scholarly journals Continuum models and Morse Theory in the simulation, design and evaluation of magnetic Skyrmion devices

2021 ◽  
Author(s):  
P. Robert Kotiuga
Keyword(s):  
Morse Theory ◽  
Lattice Models ◽  
Energy Functional ◽  
Continuum Models ◽  
Quantum Mechanical ◽  
Simulation Design ◽  
Lattice Systems ◽  
Solid Foundation ◽  
Magnetic Skyrmions ◽  
The Continuum

Nanomagnetic devices such as computer gates and memory devices based on magnetic skyrmions are close to becoming a reality. In this paper we will explore the highly nonconvex nanomagnetic energy landscape in order to draw conclusions about the complexity of magnetic phenomena. Morse theoretic arguments show that, in a bounded energy interval, the number of critical points of the energy functional grows exponentially. To show this, we introduce a hierarchy of models for the design of nanomagnetic devices to provide a solid foundation for the introduction of topological tools. To reason in terms of lattice models, one must make a distinction between two types of lattices: the quantum mechanical model of the actual physical lattice and the lattice model that can be associated with a discretization of a continuum model of the physics. By focusing on the implications of Morse theory applied to lattice systems arising from the discretization of the continuum models, and the notion of “topological frustration”, we provide a framework for understanding “complexity” in the context of nanomagnetic systems. We conclude with some suggestions for making the analysis more qualitative.

scholarly journals Continuum models and Morse Theory in the simulation, design and evaluation of magnetic Skyrmion devices

2021 ◽  
Author(s):  
P. Robert Kotiuga
Keyword(s):  
Morse Theory ◽  
Lattice Models ◽  
Energy Functional ◽  
Continuum Models ◽  
Quantum Mechanical ◽  
Simulation Design ◽  
Lattice Systems ◽  
Solid Foundation ◽  
Magnetic Skyrmions ◽  
The Continuum

Nanomagnetic devices such as computer gates and memory devices based on magnetic skyrmions are close to becoming a reality. In this paper we will explore the highly nonconvex nanomagnetic energy landscape in order to draw conclusions about the complexity of magnetic phenomena. Morse theoretic arguments show that, in a bounded energy interval, the number of critical points of the energy functional grows exponentially. To show this, we introduce a hierarchy of models for the design of nanomagnetic devices to provide a solid foundation for the introduction of topological tools. To reason in terms of lattice models, one must make a distinction between two types of lattices: the quantum mechanical model of the actual physical lattice and the lattice model that can be associated with a discretization of a continuum model of the physics. By focusing on the implications of Morse theory applied to lattice systems arising from the discretization of the continuum models, and the notion of “topological frustration”, we provide a framework for understanding “complexity” in the context of nanomagnetic systems. We conclude with some suggestions for making the analysis more qualitative.

scholarly journals Murray’s law for discrete and continuum models of biological networks

2019 ◽  
Vol 29 (12) ◽  
pp. 2359-2376
Author(s):  
Jan Haskovec ◽  
Peter Markowich ◽  
Giulia Pilli
Keyword(s):  
Biological Networks ◽  
Discrete Model ◽  
Continuum Model ◽  
Transportation Network ◽  
Continuum Models ◽  
Scaling Relation ◽  
Murray's Law ◽  
Murray’S Law ◽  
Rectangular Mesh ◽  
The Continuum

We demonstrate the validity of Murray’s law, which represents a scaling relation for branch conductivities in a transportation network, for discrete and continuum models of biological networks. We first consider discrete networks with general metabolic coefficient and multiple branching nodes and derive a generalization of the classical 3/4-law. Next we prove an analogue of the discrete Murray’s law for the continuum system obtained in the continuum limit of the discrete model on a rectangular mesh. Finally, we consider a continuum model derived from phenomenological considerations and show the validity of the Murray’s law for its linearly stable steady states.

