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. This stemmed from a ferromagnetism) have numerically equal probability distribution of block variables very general mean-field model of phase critical exponents, and both can be modelled settles down to an invariant form, peaked transitions proposed earlier by Landau, by the so-called Ising model, a lattice of on ‘black’ and ‘white’. The way in which whose contribution to this area of con- two-state spins. Commonality of critical this ‘configuration flow’ evolves with densed-matter physics was pivotal. Landau’s exponents gives rise to the idea of universali- changing length scale allows one to theory — a kind of generalized and aug- ty — that is, there are generic models in determine the precise value of the critical mented all-purpose van der Waals equation statistical physics that describe a variety of exponents — which are generally different, — remains the first port of call for any apparently different many-body systems. in one, two and three dimensions, from their simplified model of a phase transition. This means that solving one statistical mean-field values. The observation of Bose–Einstein con- mechanical problem generally delivers densation in atomic matter, seen for the first solutions for several others at the same time; Quantum transitions time in cooled sodium atoms in 19956, was it also implies that, fundamentally, many- One of the richest veins of statistical itself another instance of a quantum phase body behaviour is determined only by mechanics presently being mined is found at transition — made possible now by laser broad-brush features such as the range of its intersection with quantum mechanics. cooling techniques. interparticle forces, the dimensionality, and In particular, the many-body behaviour of But is there any systematic way to accom- the nature of the ‘order parameter’ whose electrons in condensed matter is extra- modate quantum effects into the existing abrupt change from zero to a non-zero value ordinarily rich. Correlated behaviour of framework of statistical mechanics? The defines the transition. electrons, in which they display a degree of intense interest in correlated electron Actually obtaining numerical values for collective or coherent dynamics, produces systems has now provided something of the critical exponents from first-principles for example superconductivity, the integer kind. Conventionally, phase transitions are theory is, however, another matter. Mean- and fractional quantum Hall effect (quanti- induced by changes in temperature — ice field models offer analytic solutions, but they zation of the Hall resistance, a measure of melts when heated, hot iron orders magneti- are strictly approximations in anything less the voltage generated transverse to a current cally when cooled. All this is driven by than four spatial dimensions. Lars Onsager’s in a flat conducting sheet by an applied changes in thermal fluctuations. ‘Pure’ tour de force in 1944 was the exact solution of magnetic field), heavy-fermion behaviour quantum phase transitions, meanwhile, take the two-dimensional Ising model, providing (where conduction electrons acquire a very place at zero temperature, and are induced exact numbers for the critical exponents. But large ‘effective mass’), spin density waves and by altering some other parameter, such as the three-dimensional Ising model contin- colossal magnetoresistance (CMR: a strong the strength of an applied magnetic field, ues to rebuff the advances of theoreticians, variation in electrical resistance owing to an that affects quantum fluctuations. Classical and may be analytically insoluble. applied magnetic field). All of these fluctuations are frozen out at zero kelvin, but On the other hand, Kenneth Wilson’s collective phenomena have in recent years quantum fluctuations, which are a conse- renormalization group theory provided in been shown to underlie unexpected and quence of the uncertainty principle, remain. the 1960s and early 1970s a methodology for potentially useful properties of novel materi- Their effect can be enhanced by altering computing critical exponents numerically3. als: CMR, for instance, could potentially some variable that alters the particles’ (gen- It also pointed to the scale-invariant furnish highly sensitive read-out heads for erally the electrons’) state of localization, behaviour of fluctuations at the critical magnetic memories. which is the equivalent in this case of altering point: the fact that they occur on all length Superconductivity and superfluidity were the temperature.

