Pattern Formation in Extended Continuous Systems

Abstract

Pattern formation as a self organized process in non-equilibrium systems is surely one of the most fascinating fields of research in our time. A unifying treatment developed in Synergetics [1,2,3] allows the mathematical description of the formation of spatio-temporal structures in quite different subjects, such as physics, chemistry, and biology. The behavior of a system in the vicinity of a non-equilibrium phase transition can be very often described by a few state variables which in analogy to Ginzburg-Landau theory of equilibrium thermodynamics are called order parameters. The large (in continuous systems even infinite) number of state variables assigned to the dynamics of the complete system can be expressed by the order parameters in a unique way; the order parameters enslave the many degrees of freedom that are linearly damped. By elimination of these enslaved modes [4]. a drastically reduced description in terms of few order parameter equations (OPE) is obtained. It thereby turns out that these equations have a similar form as the Ginzburg-Landau equations derived phenomenologically for phase transitions in thermal equilibrium. Beyond that, these generalized Ginzburg-Landau equations (GGLE) for phase transitions in open systems can be derived from basic physical laws, such as the Navier-Stokes equations [5] or the Maxwell equations [6] for hydrodynamic problems and for laser instabilities, respectively.