Dynamic length scales in athermal, shear-driven, jamming of frictionless disks in two dimensions

Abstract: We carry our numerical simulations of athermally sheared, bidisperse, frictionless disks in two dimensions. From an appropriately defined velocity correlation function, we determine that there are two diverging length scales, $\xi$ and $\ell$, as the jamming transition is approached. We analyze our results using a critical scaling ansatz for the correlation function, and argue that the more divergent length $\ell$ is a consequence of a dangerous irrelevant scaling variable, and that it is $\xi$ which is the correlation length that determines the divergence of the system viscosity as jamming is approached from below in the liquid phase. We find that $\xi\sim (\phi_J-\phi)^{-\nu}$ diverges with the critical exponent $\nu=1$. We provide evidence that $\xi$ measures the length scale of fluctuations in the rotation of the particle velocity field, while $\ell$ measures the length scale of fluctuations in the divergence of the velocity field.