Quantum phase transitions provide a considered to be time-reversible: an job of capturing some of this behaviour, and wider screen on which to scrutinize the encounter can be run in reverse without ‘toy’ equations such as the Swift–Hohenberg still-enigmatic high-temperature supercon- becoming a physical nonsense. So in what equation provide a general description of ductivity of layered copper oxide manner does the Second Law break down in some of the symmetry-breaking processes compounds. Superconductivity is now going from irreversible macroscopic systems that occur9; but non-equilibrium systems recognized as just one of the manifestations to reversible microscopic ones? And what seem particularly prone to the influences of of the rich physics of the correlated electrons is the relationship to the chaotic dynamics boundary conditions, defects and noise, and in these systems. There are magnetic interac- evident even in few-particle (sometimes remain resistant to too much generalization. tions between the copper ions, which even two-particle) systems? Perhaps instead All the same, some useful broad princi- bear spins by virtue of their unpaired there is some feature of the time-dependent ples have appeared, among them the idea electrons. The prototype, lanthanum stron- evolution of probability distributions that that certain non-equilibrium systems have a tium copper oxide (close to the material forces irreversibility irrespective of scale? In kind of imposed criticality said to be studied by Georg Bednorz and Alex Müller other words, might irreversibility be built in self-organized, in the sense that they will in 19867), is now recognized as an example at the microscopic level (contrary to the constantly return to a precarious critical of a diluted quantum antiferromagnet: normally accepted tenet of microscopic state following some transient instability diluted, that is, by the ‘dopant’ strontium, reversibility)? Furthermore, the Boltzmann such as a landslide10. The characteristic which contributes positively charged mobile position requires that, as Richard Feynman statistics of such systems, dubbed 1/f behav- charge carriers (holes) to the copper-oxide put it, “For some reason, the universe at one iour because of the inverse relationship layers. These holes may segregate into time had a very low entropy”. But for what between the size and frequency (f) of a stripes, and the intervening phase, an insulat- reason? fluctuation, provides a kind of fingerprint ing antiferromagnet, undergoes a magnetic Second, there is the question of whether a that alerts the observer to the likely appear- quantum phase transition at some critical thermodynamic (and corresponding micro- ance of scale-invariant structure and doping level. scopic) description of non-equilibrium dynamics. Self-organized criticality seems to The behaviour of such systems at energy systems can be developed that mirrors the be a promising candidate for a general mech- scales corresponding to temperatures above mature description of equilibrium states. As anism capable of forming the fractal 100 K or so is now well understood; but the most processes of interest, from atmospheric structures so prominent in nature, from transition to a superconducting state at circulation to cell metabolism, take place out mountain ranges to the large-scale structure lower temperatures involves additional of equilibrium, the issue is a pressing one. of the Universe. There is a clear kinship with interactions of the electronic and spin A key question is whether variational or the fractal ‘optimal channel networks’ that degrees of freedom that have yet to be minimization principles exist for selecting have been posited11 as models of real river elucidated. At least one Nobel prize lies in the most stable dynamic state, as is the case at drainage networks. These are formed under wait for this enterprise. equilibrium. Several have been suggested, the assumption that the evolution must amongst them Ilya Prigogine’s minimal rate minimize the rate of potential-energy Beyond equilibrium of entropy production for systems close to dissipation in the water flow. The resulting Some of the major unresolved questions equilibrium; but there is no consensus, and topology of the drainage basin shows non- about the behaviour of both everyday and at least some indication (the late Rolf random 1/f statistics. To what extent this exotic matter are not so much a matter of Landauer’s ‘blowtorch theorem’8) that there model captures the essential features of real being as of becoming. How does change in can in fact be no universal principle of this river systems is still a matter of debate. these many-particle ensembles occur? Tradi- sort, independent of a system’s past history, And it remains to be seen whether tionally, thermodynamics has concerned away from equilibrium. In short, it matters self-organized criticality itself will prove to itself with equilibrium states. The particles not only what state a system is in, but how it be generalizable to forest fires, epidemics, might be in frantic motion, but the macro- got there. solar flares and the countless other physical scopic properties are constant observables of In fact, for all that one can postulate that systems in which such statistics have been the states that minimize free energy: pres- entropy production