Analysis and Simulation of Long-Range Correlations in Curved Space

Abstract

Numerical simulation and analysis of long-range correlations in curved space are studied. The study is motivated by the problem of constructing accurate models of large-scale porous media which usually contain long-range correlations in their various properties (such as their permeability, porosity, and elastic moduli) within and between their strata that are typically curved layers. The problem is, however, relevant to many other important models and phenomena in which extended correlations in curved space play a prominent role. Examples include the nonlinear σ-model in a curved space, models for describing the long-range structural correlations of amorphous semiconductors that consist of polytopes (tilings of positively-curved three-dimensional space), long-range correlations in the extrapolar total zone, and models in which the Universe is created by bubble nucleations and contain long-range correlations in the fluctuations in the curved spacetime. The study is also relevant to the important industrial problem of designing highly curved objects, such as cars and ships, which use composite materials that contain extended correlations in their property values. We study such correlations along two-and three-dimensional curves, as well as curved surfaces. We show that such correlations are well-defined only on developable surfaces, i.e. those that can be flattened to form planar surfaces without any stretching or distortion, and preserve the distance between two points on such surfaces after the stretching. If a given curved surface is not developable, but can be approximated as piecewise developable, one may still define and analyze extended correlations on it. Representative examples are presented and analyzed.