My Research: FAQ

Nigel Goldenfeld

What are your areas of research?

My research is mainly (but not exclusively) in areas related to condensed matter physics, statistical physics and applied mathematics. I work closely with a number of other faculty both here and elsewhere, and both experimentalists and theorists. Some of my recent work has been in collaboration with Professors Jon Dantzig and Yoshi Oono and also the high temperature experimental groups at Urbana and elsewhere. Key themes of my research are the emphasis on describing and predicting real experiments, the development of special tools to cope with systems spanning many disparate scales of space and time, the use of renormalization group methods, and the creative use of computers to augment pencil and paper thinking.

Brief summaries of my research areas follow:

• Pattern formation in nonlinear systems far from equilibrium. For example, I have studied how snowflakes and solidifying metals or alloys form beautiful and complex spatial structures. This research has as its goal the prediction and control of material properties given the conditions of formation. We have discovered over the last decade or so how to account semi-quantitatively for many of the aspects of realistic solidification patterns, how to simulate the difficult nonlinear equations effectively on a computer, and how to predict morphologies that arise under given growth conditions.   We collaborate with experimentalists who investigate these issues from a fundamental perspective; some of the relevant experiments are performed in microgravity on the space shuttle.  These problems are interesting to physicists because they require simultaneous consideration of many disparate length and time scales, and the results yield interesting scaling behaviour.  To cope with these challenges, we have used techniques such as phase field models, adaptive mesh refinement, level set dynamics, matched asymptotic expansions, asymptotics beyond all orders, Monte Carlo simulation and the renormalization group.  Future work is aimed at including impurity and fluid flow effects in addition to heat transfer. 
  
  
• Kinetics of phase transitions. For example, how quickly does a metal change into the superconducting state from the normal state as the temperature is lowered? What time-dependent patterns would one see if one could look at (e.g.) the magnetic field or order parameter distribution in the system? How does liquid helium become a superfluid? Answer: through the creation of a complicated tangle of quantum vortices, which subsequently annihilate in a scale invariant way.  Similar physics has illuminated current discussions of the formation of structure in the early universe.    In general, phase transition kinetics are dominated by the behaviour of the symmetry-allowed topological defects which proliferate during the transition, and ultimately annihilate.  Our research focuses on the statistical properties of the patterns of defects created during the dynamics of phase transitions.  In fact, it turns out that these patterns exhibit statistical self-similarity and interesting scaling behaviour, all of which it is challenging to reproduce and explain.  We are currently attempting to predict the microstructure of the simplest pattern forming system in nature -- a mixture of He3 and He4 atoms -- starting from quantum mechanics.  Our hope is to be able to use microscopic techniques, such as path integral Monte Carlo methods, to calculate the parameters for a semi-macroscopic model of the phase separating helium binary fluid.  
  
  These systems form patterns which grow arbitrarily large as time proceeds: hence it is necessary to simulate large systems, and to develop computationally efficient tools.   We have used novel numerical techniques -- examples of underresolved computation -- to extract the universal dynamics without needing to numerically resolve all of the fine detail.  By focusing on the relevant scales of interest, computations of universal quantities have been performed and compared with experiment: the result is a perfect match, with no adjustable parameters. 
  
  
• High temperature superconductivity. My work was in part responsible for the now generally agreed upon identification of the pairing state (i.e. the orbital angular momentum state) of the Cooper pairs in the cuprate superconductor YBCO. I analysed experimental data, particularly that related to the electromagnetic penetration depth and its temperature dependence. My approach was to seek features that would be universal, and therefore independent of the specific mechanism of superconductivity. We discovered that the temperature dependence of the penetration depth at low temperatures, and the way in which it varied when small concentrations of impurities were added to the superconductor, was incompatible with s-wave pairing, and strongly suggestive of a d-wave pairing state.  This was confirmed in the now famous quantum interference experiment performed at Illinois by Dale van Harlingen's group.  
  
  We also discovered that the behaviour of the penetration depth near the critical temperature was incompatible with the predictions of the Landau-Ginzburg theory of superconductivity, and in fact exhibited the behaviour expected of the three dimensional XY model (3D XY).  This was the first time that non-trivial critical behaviour was observed in a 3D superconductor, and our original results have now been extended in super-pure YBCO so that scaling is observed over three decades of reduced temperature --- to my knowledge, a record range of scaling in a solid state system.
  