must be positive in reported. But the canonical example, the sure, density, temperature, magnetization non-equilibrium processes (so that the sand pile, points to another growth area in and so forth. But how do systems actually get Second Law is not violated if and when studies of the behaviour of matter — granu- from state to state (particularly in cases of equilibrium is reached), it is not even clear lar media. Capable of both solid-like and violent change, such as explosions or how to calculate the production rate because fluid-like behaviour, dominated by dissipa- fracture)? And what about systems that do there is no generally accepted definition tive collisions and essentially independent of not reach equilibrium at all? of entropy away from equilibrium. temperature, granular media represent a First, there is the issue of irreversibility, Boltzmann’s S = klnW (relating entropy S to new class of problem for many-body which is to say, of the Second Law of available microstates W) will not give it to us, theorists. There is still no general physical Thermodynamics. Thermodynamic change nor will any other general law. While that is framework, comparable to the kinetic model has a directionality to it: entropy increases. so, a quantitative non-equilibrium statistical of gases, able to predict the stupefying range The question is why. The answer most would mechanics remains hard to formulate. One of collective and self-organized behaviour give is Boltzmann’s, which is purely possible way forward is to use as the appro- exhibited by grains in motion: size segrega- probabilistic: entropy being related to the priate variables of dynamical steady states tion in vertical shaking and landslides, number of configurations available to the the time-invariant probability measures formation of ordered patterns12 (Fig. 2), ensemble, those states will be achieved that developed in the 1970s by Yakov Sinai, David liquefaction, hysteretic angles of instability have overwhelmingly greater probability. In Ruelle and Rufus Bowen. and so forth. Quite aside from the relevance the words of US physicist Josiah Willard But as yet, the theory of non-equilibrium of much of this for industrial powder Gibbs, “the impossibility of an uncompen- remains largely a heuristic one. It is evident processing, there is the matter of whether it sated decrease in entropy seems to be that lack of equilibrium does not imply lack might cast light on the geomorphology of reduced to an improbability”. of structure — many if not most of the grainy media distributed by wind and wave. But questions, even lingering doubts, richest patterns in nature (Fig. 2) are formed Traffic flow is now understood to be a persist. At the microscopic scale, the interac- out of equilibrium. Microscopic models special case of granular flow, and exhibits tion between two particles is generally such as reaction–diffusion schemes do a fair dynamic states that bear striking analogy to

NATURE | VOL 402 | SUPP | 2 DECEMBER 1999 | www.nature.com © 1999 Macmillan Magazines Ltd C75 impacts the equilibrium states of matter. In low- entropic effect of fluctuations on interac- density ‘free’ flow, each vehicle moves more tions of lipid membranes — there remains or less independently, like a gas. A traffic jam much scepticism as to whether any biological at high densities is resolutely solid-like, with phenomena can arise from the sort of equidistant particles trapped in near or total collective, emergent behaviour of statistical, immobility. But jams rarely nucleate from interacting ensembles rather than the closely free flow; instead, it seems that a dense, controlled protein relays to which cell biolo- congested yet mobile state intervenes, the gists are accustomed. Yet statistical physics highway equivalent of a liquid state. Changes must inevitably provide the baseline even in from one state to another seem to have the the cell: proteins may phase-separate and abrupt character of a first-order phase membranes may adopt equilibrium confor- transition (like freezing or melting), relying mations unless actively opposed. The recent on the presence of fluctuations to nucleate interest in thermal ratchets16 — Brownian the change. systems that achieve directional motion Perhaps the hardest of non-equilibrium by virtue of operating in an asymmetric problems — turbulence — remains as underlying potential — attests to the contri- obstinate as ever. The mathematician Sir butions that microscopic physical models Horace Lamb’s famous quip — that he was might make to cell biology. These models more optimistic about receiving heavenly may or may not, in the end, have much to do enlightenment on quantum electrodynamics with the way that motor proteins work; but than on turbulence — proved prophetic in they demonstrate that physics offers creative terms of which would yield first, and turbu- solutions to adaptive biological systems. The lence still retains something of its reputation same can be said for the idea that stochastic as a ‘graveyard of theories’. From a statistical resonance — the (counterintuitive) noise- physical perspective, the situation is night- Figure 2 Complex, ordered patterns form in induced amplification of a signal17 — is marish: every parcel of fluid acts non-trivially vertically oscillated thin layers of grains. (From exploited in biological