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Problems in Physics wi th Many Scales of Length Physical systems as varied as magnets and fluids are ahke J'n having fluctuations in structure over a vast range of sizes. A novel method called the renormahzation group has been invented to explain them by Kenneth G. Wilson ne of the more conspicuous prop­ temperature is raised. Near the criti­ an alloy at the temperature where two O erties of nature is the great di­ cal point water develops fluctuations in kinds of metal atoms take on an orderly versity of size or length scales in density at all possible scales. The fluctu­ distribution. Other problems that have a the structure of the world. An ocean, for ations take the form of drops of liquid suitable form include turbulent flow, the example, has currents that persist for thoroughly interspersed with bubbles of onset of superconductivity and of su­ tho usands of kilometers and has tides gas, and there are both drops and bub­ perfluidity, the conformation of poly­ of global extent; it also has waves that bles of all sizes from single molecules up mers and the binding together of the ele­ range in size from less than a centimeter to the volume of the specimen. Precisely mentary particles called quarks. A re­ to several meters; at much finer resolu­ at the critical point the scale of the larg­ markable hypothesis that seems to be tion seawater must be regarded as an est fl uctuations becomes infinite, but the confirmed by work with the renormali­ aggregate of molecules whose charac­ smaller fluctuations are in no way di­ zation group is that some of these phe­ teristic scale of length is roughly 10-8 minished. Any theory that describes wa­ nomena, which superficially seem quite centimeter. From the smallest structure ter near its critical point must take into distinct, are identical at a deeper lev­ to the largest is a span of some 17 or­ account the entire spectrum of length el. For example, the critical behavior ders of magnitude. scales. of fluids, ferromagnets, liquid mixtures In general, events distinguished by a Multiple scales of length complicate and alloys can all be described by a sin­ great disparity in size have little influ­ many of the outstanding problems in gle theory. ence on one another; they do not com­ theoretical physics and in certain other municate, and so the phenomena asso­ fields of study. Exact solutions have he most convenient context in which ciated with each scale can be treated been found for only a few of these prob­ Tto discuss the operation of the renor­ independently. The interaction of two lems, and for some others even the best­ malization group is a ferromagnet, or adjacent water molecules is much the known approximations are unsatisfac­ permanent magnet. Ferromagnetic ma­ same whether the molecules are in the tory. In the past decade a new method terials have a critical point called the Pacific Ocean or in a teapot. What is called the renormalization group has Curie point or the Curie temperature, eq ually important, an ocean wave can been introduced for dealing with prob­ after Pierre Curie, who studied the ther­ be described quite accurately as a dis­ lems that have multiple scales of length. modynamics of ferromagnets at about turbance of a continuous fluid, ignoring It has by no means made the problems the turn of the century. For iron the entirely the molecular structure of the easy, but some that have resisted all oth­ Curie temperature is \,044 degrees K. liquid. The success of almost all practi­ er approaches may yield to this one. At higher temperatures iron has no cal theories in physics depends on isolat­ The renormalization group is not a spontaneous magnetization. As the iron ing some limited range of length scales. descriptive theory of nature but a gener­ is cooled the magnetization remains If it were necessary in the eq uations of al method for constructing theories. It zero until the Curie temperature is hydrodynamics to specify the motion of can be applied not only to a fluid at the reached, and th en the material abruptly every water molecule, a theory of ocean critical point but also to a ferromagnetic becomes magnetized. If the temperature waves would be far beyond the means of material at the temperature where spon­ is reduced further, the strength of the 20th-century science. taneous magnetization first sets in, or to magnetization increases smoothly. ' A class of phenomena does exist, a mixture of liquids at the temperature Several properties of ferromagnets however, where events at many scales of where they become fully miscible, or to besides the magnetization behave oddly length make contributions of equal im­ portance. An example is the behavior of water when it is heated to boiling under MULTIPLE SCALES OF LENGTH characterize the patterns that emerge when a ferromag­ a pressure of 217 atmospheres. At that