Transition phenomena in multidimensional systems – models of evolution

Abstract

Introduction In this chapter we investigate the influence of noise in systems where many degrees of freedom are participating. The phenomenon we want to investigate is the possibility to leave and to enter attractor regions in bi- and multi-stable systems. It implies the ability to cross through separatrices (to jump over potential barriers) to alternative attractors of the systems after a certain time. Since the pioneering work of Kramers this phenomenon has been widely investigated in one- and two-dimensional bistable systems (for reviews see Hanggi, 1986, and Weiss, 1986); several papers also appeared on multistable situations (see, e.g., Buttiker and Landauer, 1982). Stochastic transitions in multistable multidimensional systems are of importance for several physical and nonphysical problems, for example for nucleation induced phase transitions, for the molecular dynamics and the kinetics of chemical reactions, for the storage of information in electronic circuits, for learning procedures in networks of bistable elements, for the search of extrema in combinatorial optimization problems, etc. All those problems possess in general many interacting degrees of freedom, have a large number of stable states, configurations, etc. Though in all the problems mentioned above stochastic transitions between stable states occur, their dynamics may be quite different. In some situations, for example chemical catalysis, it is desirable that the transition takes place very quickly. In other situations transitions should be avoided; for example, computer memory becomes meaningless if transitions occur on time scales which are comparable with the time of calculations.