Atomically Informed Continuum Models for the Elastic Contact Properties of Hollow and Coated Rigid Cylinders at the Nanoscale

2017 ◽  
Vol 84 (3) ◽  
Author(s):  
Leon Gorelik ◽  
Dan Mordehai
Keyword(s):  
Contact Stiffness ◽  
Md Simulations ◽  
Atomic Scale ◽  
Continuum Models ◽  
Elastic Contact ◽  
Pressure Profiles ◽  
Hollow Cylinders ◽  
Coated Surfaces ◽  
The One ◽  
The Continuum

Understanding the mechanical properties of contacts at the nanoscale is key to controlling the strength of coated surfaces. In this work, we explore to which extent existing continuum models describing elastic contacts with coated surfaces can be extended to the nanoscale. Molecular dynamics (MD) simulations of hollow cylinders or coated rigid cylinders under compression are performed and compared with models at the continuum level, as two representative extreme cases of coating which is substantially harder or softer than the substrate, respectively. We show here that the geometry of the atomic-scale contact is essential to capture the contact stiffness, especially for hollow cylinders with high relative thicknesses and for coated rigid cylinders. The contact pressure profiles in atomic-scale contacts are substantially different than the one proposed in the continuum models for rounded contacts. On the basis of these results, we formulate models whose solution can be computed analytically for the contact stiffness in the two extreme cases, and show how to bridge between the atomic and continuum scales with atomically informed geometry of the contact.

A Concurrent Multiscale Method for Coupling Atomistic and Continuum Models at Finite Temperatures

2009 ◽  
Vol 1229 ◽  
Author(s):  
Catalin Picu ◽  
Nithin Mathew
Keyword(s):  
Degrees Of Freedom ◽  
Interface Region ◽  
Continuum Models ◽  
Generalized Langevin Equation ◽  
Mechanical Coupling ◽  
Multi Scale ◽  
Scale Modeling ◽  
Temperature Boundary ◽  
Atomistic Region ◽  
The Continuum

AbstractA concurrent multi-scale modeling method for finite temperature simulation of solids is introduced. The objective is to represent far from equilibrium phenomena using an atomistic model and near equilibrium phenomena using a continuum model, the domain being partitioned in discrete and continuum regions, respectively. An interface sub-domain is defined between the two regions to provide proper coupling between the discrete and continuum models. While in the discrete region the thermal and mechanical processes are intrinsically coupled, in the continuum region they are treated separately. The interface region partitions the energy transferred from the discrete to the continuum into mechanical and thermal components by splitting the phonon spectrum into “low” and “high” frequency ranges. This is achieved by using the generalized Langevin equation as the equation of motion for atoms in the interface region. The threshold frequency is selected such to minimize energy transfer between the mechanical and thermal components. Mechanical coupling is performed by requiring the continuum degrees of freedom (nodes) to follow the averaged motion of the atoms. Thermal coupling is ensured by imposing a flux input to the atomistic region and using a temperature boundary condition for continuum. This makes possible a thermodynamically consistent, bi-directional coupling of the two models.

scholarly journals Julian Ernst Besag. 26 March 1945—6 August 2010

2018 ◽  
Vol 64 ◽  
pp. 27-50
Author(s):  
Peter J. Diggle ◽  
Peter J. Green ◽  
Bernard W. Silverman
Keyword(s):  
Random Fields ◽  
Markov Random Fields ◽  
Lattice Models ◽  
Field Trials ◽  
Nearest Neighbour ◽  
Agricultural Field ◽  
Lattice Systems ◽  
Markov Random ◽  
Texture Generation

Julian Besag was an outstanding statistical scientist, distinguished for his pioneering work on the statistical theory and analysis of spatial processes, especially conditional lattice systems. His work has been seminal in statistical developments over the last several decades ranging from image analysis to Markov chain Monte Carlo methods. He clarified the role of auto-logistic and auto-normal models as instances of Markov random fields and paved the way for their use in diverse applications. Later work included investigations into the efficacy of nearest-neighbour models to accommodate spatial dependence in the analysis of data from agricultural field trials, image restoration from noisy data, and texture generation using lattice models.

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