Dynamics of Order Parameters

Abstract

In the last chapter we saw that even the behavior of complex systems may, close to their instability points, be governed by very few variables, namely the order parameters. In a number of important cases, we deal with very few order parameters and we shall start here with the discussion of the behavior of a single order parameter. To describe the behavior of such an order parameter below and above the instability point of a system, a mechanical model has proved to be very useful, because it allows a simple interpretation. We identify the size of the order parameter ξ with the coordinate of a ball that slides down a hill towards the bottom of a valley. In technical terms, we are dealing with the overdamped motion of a particle under the gravitational force, but under the constraint that the particle moves on a parabola (Fig. 5.1, upper part). According to the overdamped motion, the speed with which the particle moves is proportional to the size of the slope. The steeper the slope, the higher the velocity. Quite clearly, at the bottom of the valley the slope is equal to zero and the particle has come to rest. Typical order parameters are subject to fluctuations. These fluctuations may be visualized as the kicks of soccer players hitting the ball. Quite often in actual games, the players hit the ball entirely at random, a beautiful picture of the impact of fluctuations! Because of these kicks, the ball will not remain at rest, but will go to the left- or to the right-hand side, then slide down, will be kicked up again, and so on. The situation just described applies to the region below threshold or, in other words, below the critical value of the control parameter.