Recent developments on the Kardar-Parisi-Zhang surface-growth equation

Synthetic Data in Quantitative Scanning Probe Microscopy

Synthetic data are of increasing importance in nanometrology. They can be used for development of data processing methods, analysis of uncertainties and estimation of various measurement artefacts. In this paper we review methods used for their generation and the applications of synthetic data in scanning probe microscopy, focusing on their principles, performance, and applicability. We illustrate the benefits of using synthetic data on different tasks related to development of better scanning approaches and related to estimation of reliability of data processing methods. We demonstrate how the synthetic data can be used to analyse systematic errors that are common to scanning probe microscopy methods, either related to the measurement principle or to the typical data processing paths.

Kardar-Parisi-Zhang equation with short-range correlated noise: Emergent symmetries and nonuniversal observables

We investigate the stationary-state fluctuations of a growing one-dimensional interface described by the Kardar-Parisi-Zhang (KPZ) dynamics with a noise featuring smooth spatial correlations of characteristic range ξ. We employ nonperturbative functional renormalization group methods to resolve the properties of the system at all scales. We show that the physics of the standard (uncorrelated) KPZ equation emerges on large scales independently of ξ. Moreover, the renormalization group flow is followed from the initial condition to the fixed point, that is, from the microscopic dynamics to the large-distance properties. This provides access to the small-scale features (and their dependence on the details of the noise correlations) as well as to the universal large-scale physics. In particular, we compute the kinetic energy spectrum of the stationary state as well as its nonuniversal amplitude. The latter is experimentally accessible by measurements at large scales and retains a signature of the microscopic noise correlations. Our results are compared to previous analytical and numerical results from independent approaches. They are in agreement with direct numerical simulations for the kinetic energy spectrum as well as with the prediction, obtained with the replica trick by Gaussian variational method, of a crossover in ξ of the nonuniversal amplitude of this spectrum.

Influence of individual cell motility on the 2D front roughness dynamics of tumour cell colonies

The dynamics of in situ 2D HeLa cell quasi-linear and quasi-radial colony fronts in a standard culture medium is investigated. For quasi-radial colonies, as the cell population increased, a kinetic transition from an exponential to a constant front average velocity regime was observed. Special attention was paid to individual cell motility evolution under constant average colony front velocity looking for its impact on the dynamics of the 2D colony front roughness. From the directionalities and velocity components of cell trajectories in colonies with different cell populations, the influence of both local cell density and cell crowding effects on individual cell motility was determined. The average dynamic behaviour of individual cells in the colony and its dependence on both local spatio-temporal heterogeneities and growth geometry suggested that cell motion undergoes under a concerted cell migration mechanism, in which both a limiting random walk-like and a limiting ballistic-like contribution were involved. These results were interesting to infer how biased cell trajectories influenced both the 2D colony spreading dynamics and the front roughness characteristics by local biased contributions to individual cell motion. These data are consistent with previous experimental and theoretical cell colony spreading data and provide additional evidence of the validity of the Kardar-Parisi-Zhang equation, within a certain range of time and colony front size, for describing the dynamics of 2D colony front roughness.

Statistics of interfacial fluctuations of radially growing clusters

The dynamics of fluctuating radially growing interfaces is approached using the formalism of stochastic growth equations on growing domains. This framework reveals a number of dynamic features arising during surface growth. For fast growth, dilution, which spatially reorders the incoming matter, is responsible for the transmission of correlations. Its effects include the erasing of memory with respect to the initial condition, a partial attenuation of geometrically originated instabilities, and the restoration of universality in some special cases in which the critical exponents depend on the parameters of the equation of motion. In this sense, dilution rends the dynamics more similar to the usual one of planar systems. This fast growth regime is also characterized by the spatial decorrelation of the interface, which, in the case of radially growing interfaces, naturally originates rapid roughening and scale-dependent fractality, and suggests the advent of a self-similar fractal dimension. The center-of-mass fluctuations of growing clusters are also studied, and our analysis suggests the possible nonapplicability of usual scalings to the long-range surface fluctuations of the radial Eden model. In fact, our study points to the fact that this model belongs to a dilution-free universality class.

Nonlinear dynamics in meso and nano scales: fundamental aspects and applications

This introduction to the special issue, Nonlinear dynamics in meso and nano scales: fundamental aspects and applications, gives a short overview about different contexts and current challenges posed by the emergence of nonlinearities at meso and nano characteristic sizes. It also addresses different aspects related to classical and quantum chaos. Moreover, it comments on the articles in this thematic publication, briefly summarizing their relevance in helping to understand the uprise of chaos and complex behaviour at those small scales.