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Transitions Still to Be Made Philip Ball impacts Transitions still to be made Philip Ball A collection of many particles all interacting according to simple, local rules can show behaviour that is anything but simple or predictable. Yet such systems constitute most of the tangible Universe, and the theories that describe them continue to represent one of the most useful contributions of physics. hysics in the twentieth century will That such a versatile discipline as statisti- probably be remembered for quantum cal physics should have remained so well hid- Pmechanics, relativity and the Standard den that only aficionados recognize its Model of particle physics. Yet the conceptual importance is a puzzle for science historians framework within which most physicists to ponder. (The topic has, for example, in operate is not necessarily defined by the first one way or another furnished 16 Nobel of these and makes reference only rarely to the prizes in physics and chemistry.) Perhaps it second two. The advances that have taken says something about the discipline’s place in cosmology, high-energy physics and humble beginnings, stemming from the quantum theory are distinguished in being work of Rudolf Clausius, James Clerk important not only scientifically but also Maxwell and Ludwig Boltzmann on the philosophically, and surely that is why they kinetic theory of gases. In attempting to have impinged so forcefully on the derive the gas laws of Robert Boyle and consciousness of our culture. Joseph Louis Gay-Lussac from an analysis of But the central scaffold of modern the energy and motion of individual physics is a less familiar construction — one particles, Clausius was putting thermody- that does not bear directly on the grand ques- namics on a microscopic basis. But from a tions that physicists are popularly expected modern perspective, his programme was to address but instead defines our current deeper still: he was attempting to understand understanding of phenomena at the prosaic the collective behaviour of interacting, energy and length scales characteristic of our many-body systems. This, it might be everyday experience. Statistical physics, and Figure 1 The Ising model at the critical point. argued, is the defining objective of statistical more specifically the theory of transitions Each site on this two-dimensional lattice can physics in all its guises. between states of matter, more or less defines adopt one of two states — black or white, At least with (dilute) gases one can afford what we know about ‘everyday’ matter and corresponding to ‘up’ or ‘down’ spins in a to neglect interparticle attractive forces with its transformations. ferromagnet. At the critical point, neither state some justification. Phase transitions enter Moreover, it provides the conceptual predominates, and fluctuations occur on all into the picture, however, when those forces apparatus for tackling complex collective length scales. (Courtesy of Alistair Bruce, are included. Johannes Diderik van der quantum phenomena of intense topical University of Edinburgh.) Waals, who introduced such forces in a interest such as Bose–Einstein condensation heuristic manner using what we would now (in which a collection of particles all occupy call a mean-field theory, found that he could the same quantum ground state) and high- Critical ideas describe the gas–liquid transition. In van der temperature superconductivity (that is, ‘Phase transition’ is today a debased term — Waals’ theory, the particles have a hard superconductivity above about 35 K) . Many like the classical equivalent of ‘quantum leap’, repulsive core and an infinitesimally small of the states of condensed matter that it tends to attach itself to any abrupt change in attraction of infinite range (although this is promise new technological applications, a system’s behaviour. Does a single molecule, not the way the Dutchman expressed it). ranging from block copolymers to magnetic such as a protein, undergo a phase transition Van der Waals was awarded the Nobel multilayers, can be understood as the conse- if it abruptly changes conformation? In the prize in 1910 and is regarded as something of quence of the kind of collective behaviour strict sense, no. A genuine transition requires a founding father for statistical physics. So that statistical physics describes. that there be some singularity in a thermody- far did his vision penetrate that in 1998 the There are still central issues in cosmology namic potential (such as the Gibbs free physicist Ben Widom, in his Boltzmann and high-energy physics whose solution energy), which in itself requires that one can Medal address, could still ask “what do we requires an understanding of phase transi- characterize the states of the system in a know that van der Waals did not know?”, and tions, not least the primordial symmetry- ‘thermodynamic limit’ of infinite system size. answer “not very much”1. In particular, he breaking transitions that distinguished the But too much generality may be no bad thing, was well aware not only of