Notes

Chapter 7: Mechanisms in Programs and Nature

Section 7: Origins of Discreteness


General features of phase transitions

To reproduce the Ising model, a cellular automaton must have several special properties. In addition to conserving energy, its evolution must be reversible in the sense discussed on page 435. And with the constraint of reversibility, it turns out that it is impossible to get a non-trivial phase transition in any 1D system with the kind of short-range interactions that exist in a cellular automaton. But in systems whose evolution is not reversible, it is possible for phase transitions to occur in 1D, as the examples in the main text show.

One point to notice is that the sharp change which characterizes any phase transition can only be a true discontinuity in the limit of an infinitely large system. In the case of the system on page 339, for example, it is possible to find special configurations with a finite total number of cells which lead to behavior opposite to what one expects purely on the basis of their initial density of black cells. When the total number of cells increases, however, the fraction of such configurations rapidly decreases, and in the infinite size limit, there are no such configurations, and a truly discontinuous transition occurs exactly at density 1/2.

The discrete nature of phase transitions was at one time often explained as a consequence of changes in the symmetry of a system. The idea is that symmetry is either present or absent, and there is no continuous variation of level of symmetry possible. Thus, for example, above the transition, the Ising model treats up and down spins exactly the same. But below the transition, it effectively makes a choice of one spin direction or the other. Similarly, when a liquid freezes into a crystalline solid, it effectively makes a choice about the alignment of the crystal in space. But in boiling, as well as in a number of model examples, there is no obvious change of symmetry. And from studying phase transitions in cellular automata, it does not seem that an interpretation in terms of symmetry is particularly useful.

A common feature of phase transitions is that right at the transition point, there is competition between both phases, and some kind of nested structure is typically formed, as discussed on page 273 and above. The overall form and fractal dimension of this nested structure is typically independent of small-scale features of the system, making it fairly universal, and amenable to analysis using the renormalization group approach (see page 955).


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