arXiv subjects

J. Krug

Publications and source records attributed to J. Krug.

17 recordsLinked to original sources

arXiv2012

Rounding Effects in Record Statistics

We analyze record-breaking events in time series of continuous random variables that are subsequently discretized by rounding down to integer multiples of a discretization scale $Δ>0$. Rounding leads to ties of an existing record, thereby reducing the number of new records. For an infinite number of random variables that are drawn from distributions with a finite upper limit, the number of discrete records is finite, while for distributions with a thinner than exponential upper tail, fewer discrete records arise compared to continuous variables. In the latter case the record sequence becomes highly regular at long times.

physics.data-an↗

arXiv2012

Correlations of record events as a test for heavy-tailed distributions

A record is an entry in a time series that is larger or smaller than all previous entries. If the time series consists of independent, identically distributed random variables with a superimposed linear trend, record events are positively (negatively) correlated when the tail of the distribution is heavier (lighter) than exponential. Here we use these correlations to detect heavy-tailed behavior in small sets of independent random variables. The method consists of converting random subsets of the data into time series with a tunable linear drift and computing the resulting record correlations.

physics.data-an↗

arXiv2011

Diffusion-limited reactions on disordered surfaces with continuous distributions of binding energies

We study the steady state of a stochastic particle system on a two-dimensional lattice, with particle influx, diffusion and desorption, and the formation of a dimer when particles meet. Surface processes are thermally activated, with (quenched) binding energies drawn from a \emph{continuous} distribution. We show that sites in this model provide either coverage or mobility, depending on their energy. We use this to analytically map the system to an effective \emph{binary} model in a temperature-dependent way. The behavior of the effective model is well-understood and accurately describes key quantities of the system: Compared with discrete distributions, the temperature window of efficient reaction is broadened, and the efficiency decays more slowly at its ends. The mapping also explains in what parameter regimes the system exhibits realization dependence.

cond-mat.stat-mech↗

arXiv2010

Diffusion-limited reactions on a two-dimensional lattice with binary disorder

Reaction-diffusion systems where transition rates exhibit quenched disorder are common in physical and chemical systems. We study pair reactions on a periodic two-dimensional lattice, including continuous deposition and spontaneous desorption of particles. Hopping and desorption are taken to be thermally activated processes. The activation energies are drawn from a binary distribution of well depths, corresponding to `shallow' and `deep' sites. This is the simplest non-trivial distribution, which we use to examine and explain fundamental features of the system. We simulate the system using kinetic Monte Carlo methods and provide a thorough understanding of our findings. We show that the combination of shallow and deep sites broadens the temperature window in which the reaction is efficient, compared to either homogeneous system. We also examine the role of spatial correlations, including systems where one type of site is arranged in a cluster or a sublattice. Finally, we show that a simple rate equation model reproduces simulation results with very good accuracy.

cond-mat.dis-nn↗

arXiv2007

Spiral Growth and Step Edge Barriers

The growth of spiral mounds containing a screw dislocation is compared to the growth of wedding cakes by two-dimensional nucleation. Using phase field simulations and homoepitaxial growth experiments on the Pt(111) surface we show that both structures attain the same characteristic large scale shape when a significant step edge barrier suppresses interlayer transport. The higher vertical growth rate observed for the spiral mounds on Pt(111) reflects the different incorporation mechanisms for atoms in the top region and can be formally represented by an enhanced apparent step edge barrier.

cond-mat.mtrl-sci↗

arXiv2005

Shape and scaling of moving step bunches

We study step bunching under conditions of attachment/detachment limited kinetics in the presence of a deposition or sublimation flux, which leads to bunch motion. Analysis of the discrete step dynamics reveals that the bunch velocity is inversely proportional to the bunch size for general step-step interactions. The shape of steadily moving bunches is studied within a continuum theory, and analytic expressions for the bunch profile are derived. Scaling laws obtained previously for non-moving bunches are recovered asymptotically, but singularities of the static theory are removed and strong corrections to scaling are found. The size of the largest terrace between two bunches is identified as a central scaling parameter. Our theory applies to a large class of bunching instabilities, including sublimation with attachment asymmetry and surface electromigration in the presence of sublimation or growth.

cond-mat.stat-mech↗

arXiv2004

Breakdown of step-flow growth in unstable homoepitaxy

Two mechanisms for the breakdown of step flow growth, in the sense of the appearance of steps of opposite sign to the original vicinality, are studied by kinetic Monte Carlo simulations and scaling arguments. The first mechanism is the nucleation of islands on the terraces, which leads to mound formation if interlayer transport is sufficiently inhibited. The second mechanism is the formation of vacancy islands due to the self-crossing of strongly meandering steps. The competing roles of the growth of the meander amplitude and the synchronization of the meander phase are emphasized. The distance between vacancy islands along the step direction appears to be proportional to the square of the meander wavelength

cond-mat.stat-mech↗

arXiv2002

Step bunching during Si(001) homoepitaxy caused by the surface diffusion anisotropy

