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Mathematical Physics Seminar

Predrag Cvitanovic - Spatiotemporal Cat: A Chaotic Field Theory

Location:  Zoom
Date & time: Wednesday, 16 December 2020 at 10:45AM - 11:45AM

Predrag Cvitanovic - Georgia Tech

Title: "Spatiotemporal Cat: A Chaotic Field Theory"

You find yourself face-to-face with Young-Mills or Navier-Stokes ora nonlinear PDE or a funky metamaterial or a cloudy day. And youask yourself, is this "turbulence"? What does that even mean?The goal of the seminar is to answer this question pedagogically,as a sequence of pencil and paper calculations.First I will explain what is 'deterministic chaos' by walking youthrough its simplest example, the coin toss or Bernoulli map, butreformulated as an integer lattice problem. Then I will do the samewith the 'kicked rotor', the simplest mechanical system that ischaotic, and then put infinity of these together to explain what`chaos' or `turbulence' looks like in the spacetime.What emerges is a spacetime which is very much like a big springmattress that obeys the familiar harmonic oscillator field theoryequation, the discrete Helmholtz equation / tight-binding model,but instead of being "springy", this metamaterial is adicretization of the Euclidean Klein-Gordon equation, with anunstable rotor at every lattice site, that gives, rather thanpushes back, formulated in terms of Hill determinants and zetafunctions.We call this simplest of all chaotic field theories the`spatiotemporal cat'.This is the simplest example of reformulating space and timetranslationally invariant, exponentially unstable `turbulent' fieldtheories as (D+1)-dimensional spatiotemporal systems which treatspace and time on equal footing. Here there is no `evolution intime': time evolution is replaced by the enumeration of therepertoire of the spatiotemporal solutions (invariant(D+1)-dimensional invariant tori, or `periodic orbits') of system'sequations, very much as the partition function of the Ising modelis a weighted sum formed by enumerating all its lattice states.But that is a story for another seminar.That's `turbulence'. And if you don't know, now you know.

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