Multi-Scale Phenomena in Complex Fluids

The following sections are included:

• Preface

• Contents

One of the core problems in materials science is how the interactions between electrons in a solid give rise to properties like magnetism, superconductivity, and metal-insulator transitions. Our ability to solve this central question in quantum statistical mechanics numerically is presently limited to systems of a few hundred electrons. While simulations at this scale have taught us a considerable amount about certain classes of materials, they have very significant limitations, especially for recently discovered materials which have mesoscopic magnetic and charge order.

In this paper, we begin with an introduction to the Hubbard model and quantum Monte Carlo simulations. The Hubbard model is a simple and effective model that has successfully captured many of the qualitative features of materials, such as transition metal monoxides, and high temperature superconductors. Because of its voluminous contents, we are not be able to cover all topics in detail; instead we focus on explaining basic ideas, concepts and methodology of quantum Monte Carlo simulation and leave various part for further study. Parts of this paper are our recent work on numerical linear algebra methods for quantum Monte Carlo simulations.

The following sections are included:

• Scalar diffraction theory
  • Introduction
  • Kirchhoff's theory of diffraction
  • Huygens-Fresnel principle
  • Fresnel and Fraunhofer diffraction
  • Focal spot size and resolution

• Approximations: weak fluctuation
  • Born approximation
  • Rytov approximation
  • The extended Huygens-Fresnel principle
  • Paraxial approximation

• The Wigner distribution

• Markovian approximation
  • White-noise scaling
  • Markovian limit

• Two-frequency transport theory
  • Paraxial waves
    • Two-frequency radiative transfer equations
    • The longitudinal and transverse cases
  • Spherical waves
    • Geometrical radiative transfer
    • Spatial (frequency) spread and coherence bandwidth
    • Small-scale asymptotics

• Application: time reversal
  • Spherical wave
  • Paraxial wave
  • Anomalous focal spot
  • Duality and turbulence-induced aperture
  • Coherence length
  • Broadband time reversal communications

• Application: imaging in random media
  • Imaging of phase objects
  • Long-exposure imaging
  • Short-exposure imaging
  • Coherent imaging of multiple point targets in Rician media
    • Differential scattered field in clutter
    • Imaging functions
    • Numerical simulation with a Rician medium
  • Coherent imaging in a Rayleigh medium

• Acknowledgement

• References

Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is the scale of computation. The ratio between the largest scale and the smallest scale could be as large as 10⁵ in each space dimension. From an engineering perspective, it is often sufficient to predict the macroscopic properties of the multiplescale systems, such as the effective conductivity, elastic moduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures the small scale effect on the large scales, but does not require resolving all the small scale features. This paper reviews some of the recent advances in developing systematic multiscale methods with particular emphasis on multiscale finite element methods with applications to flow and transport in heterogeneous porous media. This manuscript is not intended to be a general survey paper on this topic. The discussion is limited by the scope of the lectures and expertise of the author.

In this chapter, we consider an inverse problem of determining the corrosion occurring in an inaccessible interior part of a pipe from the measurements on the outer boundary. The problem is modeled by Laplace's equation with an unknown term γ in the boundary condition on the inner boundary. Based on the Maz'ya iterative algorithm, a regularized BEM method is proposed for obtaining approximate solutions for this inverse problem. The numerical results show that our method can be easily realized and is quite effective.

Complex fluids/soft matter are materials whose material properties are dominated by their meso-scale structures. In this chapter, we introduce some basic theories useful in modeling complex fluids/soft matter materials. We selectively focus on the kinetic theory approach to modeling complex fluids. The basic topics included are equilibrium thermodynamics, statistical mechanics, polymer physics, and the configurational space kinetic theory for transport phenomena in complex fluids.