Scanning tunneling microscopy experiments show that the unstable growth morphology observed during molecular beam homoepitaxy on slightly vicinal Si(001) surfaces consists of straight step bunches. The instability occurs under step- flow growth conditions and vanishes both during low-temperature island growth and at high temperatures. An instability with the same characteristics is observed in a 2D Kinetic Monte Carlo model of growth with incorporated Si(001)- like diffusion anisotropy. This provides strong evidence that the diffusion anisotropy destabilizes growth on Si(001) and similar surfaces towards step bunching. This new instability mechanism is operational without any additional step edge barriers.

cond-mat.mtrl-sci↗

arXiv2002

Coarsening of Sand Ripples in Mass Transfer Models with Extinction

Coarsening of sand ripples is studied in a one-dimensional stochastic model, where neighboring ripples exchange mass with algebraic rates, $Γ(m) \sim m^γ$, and ripples of zero mass are removed from the system. For $γ< 0$ ripples vanish through rare fluctuations and the average ripples mass grows as $\avem(t) \sim -γ^{-1} \ln (t)$. Temporal correlations decay as $t^{-1/2}$ or $t^{-2/3}$ depending on the symmetry of the mass transfer, and asymptotically the system is characterized by a product measure. The stationary ripple mass distribution is obtained exactly. For $γ> 0$ ripple evolution is linearly unstable, and the noise in the dynamics is irrelevant. For $γ= 1$ the problem is solved on the mean field level, but the mean-field theory does not adequately describe the full behavior of the coarsening. In particular, it fails to account for the numerically observed universality with respect to the initial ripple size distribution. The results are not restricted to sand ripple evolution since the model can be mapped to zero range processes, urn models, exclusion processes, and cluster-cluster aggregation.

cond-mat.stat-mech↗

arXiv2002

Pattern Dynamics of Vortex Ripples in Sand: Nonlinear Modeling and Experimental Validation

Vortex ripples in sand are studied experimentally in a one-dimensional setup with periodic boundary conditions. The nonlinear evolution, far from the onset of instability, is analyzed in the framework of a simple model developed for homogeneous patterns. The interaction function describing the mass transport between neighboring ripples is extracted from experimental runs using a recently proposed method for data analysis, and the predictions of the model are compared to the experiment. An analytic explanation of the wavelength selection mechanism in the model is provided, and the width of the stable band of ripples is measured.

cond-mat.stat-mech↗

arXiv2001

Second layer nucleation and the shape of wedding cakes

The rate of second layer nucleation -- the formation of a stable nucleus on top of a two-dimensional island -- determines both the conditions for layer-by-layer growth, and the size of the top terrace of multilayer mounds in three-dimensional homoepitaxial growth. It was recently shown that conventional mean field nucleation theory overestimates the rate of second layer nucleation by a factor that is proportional to the number of times a given site is visited by an adatom during its residence time on the island. In the presence of strong step edge barriers this factor can be large, leading to a substantial error in previous attempts to experimentally determine barrier energies from the onset of second layer nucleation. In the first part of the paper simple analytic estimates of second layer nucleation rates based on a comparison of the relevant time scales will be reviewed. In the main part the theory of second layer nucleation is applied to the growth of multilayer mounds in the presence of strong but finite step edge barriers. The shape of the mounds is obtained by numerical integration of the deterministic evolution of island boundaries, supplemented by a rule for nucleation in the top layer. For thick films the shape converges to a simple scaling solution. The scaling function is parametrized by the coverage $θ_c$ of the top layer, and takes the form of an inverse error function cut off at $θ_c$. The surface width of a film of thickness $d$ is $\sqrt{(1- θ_c) d}$. Finally, we show that the scaling solution can be derived also from a continuum growth equation.

cond-mat.mtrl-sci↗

arXiv2000

Asymptotic step profiles from a nonlinear growth equation for vicinal surfaces

We study a recently proposed nonlinear evolution equation describing the collective step meander on a vicinal surface subject to the Bales-Zangwill growth instability [O. Pierre-Louis et al., Phys. Rev. Lett. (80), 4221 (1998)]. A careful numerical analysis shows that the dynamically selected step profile consists of sloped segments, given by an inverse error function and steepening as sqrt(t), which are matched to pieces of a stationary (time-independent) solution describing the maxima and minima. The effect of smoothening by step edge diffusion is included heuristically, and a one-parameter family of evolution equations is introduced which contains relaxation by step edge diffusion and by attachment-detachment as special cases. The question of the persistence of an initially imposed meander wavelength is investigated in relation to recent experiments.

