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The Ising on the cake

Philip 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 insoluble.

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 magnetization.

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 theoretical physics.

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.

Nature News Service / Macmillan Magazines Ltd 2002

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