ABSTRACT
This chapter describes the fundamental principles to deal with a multiple scale system. The amplitude of the envelope reaches a saturation magnitude, in agreement with the analytical theory. Around the micrometer length and the microsecond timescale, it deals with the molecular dynamics of catalysis, nucleation, chemical reactions, transport characteristics and intrinsic material properties. Many complex systems in physics, chemistry and especially biology have wide ranges of timescales. The multiple-time methodology, which tackles the temporal evolution with the mathematical rigorousness and physical insightfulness, is by itself an exercise for understanding a complex system. The most famous example in the multiple time treatment of quantum system is the Born-Oppenheimer Approximation, which assumes the nucleons are frozen when electron wave function is examined. The chapter deals with the coupling problem of two simple harmonic oscillators, one of which has a very heavy mass.
