Catastrophes in statistical biophysics

Smooth functions may be classified into types according to the disposition of their extrema. If a function depends upon parameters, catastrophe theory describes possible changes of type as the parameters are varied. Many natural systems evolve toward an equilibrium determined by the minimization of some quantity. The application of catastrophe theory to such systems in mechanics and thermodynamics is sketched. The critical point of a gas‐liquid phase transition is shown not to behave as a cusp catastrophe because at that point the free energy whose minimization determines equilibrium has a discontinuity in its second derivative. The theory of changes of state is generalized to include those situations where the states involved are not pure phases. Such cases may arise when the role of fluctuations is appreciable, even at equilibrium. This approach is applied to the problem of protein denaturation. The resulting model incorporates statistical considerations within the framework of deterministic catastrophe theory. Because the macromolecular processes of life do not generally arrive at equilibrium, the present approach cannot be applied to living systems. Such applications must await the classification of types of evolution (differential) equations, a task that is presently far from complete.