cond-mat.mtrl-sci↗

arXiv1999

Asymmetric particle systems on R

We study interacting particle systems on the real line which generalize the Hammersley process [D. Aldous and P. Diaconis, Prob. Theory Relat. Fields 103, 199-213 (1995)]. Particles jump to the right to a randomly chosen point between their previous position and that of the forward neighbor, at a rate which may depend on the distance to the neighbor. A class of models is identified for which the invariant particle distribution is Poisson. The bulk of the paper is devoted to a model where the jump rate is constant and the jump length is a random fraction $r$ of the distance to the forward neighbor, drawn from a probability density $ϕ(r)$ on the unit interval. This is a special case of the random average process of Ferrari and Fontes [El. J. Prob. 3, Paper no. 6 (1998)]. The discrete time version of the model has been considered previously in the context of force propagation in granular media [S.N. Coppersmith et al., Phys. Rev. E 53, 4673 (1996)]. We show that the stationary two-point function of particle spacings factorizes for any choice of $ϕ(r)$. Under the assumption that this implies pairwise independence, the invariant density of interparticle spacings for the case of uniform $ϕ(r)$ is found to be a gamma distribution with parameter $ν$, where $ν= 1/2$, 1 and 2 for continuous time, backward sequential and discrete time dynamics, respectively. A heuristic derivation of a nonlinear diffusion equation is presented, and the tracer diffusion coefficient is computed for arbitrary $ϕ(r)$ and different types of dynamics.

cond-mat.stat-mech↗

arXiv1999

Damping of Growth Oscillations

Computer simulations and scaling theory are used to investigate the damping of oscillations during epitaxial growth on high-symmetry surfaces. The crossover from smooth to rough growth takes place after the deposition of (D/F)^δmonolayers, where D and F are the surface diffusion constant and the deposition rate, respectively, and the exponent δ=2/3 on a two-dimensional surface. At the transition, layer-by-layer growth becomes desynchronized on distances larger than a layer coherence length proportional l^2, where l is a typical distance between two-dimensional islands in the submonolayer region of growth.

cond-mat.stat-mech↗

arXiv1997

Damping of Oscillations in Layer-by-Layer Growth

We present a theory for the damping of layer-by-layer growth oscillations in molecular beam epitaxy. The surface becomes rough on distances larger than a layer coherence length which is substantially larger than the diffusion length. The damping time can be calculated by a comparison of the competing roughening and smoothening mechanisms. The dependence on the growth conditions, temperature and deposition rate, is characterized by a power law. The theoretical results are confirmed by computer simulations.

cond-mat.stat-mech↗

arXiv1997

Persistence exponents for fluctuating interfaces

Numerical and analytic results for the exponent θdescribing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear Langevin equations. The models are parametrized by the dynamic roughness exponent β, with 0 < β< 1; for β= 1/2 the time evolution is Markovian. Using simulations of solid-on-solid models, of the discretized continuum equations as well as of the associated zero-dimensional stationary Gaussian process, we address two problems: The return of an initially flat interface, and the return to an initial state with fully developed steady state roughness. The two problems are shown to be governed by different exponents. For the steady state case we point out the equivalence to fractional Brownian motion, which has a return exponent θ_S = 1 - β. The exponent θ_0 for the flat initial condition appears to be nontrivial. We prove that θ_0 \to \infty for β\to 0, θ_0 \geq θ_S for β< 1/2 and θ_0 \leq θ_S for β> 1/2, and calculate θ_{0,S} perturbatively to first order in an expansion around the Markovian case β= 1/2. Using the exact result θ_S = 1 - β, accurate upper and lower bounds on θ_0 can be derived which show, in particular, that θ_0 \geq (1 - β)^2/βfor small β.

cond-mat.stat-mech↗

arXiv1996

Kinetic roughening of surfaces: Derivation, solution and application of linear growth equations

We present a comprehensive analysis of a linear growth model, which combines the characteristic features of the Edwards--Wilkinson and noisy Mullins equations. This model can be derived from microscopics and it describes the relaxation and growth of surfaces under conditions where the nonlinearities can be neglected. We calculate in detail the surface width and various correlation functions characterizing the model. In particular, we study the crossover scaling of these functions between the two limits described by the combined equation. Also, we study the effect of colored and conserved noise on the growth exponents, and the effect of different initial conditions. The contribution of a rough substrate to the surface width is shown to decay universally as $w_i(0) (ξ_s/ξ(t))^{d/2}$, where $ξ(t) \sim t^{1/z}$ is the time--dependent correlation length associated with the growth process, $w_i(0)$ is the initial roughness and $ξ_s$ the correlation length of the substrate roughness, and $d$ is the surface dimensionality. As a second application, we compute the large distance asymptotics of the height correlation function and show that it differs qualitatively from the functional forms commonly used in the intepretation of scattering experiments.

cond-mat↗