Browsing by Subject "KPZ equation"

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Open Access
A continuum equation displaying Tracy-Widom distribution in the spatial and temporal fluctuations of growing interfaces
(2022-09) Liçkollari, Xhulian
A wide variety of surface growth phenomena involves random processes that result in correlated stochastic dynamics. Such dynamics is most succinctly described by a nonlinear equation known as the Kardar-Parisi-Zhang (KPZ) equation. It is of particular interest that the random fluctuations observed along a growing interface described by the KPZ equation turn out to be correlated, with statistics that match the so-called Tracy-Widom distribution. The correlated fluctuations pertain only to the space dimension, namely, along the growing interface. The fluctuations of any given point along the interface over time remain uncorrelated, thus exhibiting Gaussian fluctuations. This is to be expected since the KPZ equation and the experimental systems where the Tracy-Widom statistics have been observed lack mechanisms to induce temporal correlations. Recently, a new mechanism of dissipative self-assembly has been reported, where the self-assembly process is driven by an intrinsic feedback mechanism that is expected to induce temporal correlations. Indeed, such correlations have been experimentally observed with statistics that match the Tracy-Widom probability distribution. Here, we explore the theory of the emergence of correlated temporal fluctuations in such a system when a simplified feedback mechanism is introduced. We develop a highly simplified model, which formally constitutes a modified KPZ equation. We, then, show that this modified equation exhibits temporal fluctuations that are well described by Tracy-Widom fluctuations, up to at least the eight moment, in excellent agreement with the experimental results.
Open Access
Improvements in simulating discrete deposition models
(2021-11) Hashemi, Batoul
The main purpose of this work is to describe the surface growth phenomena. Initially, we give a brief introduction to the analytical studies, in particular, done by di erential equations such as Edward-Wilkinson and Kardar-Parisi-Zhang equations, and then introduce the discrete surface growth models as a means to describe these phenomena with a restricted number of rules. The universality properties of these models help us categorize them in three di erent classes, namely Gaussian, Edward-Wilkinson, and Kardar-Parisi-Zhang universality classes. Five di erent numerical growth models are explained, and the statistical properties of the growing interfaces in 1D systems are examined via computational studies. Growth properties of 1D structures are viewed in two di erent ways, one being the scaling exponent of roughness changing with respect to time, and the other the distribution of height uctuations. In this research, we show that our numerical simulation findings are in accord with the theoretical and analytical predictions. In addition, for the ballistic deposition and restricted solid-on-solid models, we introduce a binning method [1] which improves our numerical results. Moreover, we propose a novel modification on the ballistic deposition model by detecting the thin-deep wells on the interface and average them out which enables us to report an improvement in our numerical results with a noteworthy development on physical properties of the system. Furthermore, we investigate the e ect of the change of the deposition rate on the universal behavior of the system, as a result, we introduce critical values for the deposition rate with respect to the size of the system. With these modifications, we are one step closer to defining the complex behavior of the growth phenomena.