Laboratoire de Physique Théorique

de la Matière Condensée

Jim Sethna (Cornell University)

Using Universal Scaling Functions

Half a century ago, Ken Wilson and Leo Kadanoff introduced the renormalization-group framework for understanding systems with emergent, fractal scale invariance. For five decades, statistical physicists have applied these techniques to equilibrium phase transitions, avalanche models, glasses and disordered systems, the onset of chaos, plastic flow in crystals, surface morphologies, fracture, … But these tools have not made a substantial impact in engineering or biology.

Even now we do not have a clear understanding of the singularity at the critical point for even the traditional equilibrium phase transitions – we know the critical exponents to high precision, but are missing complete understanding of the universal scaling functions. Also, we have not developed the tools to extend our predictions of the asymptotic behavior to a systematic approximation of the phase diagrams – analytic corrections to scaling. Sethna will begin with an introduction to how normal form theory from dynamical systems can determine the arguments of scaling functions where logarithms and exponentials intrude on the traditional power laws – the 4D Ising model, the 2D random-field Ising model, and jamming and the onset of amorphous rigidity in 2D. He will then use the 2D Ising model to show how adding analytic corrections to the critical point singularity can lead to high-precision descriptions of the entire phase diagram, and present Jaron Kent-Dobias’s derivation of a universal scaling function for the free energy in an external field, accurate to seven digits.