effects of a large mutation rate near the fitness maximum. We discuss the results of a recent experiment on natural selection of E. coli in the light of our model.

Evolution on a Smooth Landscape: The Role of Bias

Abstract: The role of mutational bias in evolution on a smooth landscape is investigated. We consider both a finite-length genome where the bias increases linearly with the fitness, and an infinite genome with a fixed bias. We present simulations of finite populations in a waiting time model, showing both the nonequilibrium dynamics and the equilibrium fitness distributions that are reached. We compute the equilibrium analytically in several cases, using approximate direct solution of the master equations and truncated hierarchies.

Diffusion-limited aggregation and viscous fingering in a wedge: Evidence for a critical angle

Abstract: We show that both analytic and numerical evidence points to the existence of a critical angle of h'60° –70° in viscous fingers and diffusion-limited aggregates growing in a wedge. The significance of this angle is that it is the typical angular spread of a major finger. For wedges with an angle larger than 2 h, two fingers can coexist. Thus a finger with this angular spread is a kind of building block for viscous fingering patterns and diffusion-limited aggregation clusters in radial geometry. @S1063-651X~98!10506-8#

Level-crossing densities in random wave fields

Abstract: Level-crossing densities along two orthogonal directions in an isotropic two-dimensional Gaussian random wave field are discussed for the real and the imaginary parts of the wave function, for the intensity, for the phase, and for all the first- and second-order spatial derivatives of these functions. Analytical expressions are given for most crossing densities and are supplemented by numerical densities obtained from multidimensional Monte Carlo evaluations in cases in which analysis proved intractable. The analytical results and the Monte Carlo evaluations are generally found to be in good agreement with densities derived from a computer simulation that yields an accurate numerical representation of the wave function.

Noisy time series generation by feed-forward networks

Abstract: We study the properties of a noisy time series generated by a continuous-valued feed-forward network in which the next input vector is determined from past output values. Numerical simulations of a perceptron-type network exhibit the expected broadening of the noise-free attractor, without changing the attractor dimension. We show that the broadening of the attractor due to the noise scales inversely with the size of the system ,N, as . We show both analytically and numerically that the diffusion constant for the phase along the attractor scales inversely with N. Hence, phase coherence holds up to a time that scales linearly with the size of the system. We find that the mean first passage time, t, to switch between attractors depends on N, and the reduced distance from bifurcation as , where b is a constant which depends on the amplitude of the external noise. This result is obtained analytically for small and is confirmed by numerical simulations.

Distributions of triplets in genetic sequences

Abstract: Distributions of triplets in some genetic sequences are examined and found to be well described by a two-parameter Markov process with a sparse transition matrix. The standard deviations of all the relevant parameters are not large, indicating that most sequences gather in a small region in the parameter space. Different sequences have very similar values of the entropy calculated directly from the data and the two parameters characterizing the Markov process fitting the sequence. No relevance with taxonomy or coding/noncoding is clearly observed.

Universal gaussian falloff in soliton tails

Abstract: We show that in a large class of equations, solitons formed from generic initial conditions do not have infinitely long exponential tails, but are truncated by a region of Gaussian decay. This phenomenon makes it possible to treat solitons as localized, individual objects. For the case of the Korteweg‐de Vries equation, we show how the Gaussian decay emerges in the inverse scattering formalism. @S1063-651X~98!02212-0# PACS number~s!: 03.40.Kf, 02.30.Jr, 47.35.1i, 47.54.1r Recently, progress has been made understanding the time development of the leading exponential edge in propagating fronts @1#. Inspired by results on fronts in the Fisher equation @2,3#, it was shown in Ref. @1# that in generic reactiondiffusion equation such as the Ginzburg-Landau ~GL! equation, an initial condition with a compact front gives rise to a front with a leading exponential edge that does not extend forever, but rather is cut off a finite distance ahead of the front. The need for this is clear on physical grounds: the front, if one is sufficiently far ahead of it, has not had time to make its presence felt, and so the field exhibits the typical Gaussian falloff of the Green’s function ~which lacks an intrinsic scale!. It was discovered that there is a well-defined transition region of width O(At) wherein the field crosses over from the steady-state exponential to Gaussian falloff. This ‘‘precursor’’ transition region propagates out ahead of the front, with a velocity c* which is greater than twice the velocity c of the front itself. Given the basic underlying physics, the existence of such transition regions must be a very general phenomenon, true not just of reaction-diffusion fronts, but of other propagating solutions, such as the solitons in the Korteweg‐de Vries ~KdV! equation ut52uxxx16uux . ~1! The known exact one-soliton solutions take the form u(x,t)52 1 csech 2 @ 1 Ac(x2x 02ct)#, and exhibit exponential decay for large x. The inverse scattering transform @4# tells us that the solution of the KdV equation with generic initial condition u(x,0)5f(x) @with f(x)!0 sufficiently rapidly as uxu!‘# consists of a train of solitons moving to the right, along with a dispersive wave traveling to the left. As above, we can argue that when f(x) has compact support, the solution emerges from the rightmost soliton decaying as exp(2Acx), where c is the relevant speed, but for sufficiently largex the presence of the solitons will not yet be felt, and the behavior of the solution will be determined by the Green’s function of the linearized equation ut52uxxx , i.e., u will decay roughly as exp(22x 3/2 /3A3t) @5#. Thus there is a transition in the nature of the decay. How and where does this transition take place? If there is more than one soliton in the soliton train, say two, with speeds c 1 ,c 2 (c 2.c 1.0), then a further problem arises. The solution emerges from the faster-moving soliton decaying as exp(2Ac 2x), but because the tail of the slowermoving soliton falls slower, it is possible that it will return to dominate, i.e., the decay will slow to exp(2Ac 1x). @For very large x, as explained above, the solution must go roughly as exp(22x 3/2 /3A3t).# We note that exact two-soliton solutions

Transparent diffusion-limited aggregation in one dimension

Abstract: We introduce a transparent version of diffusion-limited aggregation (DLA) wherein the walkers stick to the aggregate with probability μ less than unity and are allowed to penetrate the aggregate. We study this model in one spatial dimension. We show that the ensemble average of this process is in the steady state only when one takes the average in the instuntaneous frame of the lead particle. We calculate this average exactly for μ near one. For small sticking probability, we introduce a mean-field treatment. As opposed to the standard mean-field treatments of DLA, our version, based on our novel ensemble average, shows no singular behaviour and no need for ad-hoc cut-off procedures. This mean-field treatment in quantitatively accurate as long as μ is not too small and qualitatively correct for all μ. We discuss the quantitative breakdown of the mean-field treatment for very small μ and directions towards improvement. The implications of these findings and future directions for research are also ...

Wrinkling of stable fronts in viscous flow

Abstract: Stable wrinkled fronts were often observed in experiments on fluid invasion into porous media, when the invaded phase is less viscous than the invading one. Recent experiments by a group at the Weizmann institute, show that capillary wetting of paper may also lead to rough self-affine fronts. A dominant factor controlling the interface roughening in these systems is the nature of the two-dimensional viscous fluid flow in the medium behind it. Since the disorder in the medium modifies this flow, it also determines the roughness of the interface. We present a theoretical analysis of the connection between the nature of randomness in the medium and the resulting roughness of the fronts.