Probabilistic Models of Critical Phenomena By Gordon Slade 1 Critical Phenomena

A population can explode if its birth rate exceeds its death rate, but otherwise it becomes extinct. The nature of the population’s evolution depends critically on which way the balance tips between adding new members and losing old ones. A porous rock with randomly arranged microscopic pores has water spilled on top. If there are few pores, the water will not percolate through the rock, but if there are many pores, it will. Surprisingly, there is a critical degree of porosity that exactly separates these behaviours. If the rock’s porosity is below the critical value, then water cannot flow completely through the rock, but if its porosity exceeds the critical value, even slightly, then water will percolate all the way through. A block of iron placed in a magnetic field will become magnetized. If the magnetic field is extinguished, then the iron will remain magnetized if the temperature is below the Curie temperature 770 ◦C, but not if the temperature is above this critical value. It is striking that there is a specific temperature above which the magnetization of the iron does not merely remain small, but actually vanishes. The above are three examples of critical phenomena. In each example, global properties of the system change abruptly as a relevant parameter (fertility, degree of porosity, or temperature) is varied through a critical value. For parameter values just below the critical value, the overall organization of the system is quite different from how it is for values just above. The sharpness of the transition is remarkable. How does it occur so suddenly?