the gas–liquid fundamental forces and gave particles their if it drives home the message that phase critical point (which his equation predicts) masses by means of the Higgs mechanism. transitions occur not only when a liquid but also of the existence of critical exponents, And in its most generalized form, statistical freezes or evaporates but also throughout the which describe mathematically how various physics is promising to offer insights into (once sub-microscopic) Universe as it cools, properties vanish or diverge at the critical phenomena once considered outside the or in a superfluid as its viscosity vanishes. The point. It is at this unique point in the ‘phase physicist’s domain: traffic flow, economics, point is that phase transitions are global and space’ of temperature, pressure and density cell biology and allometric scaling (the abrupt — they show matter behaving at its that a liquid and gas cease to be distinct and relation of biological functions to body most nonlinear, with effects quite out of separated by a phase transition: above the mass), to name a few. proportion to cause. critical temperature, there is only one fluid NATURE | VOL 402 | SUPP | 2 DECEMBER 1999 | www.nature.com © 1999 Macmillan Magazines Ltd C73 impacts phase. Thermally driven fluctuations in scales (Fig. 1). (Gaussian random noise, in always very evidently phase transitions — density of the liquid and gas (caused by the contrast, generates fluctuations of a charac- abrupt changes from a resistive to a mere randomness of particle motions) teristic average amplitude.) non-resistive state, from a viscous to a become increasingly pronounced as the The principle of renormalization is non-viscous fluid. It was not until 1938, how- critical point is approached; and their to capture the fundamental probability ever, that the connection to statistical physics range becomes infinite exactly at criticality, distribution of the different states of the was made, when Fritz London pointed out4 dragging with them so-called ‘response system by ‘coarse-graining’ — a kind of that superfluidity in liquid helium might be functions’ such as the fluid’s compressibility. mathematical squinting that eliminates the result of a kind of quantum condensation From the 1960s to the 1980s, nothing extraneous detail. In a lattice model such as transition, in which the particles of the sys- obsessed statistical physicists more than the Ising model, where the particles occupy tem become bosonic (that is, having integer critical points. It seems strange, at first glance, sites on a regular grid, this involves calculat- spin) and so capable of occupying a single that so much attention should be focused on a ing the average state of blocks of sites of quantum state. It became generally accepted specific location in the phase diagram; but the specified size. Thus, whereas in the Ising that these phenomena were examples of reasons are twofold. First, the behaviour of a model each site is assigned a two-state Bose–Einstein condensation, although the system at its critical point also determines variable (‘up’ or ‘down’ spin, say, represented exact connection remains murky. its behaviour in the broad vicinity too, within by the black and white squares in Fig. 1), the In superconductivity the fermionic the so-called critical region. The fluctuations renormalized system contains a broader (spin-1/2) electrons become bosons by that overwhelm the system at the critical spectrum of averaged ‘block variables’. The forming Cooper pairs, a many-body effect point remain significant well beyond it; one of interaction strength between blocks is that (in the conventional low-temperature the reasons why the (controversial) idea of rescaled accordingly. superconductors) results from an effective a high-pressure, low-temperature liquid– Progressive rescaling at different block attraction mediated by the electrons’ inter- liquid critical point in water2 is so stimulating sizes smoothes out ever-larger fluctuations. action with lattice vibrations (phonons) in is that it might be expected to affect the liquid’s At temperatures either side of the critical the crystal. The full details of that process behaviour under everyday conditions. temperature, this makes the system ‘look’ were determined in the 1950s by John But second, behaviour of a system at a ever further from criticality — it begins to Bardeen, Leon Cooper and Bob Schrieffer5. critical point is like a badge of identification: resolve itself into one equilibrium state or the It is significant that this is one of the many it reveals kinships between different systems. other. Exactly at the critical point, however, phase transitions that can be described in an Liquid–gas criticality and the behaviour of rescaling creates a patchwork rather like that approximate way by the phenomenological some magnets at their Curie point (the in Fig. 1 (but with grey squares too) no theory developed by Lev Landau and Vitaly temperature above which they lose their matter how large the blocks become. The Ginzburg in the 1950s.
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