signal transduction. on the others, there is structure at all length ref. 12.) There now seems to be good evidence that scales, dissipation is very strong, and the this mechanism has some role in neural pro- solutions are wholly time-dependent. Ideas cessing, and one might even be surprised if it from critical phenomena might prove helpful magnetic spin glasses known in condensed- did not turn out to be more general. In both for characterizing the scaling behaviour of matter physics has helped to explain the wide of these cases, we see how noise — inevitable turbulent flows. Yet questions remain, for range of protein folding speeds (a function of in any environment — can, with the right example, about just how many distinct the degree of frustration) and has provided adaptation, serve a functional purpose. regimes of thermal turbulence (as a function the useful concept of a folding ‘funnel’, the The considerable redundancy evident in of Rayleigh number, proportional to the broad basin in the energy landscape that the cell’s machinery — so that cells continue temperature gradient) there are. Each regime surrounds the deep well of the native fold14. happily with disabled genes that were sup- is characterized by a specific scaling law posedly central to their survival — suggests relating Rayleigh number to heat transport; Is physics rigorous? the need for some kind of nonlinear collec- recent predictions of an ‘ultrahard’ regime It can be remarkably hard to prove rigorously tive modelling of the interactions among its relevant to atmospheric dynamics have been the development of genuine long-ranged components. This is the kind of thing that met with conflicting experimental results13. (crystal-like or ferromagnetic) order in an physicists have been doing for years, and in Even in equilibrium, a strong element of equilibrium phase transition — particularly increasingly complex systems. Cells provide disorder in a system complicates the picture. if the degrees of freedom are continuous perhaps the ultimate challenge, although the The glass transition withholds some of its rather than discrete (for example, in Heisen- modelling will have to be dosed with a strong mysteries still, exemplifying a whole range of berg rather than Ising models of magnetic element of biological good sense. systems — among them spin glasses and states, where the spins can take any orienta- Philip Ball is a consultant editor for Nature. folded proteins — in which the problem is tion). We ‘know’ from numerical modelling e-mail: [email protected] that of finding a global energetic minimum that such systems do adopt long-range 1. Widom, B. Physica A 263, 500 (1999). in a rugged ‘energy landscape’. How this order in appropriate circumstances; but we 2. Mishima, O. & Stanley, H. E. Nature 396, 329 (1998). 3. Bruce, A. & Wallace, D. in The New Physics (ed. Davies, P.) landscape is explored as a liquid is lowered cannot prove that they do so rigorously. (Cambridge Univ. Press, 1989). towards its glass transition is in need of Physicist Elliott Lieb’s comment on this issue 4. London, F. Nature 141, 643 (1938). further clarification before a truly thermo- applies equally to many are§as of physics: 5. Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Phys. Rev. 108, 1175 dynamic description of this transition can be “One might ask why one should bother (1957). 6. Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E. developed. The conceptual tools needed for to prove rigorously what is ‘physically & Cornell, E. A. Science 269, 198 (1995). the job are also being usefully brought to obvious’ and the answer is that ‘Not 7. Bednorz, J. G. & Müller, K. A. Z. Phys. B – Cond. Matt 64, 189- bear on the vexed issue of protein folding. everything that is obvious is true and the 93 (1986). This compares with a glass in the sense that ability to prove something interesting about 8. Landauer, R. Physical Rev. A 12, 636 (1975). 9. Cross, M. C. & Hohenberg, P. Rev. Mod. Phys. 65, 851 (1993). the system of particles (in this case interact- a model usually requires an additional 10.Bak, P., Tang, C. & Weisenfeld, K. Phys. Rev. Lett. 59, 381 ing residues on the polypeptide chain) has a degree of physical understanding that goes (1987). great many configurations that correspond beyond intuition; in other words, we learn 11.Rodriguez-Iturbe, I. & Rinaldo, A. Fractal River Basins to local free-energy minima, but only one — something new about physics’.”15 (Cambridge Univ. Press, 1997). 12.Melo, F., Umbanhowar, P. B. & Swinney, H. L. Phys. Rev. Lett. the native fold — that equates with the global Such rigour is, however, surely an impos- 75, 3838 (1995). minimum. The protein experiences ‘frustra- sible dream when one comes to apply the 13.Glazier, J. A., Segawa, T., Naert, A. & Sano, M. Nature 398, 307 tion’ as it folds: an amino-acid residue elegance of statistical physics to the orches- (1999). cannot simultaneously optimize interac- trated chaos we call life. Despite the proven 14.Wolynes, P. G. & Eaton, W. A. Physics World 12(9), 39 (1999). 15.Lieb, E. Physica A 263, 491 (1999). tions with all its neighbours. Mapping value to cell biology of some concepts from 16.Astumian, R. D. Science 276, 917 (1997). this situation onto the case of frustrated the study of phase transitions — such as the 17.Wiesenfeld, K. & Moss, F. Nature 373, 33 (1995).