  The question about the behavior of gaps between zeros of polynomials under
differentiation is classical and goes back to Marcel Riesz. In this paper, we
analyze a nonlocal nonlinear partial differential equation formally derived by
Stefan Steinerberger to model dynamics of roots of polynomials under
differentiation. Interestingly, the same equation has also been recently
obtained formally by Dimitri Shlyakhtenko and Terence Tao as the evolution
equation for free fractional convolution of a measure - an object in free
probability that is also related to minor processes for random matrices. The
partial differential equation bears striking resemblance to hydrodynamic models
used to describe the collective behavior of agents (such as birds, fish or
robots) in mathematical biology. We consider periodic setting and show global
regularity and exponential in time convergence to uniform density for solutions
corresponding to strictly positive smooth initial data. In the second part of
the paper we connect rigorously solutions of the Steinerberger's PDE and
evolution of roots under differentiation for a class of trigonometric
polynomials. Namely, we prove that the distribution of the zeros of the
derivatives of a polynomial and the corresponding solutions of the PDE remain
close for all times. The global in time control follows from the analysis of
the propagation of errors equation, which turns out to be a nonlinear
fractional heat equation with the main term similar to the modulated
discretized fractional Laplacian $(-\Delta)^{1/2}$.
