Introduction

The two most prominent branches of macroscopic physics, reversible classical mechanics and reversible equilibrium thermodynamics, are both characterized by extremum principles – the principle of stationary action for a classical trajectory, and the principle of maximum entropy for thermodynamic equilibrium in a closed system. It is well known how both of these extremum principles arise by taking the macroscopic limit of more fundamental underlying microscopic theories, in which these extremum principles do not hold: quantum mechanics and statistical mechanics, respectively. In these microscopic theories the extremum principles are violated by the occurrence of fluctuations; quantum fluctuations by which finite probability amplitudes are assigned to nonclassical trajectories, and classical fluctuations which assign nonzero probabilities to states with less than the maximum entropy. Hence, there is a deep connection in physics between fluctuation phenomena and extremum principles, the former providing a mechanism allowing the system to explore a neighbourhood of the extremizing state and thereby to identify the extremum.

The last decade has seen a considerable increase of interest in non-equilibrium phenomena in macroscopic systems, as it was realized that simple and rather general mechanisms of self-organization exist in such systems (cf., e.g. Haken, 1977, 1983; Nicolis and Prigogine, 1977), leading to the selection, formation and competition of patterns in space and (or) time. The evolution equations governing these nonequilibrium phenomena neither belong to the realm of equilibrium thermodynamics, therefore thermodynamic extremum principles are not applicable, nor to the realm of reversible classical mechanics covered by the principle of least action.