A Simple Model’s Best Hope:A Brief Introduction to UniversalityBenjamin GoodSwarthmore College(Dated: May 5, 2008)For certain classes of systems operating at a critical point, the concept of universality can offer away out of the reductive dilemma of modeling systems with many interacting parts. In these cases,simple models can b e used to classify the large scale behavior of an entire universality class. Thispap e r provides a brief introduction to the idea of universality and the renormalization group ideasused to derive it, along with a derivation of the power law behavior of certain observables.I. INTRODUCTIONIf there is one lesson to take home from Physics 120,it is that the quantitative study of biological systems –and complex systems in general – is hard. In fact, it isextremely hard. No doubt this stems in part from theexotic and diverse behavior that these systems display,with individual problems as different from each other aswhole disciplines. Yet lying underneath the teeming di-versity of the individual examples is a common theme.Each of these systems arises from the countless interac-tions of many small parts, all operating according to acertain set of rules. We se e this theme at work in systemsranging from swarms of insects to the inner workings ofthe cell — even in the large scale actions of evolution.Part of the reason that these systems are so hard tostudy is that our normal approach to modeling breaksdown. In order to accurately predict the overall behaviorof the system, we would like to specify the behaviors ofthe individual parts as precisely as possible. However,the more precisely we specify individual behavior, themore intractable the analysis bec omes when we considerthe simultaneous interaction of all the constituents. Thisdilemma clearly poses a problem for the study of anycomplex s ystem , and it has taken the work of some of thebrightest minds of the century to even begin to overcomeit in a few scattered cases.Fortunately, it can be shown that for a small set ofsystems operating at a so-called critical point, the large-scale behavior does not necessarily depend on the pre-cise nature of the individual parts. According to a con-cept from statistical physics called universality, systemsas disparate as magnets and fluids behave almost iden-tically near a critical point, and these behaviors in turnare almost identical to even simpler abstract models. Theconcept of universality can offer a way out of the model-ing problem, as the overall behavior of systems belongingto the same universality class can be deduced from thestudy of extremely simple models that can often be solvedexactly.A thorough understanding of what universality is –and what it isn’t – could be an invaluable tool for thoseembarking on the study of the types of complex systemswe see around us every day. And while the language ofthe theory is currently couched in the mindset of sta-tistical physics, the underlying concepts are pertinent toa much broader range of problems and really say some-thing about the nature of modeling process itself. Thepurp ose of this paper is to provide the quickest possibleintroduction to the theory behind universality using onlyconcepts usually encountered in the study of simple pop-ulation dynamics. In order to do so, some of the morecomplicated features of the theory must be ignored, butthis does not hinder the the basic concepts. For a morecomplete introduction to the theory of universality, aswell as an analysis of some specific examples, the readershould consult the references provided at the end of thepaper.II. THE RENORMALIZATION GROUPAlthough certain experiments in statistical physicshinted at the notion of universality early in the century,it was K.G. Wilson’s Nobel prize-winning application ofthe Renormalization Group in 1971 that finally put theconcept of universality on a firm theoretical foundation.A thorough understanding of universality requires us tolay out some of the basics of this theory, which we elab-orate in the following sections.A. A “Simple” Complex SystemWe begin by considering a system made out of N inter-acting parts, where N is some large number. For the sakeof simplicity, each part will be represented by a variables which can take on the values ±1. The interactions be-tween these parts are mo deled by a “Hamiltonian” func-tion [1] given byH = hXisi+Xi,jKi,jsisj+Xi,j,kKi,j,ksisjsk+ . . . (1)where h and K represent the coupling strength of the in-teractions. In ge neral, these coupling constants would becomplex functions of the parameters of the system, suchas temperature, magnetic field, etc. As a specific exam-ple, a one-dimensional system with only nearest neighbor2interactions would have a Hamiltonian given byH =XiKsisi+1. (2)Once we have a Hamiltonian, we can define a “partitionfunction”Z = Trse−H, (3)where the operation Trsmeans to sum over all the possi-ble combinations of values for all the various si. This par-tition function is defined such that the probability of find-ing the system in a microstate x (i.e., s1= 1,s2= −1,. . . )is given byP (x) =e−H(x)Z. (4)To complete the description of our simple system,imagine that we can define a “free-energy” function ofthe formf = −ln ZN(5)and that there exists some measurable large-scale prop-erty of the system given byC =∂2f∂T2, (6)where T some adjustable parameter like temperature.Such properties are common to the systems discussed instatistical physics, although their immediate applicationto more general systems is not always clear.In addition to this large-scale property C, we can alsodescribe the state of the system by means of a “correla-tion function” Γ(~r) which measures the tendency for anytwo variables separated by a “distance” ~r to have thesame value. A simple correlation function is given byΓ(~r) = hsisji − hsi2, (7)where the h. . .i denote the average taken over all suchvariables. Since we might normally expect the correlationto decay to zero with increasing distance, we can write Γin the formΓ(~r) ∼ r−τe−r/ξ, (8)where ξ is the correlation length. One definition of a crit-ical state is one in which the correlation length becomesinfinite and meaningful correlations be tween variables ex-ist at all length scales. We will see how this arises fromthe general theory later on.B. Renormalization Group TransformationsThe renormalization group ideas are principally con-cerned with how the behavior of a system