Chapter 7: Mechanisms in Programs and Nature

Section 7: Origins of Discreteness

Phase transitions

The discrete transitions shown in cellular automata in this section are examples of general phenomena known in physics as phase transitions. A phase transition can be defined as any discontinuous change that occurs in a system with a large number of components when a parameter associated with that system is varied. (Some physicists might argue for a somewhat narrower definition that allows only discontinuities in the so-called partition function of equilibrium statistical mechanics, but for many of the most interesting applications, the definition I use is the appropriate one.) Standard examples of phase transitions include boiling, melting, sublimation (solids such as dry ice turning into gases), loss of magnetization when a ferromagnet is heated, alignment of molecules in liquid crystals above a certain electric field (the basis for liquid crystal displays), and the onset of superconductivity and superfluidity at low temperatures.

It is conventional to distinguish two kinds of phase transitions, often called first-order and higher-order. First-order transitions occur when a system has two possible states, such as liquid and gas, and as a parameter is varied, which of these states is the stable one changes. Boiling and melting are both examples of first-order transitions, as is the phenomenon shown in the cellular automaton in the main text. Note that one feature of first-order transitions is that as soon as the transition is passed, the whole system always switches completely from one state to the other.

Higher-order transitions are in a sense more gradual. On one side of the transition, a system is typically completely disordered. But when the transition is passed, the system does not immediately become completely ordered. Instead, its order increases gradually from zero as the parameter is varied. Typically the presence of order is signalled by the breaking of some kind of symmetry—say of rotational symmetry by the spontaneous selection of a preferred direction.

From Stephen Wolfram: A New Kind of Science [citation]