Models of Dynamics

Abstract

Our concern will mainly be with mesoscopic systems. These are systems whose length and time scales are significantly larger (a few orders of magnitude) than atomic scales but still small compared to macroscopic scales (system size etc.). A typical example is provided by a system near a second order phase transition point, e.g. a magnet being cooled towards the Curie point. For temperatures far above the Curie point the individual magnetic moments inside the magnet are moving around randomly and the overall magnetization of the sample is zero. As the temperature is lowered and the Curie point is approached, the individual magnetic moments become more correlated. The energy is lowered if the moments are aligned and as the temperature decreases, the entropy effects become smaller and the gradually dominating energy part of the thermodynamic free energy causes the correlations to build up. The distance over which the correlations exist is called the correlation length ξ. As the temperature reaches the Curie point, the correlation length becomes infinitely big, leading to correlation functions which become infinitely long ranged. For temperatures very close to the critical point (say a millikelvin away from the critical point), the correlation length is of the order of a few microns which is about three or four orders of magnitude larger than the atomic scale which is of the order of a few angstroms. This makes for an ideal mesoscopic system. If we are to describe the statics or dynamics of such a system, the task would be quite difficult if it were to be in terms of individual atoms or molecules. Consequently, one uses a coarse grained description.