Comment on "Dynamic Scaling of Non-Euclidean Interfaces" [arXiv:0804.1898]

This is the revised version of a Comment on a paper by C. Escudero (Phys. Rev. Lett. 100, 116101, 2008; arXiv:0804.1898)

Phase separation in disordered exclusion models

The effect of quenched disorder in the one-dimensional asymmetric exclusion process is reviewed. Both particlewise and sitewise disorder generically induces phase separation in a range of densities. In the particlewise case the existence of stationary product measures in the homogeneous phase implies that the critical density can be computed exactly, while for sitewise disorder only bounds are available. The coarsening of phase-separated domains starting from a homogeneous initial condition is addressed using scaling arguments and extremal statistics considerations. Some of these results have been obtained previously in the context of directed polymers subject to columnar disorder.Comment: 15 pages, 4 figure

Coarsening of vortex ripples in sand

The coarsening of an array of vortex ripples prepared in an unstable state is discussed within the framework of a simple mass transfer model first introduced by K.H. Andersen et al. [Phys. Rev. E 63, 066308 (2001)]. Two scenarios for the selection of the final pattern are identified. When the initial state is homogeneous with uniform random perturbations, a unique final state is reached which depends only on the shape of the interaction function $f(\lambda)$. A potential formulation of the dynamics suggests that the final wavelength is determined by a Maxwell construction applied to $f(\lambda)$, but comparison with numerical simulations shows that this yields only an upper bound. In contrast, the evolution from a perfectly homogeneous state with a localized perturbation proceeds through the propagation of wavelength doubling fronts. The front speed can be predicted by standard marginal stability theory. In this case the final wavelength depends on the initial wavelength in a complicated manner which involves multiplication by factors of 2 and rational ratios such as 4/3.Comment: 7 pages, 4 figure

Introduction to Step Dynamics and Step Instabilities

This paper provides an elementary introduction to the basic concepts used in describing epitaxial crystal growth in terms of the thermodynamics and kinetics of atomic steps. Selected applications to morphological instabilities of stepped surfaces are reviewed, and some open problems are outlined.Comment: To appear in the Proceedings of the Oberwolfach workshop on Multiscale Modeling in Epitaxial Growt

Scaling regimes for second layer nucleation

Nucleation on top of two-dimensional islands with step edge barriers is investigated using scaling arguments. The nucleation rate is expressed in terms of three basic time scales: The time interval between deposition events, the residence time of atoms on the island, and the encounter time required for $i^\ast + 1$ atoms forming a stable nucleus to meet. Application to the problem of second-layer nucleation on growing first layer islands yields a sequence of scaling regimes with different scaling exponents relating the critical island size, at which nucleation takes place, to the diffusion and deposition rates. Second layer nucleation is fluctuation-dominated, in the sense that the typical number of atoms on the island is small compared to $i^\ast + 1$, when the first layer island density exponent $\chi$ satisfies $\chi (i^\ast + 1) < 2$. The upper critical nucleus size, above which the conventional mean-field theory of second layer nucleation is valid, increases with decreasing dimensionality. In the related case of nucleation on top of multilayer mounds fluctuation-dominated and mean-field like regimes coexist for arbitrary values of the critical nucleus size $i^\ast$.Comment: 11 pages, 3 figure

Dynamic phase transitions in electromigration-induced step bunching

Electromigration-induced step bunching in the presence of sublimation or deposition is studied theoretically in the attachment-limited regime. We predict a phase transition as a function of the relative strength of kinetic asymmetry and step drift. For weak asymmetry the number of steps between bunches grows logarithmically with bunch size, whereas for strong asymmetry at most a single step crosses between two bunches. In the latter phase the emission and absorption of steps is a collective process which sets in only above a critical bunch size and/or step interaction strength.Comment: 4 pages, 4 figure

Deterministic and stochastic regimes of asexual evolution on rugged fitness landscapes

We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length $L$. The population evolves in a maximally rugged fitness landscape with a large number of local optima. We find that whether the evolutionary trajectory is deterministic or stochastic depends on the effective mutational distance $d_{\mathrm{eff}}$ upto which the population can spread in genotype space. For $d_{\mathrm{eff}}=L$, the deterministic quasispecies theory operates while for $d_{\mathrm{eff}} < 1$, the evolution is completely stochastic. Between these two limiting cases, the dynamics are described by a local quasispecies theory below a crossover time $T_{\times}$ while above $T_{\times}$, the population gets trapped at a local fitness peak and manages to find a better peak either via stochastic tunneling or double mutations. In the stochastic regime $d_\mathrm{eff} < 1$, we identify two subregimes associated with clonal interference and uphill adaptive walks, respectively. We argue that our findings are relevant to the interepretation of evolution experiments with microbial populations.Comment: Revised version, to appear in Genetics. Note on the role of selection in defining d_eff added; new figure 4 include