2.5 Phase Transitions
21
Fig. 2.7 Left: The Ising model on a square grid. Right: Schelling’s segregation model
ban neighborhoods (Fig. 2.7, right). The model is also set on a square grid, and
oppositely oriented spins are replaced by agents of two kinds, which are supposed
to become uncomfortable when the number of neighbors of another kind exceeds a
certain number; some cells are left empty. The model was evidently aimed at racial
segregation, and as these agents, unlike spins, cannot change their color from pink
to blue or the other way around, they ease their discomfort by moving to an empty
cell. Quite naturally, this leads to a phase transition to a segregated state, with blue
and pink “races” occupying distinct clusters. What was considered special in this
model, and won Schelling a quasi-Nobel prize in economics (Onsager got the real
thing – in physics), was a demonstration that segregation can be totally spontaneous,
without any collusion or command.
The idea of more general models of this kind, based on agents moving on a twodimensional grid according to set rules, goes back to Stanislaw Ulam and John von
Neumann in the early postwar years, but they wouldn’t engage in this seriously.
Such “cellular automata”, easy to design and implement even on the computers of
yesteryear, have been used to design a plethora of games which bear little relation
to reality but are capable of generating all kinds of dynamic patterns never reaching
equilibrium. The first and most famous of them was John Conway’s Game of Life
(Gardner, 1970). Steven Wolfram (2002) attributed reality to such games, proclaiming them “a new kind of science”, to the bewilderment of mainstream physicists
expecting illuminations of another kind from the former young genius.
21
Fig. 2.7 Left: The Ising model on a square grid. Right: Schelling’s segregation model
ban neighborhoods (Fig. 2.7, right). The model is also set on a square grid, and
oppositely oriented spins are replaced by agents of two kinds, which are supposed
to become uncomfortable when the number of neighbors of another kind exceeds a
certain number; some cells are left empty. The model was evidently aimed at racial
segregation, and as these agents, unlike spins, cannot change their color from pink
to blue or the other way around, they ease their discomfort by moving to an empty
cell. Quite naturally, this leads to a phase transition to a segregated state, with blue
and pink “races” occupying distinct clusters. What was considered special in this
model, and won Schelling a quasi-Nobel prize in economics (Onsager got the real
thing – in physics), was a demonstration that segregation can be totally spontaneous,
without any collusion or command.
The idea of more general models of this kind, based on agents moving on a twodimensional grid according to set rules, goes back to Stanislaw Ulam and John von
Neumann in the early postwar years, but they wouldn’t engage in this seriously.
Such “cellular automata”, easy to design and implement even on the computers of
yesteryear, have been used to design a plethora of games which bear little relation
to reality but are capable of generating all kinds of dynamic patterns never reaching
equilibrium. The first and most famous of them was John Conway’s Game of Life
(Gardner, 1970). Steven Wolfram (2002) attributed reality to such games, proclaiming them “a new kind of science”, to the bewilderment of mainstream physicists
expecting illuminations of another kind from the former young genius.