Scaling Self-Similarity and Intermediate Asymptotics

Scaling (power-type) laws reveal the fundamental property of the phenomena--self similarity. Self-similar (scaling) phenomena repeat themselves in time and/or space. The property of self-similarity simplifies substantially the mathematical modeling of phenomena and its analysis--experimental analytical and computational. The book begins from a non-traditional exposition of dimensional analysis physical similarity theory and general theory of scaling phenomena. Classical examples of scaling phenomena are presented. It is demonstrated that scaling comes on a stage when the influence of fine details of initial and/or boundary conditions disappeared but the system is still far from ultimate equilibrium state (intermediate asymptotics). It is explained why the dimensional analysis as a rule is insufficient for establishing self-similarity and constructing scaling variables. Important examples of scaling phenomena for which the dimensional analysis is insufficient (self-similarities of the second kind) are presented and discussed. A close connection of intermediate asymptotics and self-similarities of the second kind with a fundamental concept of theoretical physics the renormalization group is explained and discussed. Numerous examples from various fields--from theoretical biology to fracture mechanics turbulence flame propagation flow in porous strata atmospheric and oceanic phenomena are presented for which the ideas of scaling intermediate asymptotics self-similarity and renormalization group were of decisive value in modeling.