Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

101 geodynamic modelling: how to design, interpret, and communicate numerical studies of the solid Earth

Geodynamic modelling provides a powerful tool to investigate processes in the Earth's crust, mantle, and core that are not directly observable. However, numerical models are inherently subject to the assumptions and simplifications on which they are based. In order to use and review numerical modelling studies appropriately, one needs to be aware of the limitations of geodynamic modelling as well as its advantages. Here, we present a comprehensive yet concise overview of the geodynamic modelling process applied to the solid Earth from the choice of governing equations to numerical methods, model setup, model interpretation, and the eventual communication of the model results. We highlight best practices and discuss their implementations including code verification, model validation, internal consistency checks, and software and data management. Thus, with this perspective, we encourage high-quality modelling studies, fair external interpretation, and sensible use of published work. We provide ample examples, from lithosphere and mantle dynamics specifically, and point out synergies with related fields such as seismology, tectonophysics, geology, mineral physics, planetary science, and geodesy. We clarify and consolidate terminology across geodynamics and numerical modelling to set a standard for clear communication of modelling studies. All in all, this paper presents the basics of geodynamic modelling for first-time and experienced modellers, collaborators, and reviewers from diverse backgrounds to (re)gain a solid understanding of geodynamic modelling as a whole

Novel excitations in driven vortex channels in a superconductor, and solitary waves of light and atoms in photonic crystal fibres

This is a thesis in two parts. In Part I, we will study the shear response of confined vortices. In Part 2, we will study light and matter interactions in photonic crystal fibres. Whilst the approaches of each are completely different, they both have the same central theme: solitons. In the first part of this thesis we study the static and dynamic properties of vortices within a Type-II superconductor, confined within a channel. The channel comprises a collection of pinned vortices, which form the perfect triangular lattice in the boundary, and rows of “free” particles which are driven via an external force. We provide two main results within this system. First we calculate the potential stemming from the boundary, and derive (under certain approximations) the phenomenologically accepted result for the critical shear dependence on the system width. We then study a novel system in which a defect is placed in a deformable potential; specifically a system comprised of two channels where one or both channels have a defect. This system provides a mechanism for the proliferation of kink/kink and anti-kink/anti-kink pairs as the defect binds to a local excitation in the form of a “breather”. We observe and explain what appears to be an action at a distance style interaction between excitations. In Part II, we will utilise the nonlinear effects of a Bose condensate and the unique optical properties of a photonic crystal fibre to demonstrate there are nonlinearly stable configurations which exist in the vicinity of an optical mode with a cut-off. These are solitary waves, whose relative composition of atoms and photons may be changed via altering the detuning of light from an atomic transition and Feshbach resonances

Microstructure and rheology of soft particle glasses

textSoft particle glasses like microgels and compressed emulsions are densely packed, disordered suspensions of deformable particles. Quantitative relationships among the constituent properties and the macroscopic properties of the suspension are determined for their customized design as rheological additives. The microscopic origin of their macroscopic properties is also determined. Advanced characterization techniques like Large Amplitude Oscillatory Shear (LAOS) and microrheology are studied to use them efficiently to characterize these materials. Their microstructure and rheology are investigated through theory, simulations and experiments. Soft particle glasses are used as rheological additives in many applications including coatings, solid inks and textured food and cosmetic products but their formulation is largely empirical. A quantitative connection between their formulation and rheology is critical to enable their rational design. Their microstructure will lead to the microscopic origin of some unique properties in common with other soft crowded materials like intracellular cytoplasm and clays. These are complex fluids and require novel techniques to characterize them. A study of these techniques is essential to efficiently interpret the observations in terms of their macroscopic properties and the microscopic dynamics involved. Particle scale simulations of steady and oscillatory shear flow are developed to predict the nonlinear rheology and microstructure of these glasses. The origin of yielding is determined as escape of particles from their cages giving rise to a shear induced diffusion. Microrheology is studied by developing simulations of a probe particle being pulled at a constant force and the rheological information from microrheology is quantitatively connected to that from bulk rheological measurements. Soft particle glasses develop internal stresses when quenched to a solid state by flow cessation during processing. Experiments are performed to characterize and a priori predict these stresses. Simulations are used to determine the particle scale mechanisms involved in the stress relaxation on flow cessation and the microstructural origin of internal stresses. A pairwise interaction theory is developed for quiescent glasses to quantitatively predict their microstructure and elastic properties. The theory is then extended to sheared glasses to quantitatively predict their nonlinear rheology. The implementation of the pairwise theories is computationally much faster than the full three-dimensional simulations.Chemical Engineerin

Numerical Simulations

This book will interest researchers, scientists, engineers and graduate students in many disciplines, who make use of mathematical modeling and computer simulation. Although it represents only a small sample of the research activity on numerical simulations, the book will certainly serve as a valuable tool for researchers interested in getting involved in this multidisciplinary field. It will be useful to encourage further experimental and theoretical researches in the above mentioned areas of numerical simulation