arXiv subjects

M. Grinfeld

Publications and source records attributed to M. Grinfeld.

7 recordsLinked to original sources

arXiv2021

Patterns formed in a thin film with spatially homogeneous and non-homogeneous Derjaguin disjoining pressure

We consider patterns formed in a two-dimensional thin film on a planar substrate with a Derjaguin disjoining pressure and periodic wettability stripes. We rigorously clarify some of the results obtained numerically by Honisch et al. and embed them in the general theory of thin-film equations. For the case of constant wettability, we elucidate the change in the global structure of branches of steady state solutions as the average film thickness and the surface tension are varied. Specifically we find, by using methods of local bifurcation theory and the continuation software package AUTO, both nucleation and metastable regimes. We discuss admissible forms of spatially non-homogeneous disjoining pressure, arguing for a form that differs from the one used by Honisch et al. , and study the dependence of the steady state solutions on the wettability contrast in that case.

nlin.PS↗

arXiv2018

Point island dynamics under fixed rate deposition

In this paper we consider the dynamics of point islands during submonolayer deposition, in which the fragmentation of subcritical size islands is allowed. To understand asymptotics of solutions, we use methods of centre manifold theory, and for globalisation, we employ results from the theories of compartmental systems and of asymptotically autonomous dynamical systems. We also compare our results with those obtained by making the quasi-steady state assumption.

math.CA↗

arXiv2011

Gap size and capture zone distributions in one-dimensional point island nucleation and growth simulations: asymptotics and models

The nucleation and growth of point islands during submonolayer deposition on a one-dimensional substrate is simulated for critical island size $i=0,1,2,3$. The small and large size asymptotics for the gap size and capture zone distributions (GSD and CZD) are studied. Comparisons to theoretical predictions from fragmentation equation analyses are made, along with those from the recently proposed Generalised Wigner Surmise (GWS). We find that the simulation data can be fully understood in the framework provided by the fragmentation equations, whilst highlighting the theoretical areas that require further development. The GWS works well for the small-size CZD behaviour, but completely fails to describe the large-size CZD asymptotics of the one-dimensional system.

nlin.AO↗

arXiv2011

Capture-zone distribution in one-dimensional sub-monolayer film growth: a fragmentation theory approach

The distribution of capture zones formed during the nucleation and growth of point islands on a one-dimensional substrate during monomer deposition is considered for general critical island size $i$. A fragmentation theory approach yields the small and (for $i=0$) large size asymptotics for the capture zone distribution (CZD) under the assumption of no neighbour-neighbour gap size correlation. These CZD asymptotic forms are different to those of the Generalised Wigner Surmise which has recently been proposed for island nucleation and growth models, and we discuss the reasons for the discrepancies.

nlin.AO↗

arXiv2010

Finite to infinite steady state solutions, bifurcations of an integro-differential equation

We consider a bistable integral equation which governs the stationary solutions of a convolution model of solid--solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is varied to examine the transition from an infinite number of steady states to three for the continuum limit of the semi--discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem.

math.DS↗

arXiv2008

Bifurcations in the regularized Ericksen bar model

We consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation methods, the structure of bifurcations from double zero eigenvalues. Our results provide evidence in support of Müller's conjecture \cite{Muller} concerning the symmetry of local minimizers of the associated energy functional and describe in detail the structure of the primary branch connections that occur in this problem. We give a reformulation of Müller's conjecture and suggest two further conjectures based on the local analysis and numerical observations. We conclude by analysing a ``loop'' structure that characterizes $(k,3k)$ bifurcations.

math.DS↗

arXiv2006

Stylized facts from a threshold-based heterogeneous agent model

A class of heterogeneous agent models is investigated where investors switch trading position whenever their motivation to do so exceeds some critical threshold. These motivations can be psychological in nature or reflect behaviour suggested by the efficient market hypothesis (EMH). By introducing different propensities into a baseline model that displays EMH behaviour, one can attempt to isolate their effects upon the market dynamics. The simulation results indicate that the introduction of a herding propensity results in excess kurtosis and power-law decay consistent with those observed in actual return distributions, but not in significant long-term volatility correlations. Possible alternatives for introducing such long-term volatility correlations are then identified and discussed.

physics.comp-ph↗