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Register a new userGeneric equations for pattern formation in evolving interfaces
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Added by
Javier Mu\\~noz-Garc\\'ia
Publication date
2007
fields
Physics
and research's language is
English
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We present a general formalism which allows us to derive the evolution equations describing one-dimensional (1D) and isotropic 2D interfacelike systems, that is based on symmetries, conservation laws, multiple scale arguments, and exploits the relevance of coarsening dynamics. Our approach becomes especially significant in the presence of surface morphological instabilities and allows us to classify the most relevant nonlinear terms in the continuum description of these systems. The formalism applies to systems ranging from eroded nanostructures to macroscopic pattern formation. In particular, we show the validity of the theory for novel experiments on ion plasma erosion.
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Partial differential equations (PDE) have been widely used to reproduce patterns in nature and to give insight into the mechanism underlying pattern formation. Although many PDE models have been proposed, they rely on the pre-request knowledge of physical laws and symmetries, and developing a model to reproduce a given desired pattern remains difficult. We propose a novel method, referred to as Bayesian modelling of PDE (BM-PDE), to estimate the best PDE for one snapshot of a target pattern under the stationary state. We show the order parameters extracting symmetries of a pattern together with Bayesian modelling successfully estimate parameters as well as the best model to make the target pattern. We apply BM-PDE to nontrivial patterns, such as quasi-crystals, a double gyroid and Frank Kasper structures. Our method works for noisy patterns and the pattern synthesised without the ground truth parameters, which are required for the application toward experimental data.
We study a $mathcal PT$-symmetric scalar Euclidean field theory with a complex action, using both theoretical analysis and lattice simulations. This model has a rich phase structure that exhibits pattern formation in the critical region. Analytical results and simulations associate pattern formation with tachyonic instabilities in the homogeneous phase. Monte Carlo simulation shows that pattern morphologies vary smoothly, without distinct microphases. We suggest that pattern formation in this model may be regarded as a form of arrested spinodal decomposition. We extend our theoretical analysis to multicomponent $mathcal PT$-symmetric Euclidean scalar field theories and show that they give rise to new universality classes of local field theories that exhibit patterned behavior in the critical region. QCD at finite temperature and density is a member of the $Z(2)$ universality class when the Polyakov loop is used to distinguish confined and deconfined phases. This suggests the possibility of the formation of patterns of confined and deconfined matter in QCD in the critical region in the $mu-T$ plane.
Inspired by recent experimental observation of patterning at the membrane of a living cell, we propose a generic model for the dynamics of a fluctuating interface driven by particle-like inclusions which stimulate its growth. We find that the coupling between interfacial and inclusions dynam- ics yields microphase separation and the self-organisation of travelling waves. These patterns are strikingly similar to those detected in the aforementioned experiments on actin-protein systems. Our results further show that the active growth kinetics does not fall into the Kardar-Parisi-Zhang universality class for growing interfaces, displaying instead a novel superposition of equilibrium-like scaling and sustained oscillations.
Active force generation by actin-myosin cortex coupled to the cell membrane allows the cell to deform, respond to the environment, and mediate cell motility and division. Several membrane-bound activator proteins move along it and couple to the membrane curvature. Besides, they can act as nucleating sites for the growth of filamentous actin. Actin polymerization can generate a local outward push on the membrane. Inward pull from the contractile actomyosin cortex can propagate along the membrane via actin filaments. We use coupled evolution of fields to perform linear stability analysis and numerical calculations. As activity overcomes the stabilizing factors such as surface tension and bending rigidity, the spherical membrane shows instability towards pattern formation, localized pulsation, and running pulsation between poles. We present our results in terms of phase diagrams and evolutions of the coupled fields. They have relevance for living cells and can be verified in experiments on artificial cell-like constructs.
We present a theory to describe domain formation observed very recently in a quenched Rb-87 gas, a typical ferromagnetic spinor Bose system. An overlap factor is introduced to characterize the symmetry breaking of M_F=pm 1 components for the F=1 ferromagnetic condensate. We demonstrate that the domain formation is a co-effect of the quantum coherence and the thermal relaxation. A thermally enhanced quantum-oscillation is observed during the dynamical process of the domain formation. And the spatial separation of domains leads to significant decay of the M_F=0 component fraction in an initial M_F=0 condensate.
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