The Ising on the cakePhilip Ball explains why those trying
to solve one of the hardest problems in physics may have
been wasting their time. .
26 April 2000
Ever since Ernst Ising formulated it in 1920s, the
problem of the 'Ising model' has given some of the
finest minds in physics headaches and sleepless nights.
Now a researcher at Sandia National Laboratories in
Albuquerque, New Mexico, claims that they have all been
on a hopeless quest. The problem, he says, is literally
The Ising model has some robustly practical
applications. It was conceived as a description of
magnetism in crystalline materials, but it can also be
applied to phenomena as diverse as the freezing and
evaporation of liquids, the 'folding' of protein
molecules into their biologically active forms, and the
behaviour of glassy substances.
The model aims to describe how a large collection of
'agents' will act when each can exist in two different
states, they are arranged on a regular lattice (like a
chess board), and the state of each 'agent' influences
that of its immediate neighbours.
For example, in the Ising model of a magnet, the
agents are magnetic atoms, each of which can have its
magnetic 'poles' pointing in one of two opposite
orientations. In a so-called ferromagnet, such as iron,
the interactions between neighbouring atoms favour the
alignment of their magnetic poles in the same direction.
When all the 'atomic magnets' are orientated in the same
way, they add up to give the material a net
At high temperatures, however, the orientations get
randomized because of the thermal motions of the atoms,
and the material loses its magnetization. This happens
to iron at around 770 °C. The Ising model describes how,
as the temperature falls, certain kinds of material
switch in a sudden 'phase transition' from a
non-magnetic to a ferromagnetic state. It can also
provide a model for the separation of a hot fluid into
liquid and gas as it cools, or the transition from an
unfolded to a folded protein molecule.
Ising's original formulation of his model was
one-dimensional -- he postulated a linear chain of
equally spaced magnetic atoms. It is relatively easy to
figure out how such a system behaves: perfect alignment
of all the atomic magnets happens only if they are
cooled to absolute zero. The smallest thermal motion is
sufficient to break the cooperativity between them.
In two dimensions, with atoms arranged on a flat
grid, things are more complicated. In the 1940s, the
Norwegian physicist Lars Onsager showed that there is a
transition to a magnetic state at some temperature above
absolute zero, and he indicated how this temperature
could be calculated from the basic features of the
model, such as the strength of the interaction between
neighbouring atoms. Onsager's theoretical 'solution' of
the two-dimensional Ising model was tremendously
difficult, and is widely regarded as a milestone in
But of course most real systems are
three-dimensional. In a 3D Ising model, the atoms occupy
blocks on a 3D grid, like a collection of stacked boxes.
It is easy enough to simulate such a system on the
computer, whereupon one finds that this too undergoes a
phase transition. The 3D Ising model is now widely used
to model magnetism and other 'collective' behaviour of
atoms on computers. But can one calculate the transition
temperature knowing just the fundamental characteristics
of the individual atomic components, as Onsager did in
two dimensions? This is the challenge of the
three-dimensional Ising model.
Now Sorin Istrail, a computational biologist, says it
can't be done. He claims that the problem falls into the
'computationally intractable' class of conundrums that
are too complex to be solved on any realistic timescale.
Several thousand such problems are already known. It is
not the same as saying that a computer can't find the
transition by sheer number-crunching -- clearly it can.
Rather, Istrail's work demonstrates that any attempt to
formulate an exact equation relating the transition
temperature to the model's basic parameters would take
longer than is humanly feasible.