  Our current research is directed towards three ends:
  
          (1) Understanding of the role of phase fluctuations at low temperatures
  
          (2) Understanding the dynamic critical behaviour of the optimally-doped high temperature superconductors
  
          (3) Exploring the nature of the pseudo-gap state in under-doped cuprates, using quantum interference probes
  
  
  
• Renormalisation group theory applied to partial differential equations. In trying to understand how scaling phenomena can occur in nonequilibrium systems, such as turbulent fluids, I discovered that the renormalisation group (RG) can be used to do asymptotic analysis of differential equations. The RG was originally developed in quantum electrodynamics and was extended to equilibrium statistical physics and critical phenomena about 25 years ago. Remarkably, there are phenomena in nonequilibrium systems that ressemble critical phenomena; these and other singular mathematical behaviour can be fruitfully described by RG.  During the course of our project we have discovered how to compute anomalous scaling or power law behaviour in partial differential equations, such as those describing the flow of fluids in porous media or the propagation of turbulent fluids; we have discovered how to systematically analyse front propagation as a stable state invades a dynamically unstable state; we have discovered how to use renormalization group techniques to simplify and improve the analysis of singular perturbations, conventionally treated by matched asymptotic expansions and other methods; and we have developed novel numerical techniques based on the renormalization group for dealing with singular behaviour in spatially-extended dynamical systems.  Our work has been used to study systems as disparate as the pattern formation of particle beams inside high energy physics accelerators, the dynamics of patterns in Rayleigh-Benard convection experiments, and the formation of black holes.
  
  
• Turbulence in fluids. This is a long-unsolved problem, a graveyard for physicists. I am trying to approach it from a new perspective. To this end, I have participated in an experimental project to use superfluid helium as the test fluid. Superfluid helium was studied by Landau and others at low Reynolds numbers, where it flows without dissipation. But at high Reynolds numbers, quantum vortices penetrate the fluid and there is dissipation and viscosity. Superfluid helium has the lowest kinematic viscosity of any fluid, and so offers the best prospect of achieving very turbulent flows. With Russell Donnelly's group at the University of Oregon, we have studied how turbulence dies away once the forcing element is removed, and how it propagates into quiescent regions of a fluid.  With Grigory Barenblatt at UC Berkeley, we have discussed how fully-developed turbulence may only exist in a literal sense at infinite Reynolds number; and with Gregory Eyink, we have proposed how turbulent behaviour may exhibit scaling related to the finite-size scaling of equilibrium phase transitions.  These latter ideas may be borne out in recent experiments on the probability distribution of fluctuations in a closed turbulent flow. 
  
  Ongoing and future work, in collaboration with Alan McKane at the University of Manchester concerns the use of renormalization group methods to solve stochastic differential equations arising in turbulence and other problems.
  
  
• Statistical mechanics of polymers and liquid crystals. What is the equilibrium phase diagram of a complex fluid, such as one composed of liquid crystals, polymers and other materials? The important point here is that the primitive objects are not point particles but have other degrees of freedom: they may be rod-like, spaghetti-like, or have different types of specific chemical activity (such as hydrophobic groups etc.). What is the kinetics of such phase transitions? How does the non-trivial topological structure arising from the complexity of the fluid influence the kinetics? Another area is the physics of rubber or gels: how do such systems become solid as they are cross-linked? Recall that such materials are not built out of rigid objects; they are built from microscopically flexible objects (polymer chains) but nevertheless become macroscopically rigid after cross-linking. Such surprising behaviour is a consequence of spontaneous symmetry breaking, a well-worn theme in condensed matter physics, but technically intricate in a disordered, spatially random system such as a gel or rubber.  
  

In addition to these areas, I have worked on a number of other diverse projects including the mechanism of hearing, dynamics of steps on crystal surfaces, and cosmological phase transitions.

What funding is available?

At present I have three students and two postdocs, fully funded through NSF and NASA grants.

What preparation should I have?

My research invariably involves a mixture of computational and analytical work. I do not generally use computers to do conceptually simple problems that happen to be complicated due to excessive realism. Instead, I try to use computers as a tool to uncover the qualitative properties and mathematical structure of problems. Prospective students must be comfortable with treating computational physics and analytical physics on an equal footing. The language of modern day condensed matter physics includes quantum field theory, statistical mechanics, fluid dynamics and partial differential equations. I hope that my students will learn these subjects as they embark on research. Although some basic knowledge is required, I believe that one should develop or learn skills on the job. I don't expect my students to ``learn everything first, then apply it".

The most important preparation is to develop a sense of curiosity and fun. You cannot do physics without wanting to ask questions. It is almost certain that all the ideas I have about future research directions will not be right. So students must be prepared to face tough challenges. I always try to start off my students with what looks like a relatively simple problem, to gain confidence and get some results relatively quickly.

How long do your students take to graduate?

I try to have my students graduate in four years. If a student has a particular (e.g. personal) reason to want to stay a fifth year, I will try to accomodate that. Four years should be sufficient to get worthwhile scientific results, and attain enough scientific maturity to compete effectively in either the academic or real worlds.

What do your students do after they graduate?

So far, I have supervised 6 students to completion of Ph.D. Two others have left physics for other careers early on in their research. The students who completed a Ph.D all had the opportunity to do good postdocs (three were accepted for their first choice postdoctoral position). One former student is raising his children and working on physics education, one works for Hughes Research Labs, and four others work on Wall Street.

Updated by Nigel Goldenfeld
Jan 2000