netic solid is cooled to the temperature at which it becomes spontaneously magnetized. Each pressure water does not boil until the square represents the magnetic moment associated with a single atom in the solid, and each temperature reaches 647 degrees Kel­ moment is assumed to have only two possible orientations, labeled "up" (black sqllares) and vin. This combination of pressure and "down" (opell sqllares). At high temperature (top) the orientation of the magnetic moments is temperature defines the critical point of essentially random, and so there is only short-range order in the pattern. As the temperature is reduced (middle) somewhat larger patches in which most of the magnetic moments are lined water, where the distinction between liq­ np in the same direction begin to develop. When the temperature reaches a critical value called uid and gas disappears; at higher pres­ the Curie temperature, or Tc (bottom), these patches expand to infinite size; significantly, sures there is only a single, undifferenti­ however, fluctuations at smaller scales persist. As a result all scales of length must be included ated fluid phase, and water cannot be in a theoretical description of the ferromagnet. This simulation of a ferromagnet was carried made to boil no matter how much the out with the aid of a computer by Stephen Shenker and Jan Toboch.nik of Cornell University. 158 © 1979 SCIENTIFIC AMERICAN, INC 159 © 1979 SCIENTIFIC AMERICAN, INC near the Curie point. Another property which defines the direction of the elec­ other materials is a coupling between of interest is the magnetic susceptibility, tron's magnetic field. nearby spins that makes them tend to or the change in magnetization induced A real ferromagnet has a complex line up in the same direction. This ten­ by a small applied field. Well above the atomic structure, but all the essential dency can be stated more precisely by Curie point the susceptibility is small properties of the system of spins can be pointing out that the total energy of any because the iron cannot retain any mag­ illustrated by a quite simple model. two adjacent spins is smaller when the netization; well below the Curie temper­ Indeed, I shall describe a model that spins are parallel than it is when they are ature the susceptibility is small again be­ includes no atoms or other material par­ anti parallel. The interaction responsible cause the material is already magnetized ticles but consists only of spin vectors for the coupling of the spins has a short and a weak applied field cannot change arranged in a lattice. For the sake of range, which is reflected in the model by the state of the system very much. At simplicity I shall deal with a two-di­ specifying that only nearest-neighbor ' temperatures close to 1,044 degrees, mensional lattice: a rectilinear grid of spins are coupled to each other. In the however, the susceptibility rises to a uniformly spaced lines in a plane, with two-dimensional rectilinear lattice each sharp peak, and at the Curie point itself a spin vector at each intersection of the spin is influenced by four nearest neigh­ the susceptibility becomes infinite. grid lines. Furthermore, I shall assume bors; no other spins have any direct ef­ The ultimate source of ferromagnet­ that each spin can point in only two fect on it. ism is the quantum-mechanical spinning possible directions, designated up and of electrons. Because each electron ro­ down. The model lattice is said to be rom the nature of the interaction be­ tates it has a small magnetic dipole mo­ magnetized whenever more than half of F tween spins in a ferromagnet one ment; in other words, it acts as a magnet the spins point in the same direction. might well predict that all the spins with one north pole and one south pole. The magnetization can be defined as the would always be parallel and the mate­ How the spin of the electron gives rise to number of up spins minus the number of rial would always have its maximum the magnetic moment will not concern down spins. magnetization. That is the state of low­ me here. It is sufficient to note that both Every electron has the same spin est energy, and in the absence of any the spin and the magnetic moment can and the same magnetic dipole moment. perturbing effects it would be the fa­ be represented by a vector, or arrow, What distinguishes a ferromagnet from vored state. In a real ferromagnet, how­ ever, there is one perturbation that can­ not be neglected: the thermal motion of " f\ I f\ f\ the atoms and the electrons. At any tem­ t perature above absolute zero thermal excitations of the solid randomly flip M = +4 M = +2 M = +2 M =0 ]'IP = .273 P = .037 P = .037 P = .037 some of the spins so that the direction of " " the spin vector is reversed, even when " " reversing the spin puts the magnet in a 1 state of higher energy.

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