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Predrag Cvitanovic - Georgia Tech
Title:
"Spatiotemporal Cat: A Chaotic Field Theory"
You find yourself face-to-face with Young-Mills or Navier-Stokes or
a nonlinear PDE or a funky metamaterial or a cloudy day. And you
ask 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 you
through its simplest example, the coin toss or Bernoulli map, but
reformulated as an integer lattice problem. Then I will do the same
with the 'kicked rotor', the simplest mechanical system that is
chaotic, 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 spring
mattress that obeys the familiar harmonic oscillator field theory
equation, the discrete Helmholtz equation / tight-binding model,
but instead of being "springy", this metamaterial is a
dicretization of the Euclidean Klein-Gordon equation, with an
unstable rotor at every lattice site, that gives, rather than
pushes back, formulated in terms of Hill determinants and zeta
functions.
We call this simplest of all chaotic field theories the
`spatiotemporal cat'.
This is the simplest example of reformulating space and time
translationally invariant, exponentially unstable `turbulent' field
theories as (D+1)-dimensional spatiotemporal systems which treat
space and time on equal footing. Here there is no `evolution in
time': time evolution is replaced by the enumeration of the
repertoire of the spatiotemporal solutions (invariant
(D+1)-dimensional invariant tori, or `periodic orbits') of system's
equations, very much as the partition function of the Ising model
is 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.