Georgia Institute of Technology College of Sciences

I will describe my recent joint work with Jacob Bedrossian and Sam Punshon-Smith on the formation of small scales in passively-advected scalars being mixed by a fluid evolving by the Navier-Stokes equation. Our main result is a confirmation of Batchelor's law, a power-law for the spectral density of a passively advected scalar in the so-called Batchelor regime of infinite Schmidt number. Along the way I will describe how this small-scale formation is intimately connected with dynamical questions, such as the connection between shear-straining in the fluid and sensitive dependence on initial conditions (Lyapunov exponents). Time-permitting I will describe some work-in-progress as well as interesting open problems in the area.

A central question in ergodic theory is whether sequences obtained by sampling along the orbits of a given dynamical system behave similarly to sequences of i.i.d. random variables. Here we consider this question from a spectral-theoretic perspective. Specifically, we study large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift on the 2-torus with irrational frequency. We prove that their global eigenvalue distribution converges to the Wigner semicircle law, a hallmark of random matrix statistics, which evidences the quasi-random nature of the skew-shift dynamics. This is joint work with Arka Adhikari and Horng-Tzer Yau.

Consider a system of rotators subject to a small quasi-periodic forcing which (1) is analytic, (2) satisfies a time-reversibility property, and (3) has a Bryuno frequency vector. Without imposing any non-degeneracy condition, we prove that there exists at least one quasi-periodic solution with the same frequency vector as the forcing. The result can be interpreted as a theorem of persistence of lower-dimensional tori of arbitrary codimension in degenerate cases. This is a joint work with Livia Corsi.

We show that the dynamics of nonlinear dynamical systems with many degrees of freedom (possibly infinitely many) can be similar to that of ordered system in a surprising fashion. To this aim, in the literature one typically uses techniques from perturbation theory, such as KAM theorem or Nekhoroshev theorem. Unfortunately they are known to be ill-suited for obtaining results in the case of many degrees of freedom. We present here a probabilistic approach, in which we focus on some observables of physical interest (obtained by averaging on the probability distribution on initial data) and for several models we get results of stability on long times similar to Nekhoroshev estimates. We present the example of a nonlinear chain of particles with alternating masses, an hyper-simplified model of diatomic solid. In this case, which is similar to the celebrated Fermi-Pasta-Ulam model and is widely studied in the literature, we show the progress with respect to previous results, and in particular how the present approach permits to obtain theorems valid in the thermodynamic limit, as this is of great relevance for physical implications.