Probability theory was first formalized, using the new measure theory of Lebesgue, by A. Kolmogorov in 1939. It was soon used to understand ideas in quantum mechanics, statistical mechanics, stochastics, and other parts of applied science. Today an increasing number of phenomena in nature are best described using probability theory. Consider a total system consisting of a large number particles (say grains of sand or perhaps atoms). There are many variables, and many state variables (such as position, velocity) for each particle. The system falls under the sway of certain control parameters such as temperature, pressure, density. These are thermodynamic properties. We wish to study certain abrupt changes in such a system. An elegant example is given by water turning into ice. There is no gradual transition from the first state to the second one. We instead think about a phase, which is a collection of states. We have a large number of water molecules and we wish to examine ``bulk properties'' of this system. In particular, we would like to have an order parameter that explains or measures the change from the one state (water, in which the molecules are chaotic) and the second state (ice, in which the molecules cooperate). We may also say that water has no long range order while ice does have long range order. We think of water and ice as different phases. The study of a system like this was initiated by Lev Landau. In this system, the state of matter is described by temperature and pressure. Each pair of these parameters gives a probability distribution. The order parameter for the water/ice system is then the probability of the joint event minus the probability of the product event. One can think of the phase transition as being described by a real analytic function φ (i.e., one with a local power series expansion in the space variables). When the system is in the water state, the function φ is identically 0. When the system is in the ice state, the function φ is identically 1. The important mathematical fact is that there is no real analytic function that can be equal to 0 on one open set and equal to 1 on another open set. The order parameter measures the obstruction to the existence of such a real analytic function. The water-to-ice example is but one of many in which phase transitions arise naturally. Workshop participant Susan Coppersmith described another. Recall that Mitchell Feigenbaum considered the logistic equation φ (x) = λ x(1 - x) \, . Feigenbaum is interested in dynamical systems, so he considered iterations (i.e., compositions) of φ with itself. This was in the early days of handheld calculators, and he performed his calculations on (what is by today's standards) a rather primitive HP calculator. Feigenbaum found these results:

If λ is small, the values of the iteration eventually just repeat.

As λ increases, the system becomes periodic. Eventually there is a bifurcation and the periodicity doubles.

As λ continues to increase, periodicity continues to double until, when λ > 3.57, the system is chaotic.