Nigel Goldenfeld's Group: Key Accomplishments






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Our key accomplishments are summarized briefly here:

  1. Discovery of the role of anisotropy and the steady state selection mechanism in diffusion-controlled growth, stability analysis of dendrites, in collaboration with Eshel Ben-Jacob, Jim Langer.

  2. First experimental exploration of the pattern phase diagram for diffusion-controlled growth, from our identification of the analogy with Hele-Shaw flow, in collaboration with Eshel Ben-Jacob.

  3. First large-scale, systematic simulations of phase ordering in 2 and 3 dimensions for quenched systems with continuous symmetries, including the calculation of universal scaling functions for topological defect correlation functions. Systems studied include XY models (superfluids, magnets), liquid crystals (isotropic-nematic), superconductors, and miscut crystal surfaces. This program was influenced by collaboration with Yoshi Oono at Illinois.

  4. First multiscale phase field model code, using adaptive mesh refinement and finite elements. Capable of simulating complex, realistic solidification microstructures, even in 3 dimensions, with fluid flow effects. In collaboration with Jon Dantzig.

  5. Development of the renormalization group for deterministic partial differential equations (PDEs) describing non-equilibrium phenomena. First analytic calculation of anomalous dimensions in a nonlinear diffusion equation. Development of numerical RG methods for partial differential equations. In collaboration with Yoshi Oono.

  6. Development of the renormalization group for singular perturbation problems, unifying boundary layer, matched asymptotics, multiple scales analysis and reductive perturbation theory, and with many potential applications in fluid dynamics and other fields. Our theory enables coarse-grained amplitude and phase equations to be derived from underlying PDEs. In collaboration with Yoshi Oono.

  7. Developed the first order parameter theory for the vulcanization transition in cross-linked polymers, which proved to be capable of predicting architectural, thermodynamic and dynamic scaling properties of cross-linked systems. In collaboration with Paul Goldbart at Illinois.

  8. Proposed a d-wave pairing state for the cuprate superconductor YBCO, and provided the first comprehensive review of experimental evidence to test this hypothesis. Discovery of the power law nature of the penetration depth at low temperatures in data previously thought to be iron-clad evidence for s-wave pairing. Calculation of the influence of impurity scattering on the low temperature penetration depth and successful comparison with experimental data ruled out s-wave pairing, and was quantitatively consistent with d-wave pairing (see also this review).

  9. First to observe a convincing critical fluctuation regime in YBCO (now known as 3D XY scaling) in data analysis of experimental work with the University of British Columbia superconductivity group. In the best samples, an unprecedented 3 decades of power law scaling near the superconducting transition can be observed, a record (I believe) for any solid state system.

  10. First comprehensive treatment of the dynamic critical phenomena at the superconducting transition, resolving puzzles in data from computer simulation and experiment; nonlinear scaling theory for the dynamic critical behavior of the superconducting transition, verified in collaborative work with Dale van Harlingen's experimental group at Illinois.

  11. First measurements and theoretical analysis of fully-turbulent superfluid helium, with prediction and observation of the decay of turbulence and the demonstration that a turbulent superfluid behaves like a turbulent normal fluid. In collaboration with Russell Donnelly.

  12. Discovery of roughness-induced criticality in pipe flow turbulence, a scaling theory for the way in which the friction factor of turbulent pipe flow veries with Reynolds number and roughness. Theory and computation for the case of turbulent soap-films, highlighting the role of enstrophy and demonstrating that the Prandtl theory of turbulent boundary layers is incomplete.

  13. Development of statistical mechanics of genes, with particular emphasis on horizontal gene transfer and evolution. Proposed mechanism for microbial speciation following horizontal gene transfer event, together with a comparative genomics test of the theory. Theory for the effects of horizontal gene transfer on the phase behavior of biofilms. Identification of a nonlinear instability over evolutionary time in the evolution of microbial genomes, explaining the observed pattern of genome biases.

  14. Proposed dynamical mechanism for the evolution of the genetic code, leading to its optimality and universality, and suggesting the collective nature of early life. In collaboration with Carl Woese.

  15. Theory of carbonate precipitation pattern formation to describe the evolution of landscapes at geothermal hot springs. First time-lapse movie of rocks growing, and the determination of scaling laws in the pond/terrace statistics.

Note: I like to collaborate with my students, postdocs and faculty colleagues, and the full credits for these works can be found from the publications themselves, where typically, students are listed first. Principal collaborators are mentioned here. Links above are to some but not all of the relevant publications. Full references are obtainable here.

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