Using Universal Scaling Functions

by James P. Sethna (Cornell University)

Thursday 3 Nov 2022, 14:00 → 16:00 Europe/Rome

Aula Careri (Dipertimento di Fisica - Ed. G.Marconi)

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. The talk 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. The 2D Ising model will be used
to show how adding analytic corrections to the critical point
singularity can lead to high-precision descriptions of the entire
phase diagram, and Jaron Kent-Dobias’s derivation of a universal
scaling function for the free energy in an external field, accurate to
seven digits will be discussed.