The Challenge of Different Scales in Nature

Abstract

Between quantum length scales (atomic diameters of about 10 −10 m) and the earth's diameter (10 6 m) there are about 16 length scales. Most of technology and much of science occurs in this range. Between the Planck length (10 −35 m) and the diameter of the visible universe there are 70 length scales; 70, 16, or even 2 is a very large number. Most theories become intractable when they require coupling between even two adjacent length scales. Computational resources are generally not sufficient to resolve multiple length scales in three-dimensional problems and even in many two-dimensional problems. The problem is not merely one of presently available computational resources, which are growing at a rapid rate. To obtain an extra factor of 10 in computational resolution requires in the most favorable case a factor 10 4 in computational resources for time-dependent three-dimensional problems. When multiple length scales are in question, the under-resolution of computations performed with today's algorithms will be with us for some time to come, and the essential role which must be assigned to theory, and to the design of algorithms of a new nature, becomes evident. It is for this reason that nonlinear and stochastic phenomena, often described by the theory of coherent and chaotic structures, coupling adjacent and multiple length scales, is a vital topic.