Non-Linear Dynamics Near and Far from Equilibrium

We will be concerned mainly with systems with in?nite degrees of freedom which can however, be described by a few variables. These variables must necessarily be ?elds i. e. functions of space and time. A typical example would be to try to describethe?owofairaroundus. Thevariablesthatwouldbenecessarytodescribe the state of air would certainly be its density, its temperature and its velocity. All these variables (density, temperature and velocity) are, in general, functions of space and time. They are mesoscopic variables. They do not re?ect the variations occurring at the molecular level. To de?ne a density, it should be recalled, we take a small volume (small compared to the total system size, yet large compared to atomic dimensions) and consider the mass of this small volume. The ratio of mass tovolumeremainsconstantforareasonablylargevariationinthesizeofthevolume chosen and de?nes the density of the system. It fails to be a constant if the volume becomessosmallthatitcontainsonlyafewmolecules. Inthatcaseourdescription in terms of a density fails. All the systems that we will talk about can be described in terms of a coarse grained ?eld like the density. Because of the smallness (at the macroscopic level) of the volume used in de?ning density it can be considered a local variable. This is what makes it a ?eld. Similarly we can talk about the local temperature and local velocity. The local velocity is not the velocity of an individual molecule but the velocity associated with a macroscopically small, yet microscopicallylargevolumeofair.