Tuesday, November 14, 2006

Catastrophe Means Change Of State, Like Water Into Ice

It is essential for catastrophe theory that a small number of external variables be chosen, say three, which will be the only ones to vary, all other parameters being kept fixed. The values given to these three external variables will then be registered as a point in three-dimensional space. If these values are changed, the corresponding point moves along, and the potential of the system gets modified in some (unknown) fashion. The system, initially at a stable equilibrium, follows the latter in its variations. It may happen that, for certain values of the parameter, this particular equilibrium vanishes. The system then jumps to another stable equilibrium; that is, we get a discontinuous response to a continuous change in the external variables. The set of parameter values where this happens is called the catastrophe set. It is a kind of boundary which stretches across three-dimensional space: if the point representing the values of the external variables crosses this boundary, the system will jump from one equilibrium to another. This change of equilibrium will be perceived by the observer as a sudden and substantial change in the properties of the system—a phase transition, for instance, like water solidifying into ice. If one lowers the temperature of water regularly, nothing much happens, until the critical temperature of 0˚ is reached, at which time ice appears. Here there is just one external variable, temperature, and one catastrophic value, 0˚ C. The parameter space is one-dimensional, and the catastrophe set consists of one point, 0, which acts as a boundary between negative temperatures (ice only) and positive ones (water only).

Ivar Ekeland
Mathematics And The Unexpected

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