Exploiting emergent scale invariance

Cornell University, Ithaca NY

Investigators

Abstract

NONTECHNICAL Summary This award supports theoretical and computational research and education that aims to hone a powerful fundamental theory into a practical approach toward a more realistic description of a wide variety of phenomena. Important parts of our world fluctuate on many scales. Clouds are wispy, fractals are bumpy, and Rice Krispies crackle. (So does the Earth - earthquakes). Materials, like metal alloys and the membranes of biological cells, can fluctuate on many scales, when their compositions are set near special points. Fifty years ago, the explanation for these fluctuations was discovered to be a kind of scale invariance: these systems look nearly the same after they are magnified, for example big and small cloud wisps look similar. This scale invariance was explained by a theory, describing how the laws governing these systems change when the systems are magnified. The wisps of clouds and the milk in puffed rice have rules that become the same upon changing scale. But this elegant theory has not reached its potential, for example in predicting the weather, or in describing how materials like bone, concrete, and sea-shells form and break. Science can explain the whorls within whorls of turbulence, but it needs to mesh the biggest whorls into the engineer's models designing airplane wings, or the meteorologist's seashores and mountains. This project will build on the PI's recently developed methods for solving this theory more completely and systematically. The PI aims to turn this elegant theory of physics into the bones of a practical engineering tool. In particular, far from these special, scale-invariant points there are well-established methods to calculate the properties of materials and alloys. The PI will develop tools to merge these two descriptions into an integrated, powerful tool for describing complex materials and systems. TECHNICAL SUMMARY This award supports theoretical and computational research and education that aims to extend the quantitative validity of the renormalization group and enable it to engage more realistic problems. Understanding critical phenomena and emergent scale invariance is a key to many of our scientific and engineering challenges. The current formulation of the field is split between an admiration of universal scaling laws and dense, inscrutable calculations for exactly solvable systems, for example the 2D Ising model, and systems near special points, for example the Ising model in 1D and 4D. These latter calculations often violate the naive power laws and Widom scaling usually expected. The PI has recently used the mathematics of bifurcation theory to arrange these anomalous universality classes into universality families, characterizing the logarithms, exponentials, and invariant combinations of scaling variables, many for the first time. These new methods are closely associated with known techniques for incorporating analytic and singular corrections to scaling, and the PI will combine the new correct critical-point singularities with corrections to scaling to dramatically extend the realm of quantitative validity. The project will use the 2D nonequilibrium random-field Ising model to illustrate both the new predicted scaling forms and the role of corrections to scaling, generating a quantitative theory predicting avalanche sizes and magnetization curves over an enormous range of disorder, until the size scale becomes as small as that of the lattice structure. The project will also develop tools to extract entire phase diagrams and concomitant materials behavior for experiments on cellular membranes, honing and validating the tools using the complex phase diagram of a multiparameter membrane model for microemulsions and tricritical Ising phenomena.