  Rate-effects in sheared disordered solids are studied using molecular
dynamics simulations of binary Lennard-Jones glasses in two and three
dimensions. In the quasistatic (QS) regime, systems exhibit critical behavior:
the magnitudes of avalanches are power-law distributed with a maximum cutoff
that diverges with increasing system size $L$. With increasing rate, systems
move away from the critical yielding point and the average flow stress rises as
a power of the strain rate with exponent $1/\beta$, the Herschel-Bulkley
exponent. Finite-size scaling collapses of the stress are used to measure
$\beta$ as well as the exponent $\nu$ which characterizes the divergence of the
correlation length. The stress and kinetic energy per particle experience
fluctuations with strain that scale as $L^{-d/2}$. As the largest avalanche in
a system scales as $L^\alpha$, this implies $\alpha < d/2$. The diffusion rate
of particles diverges as a power of decreasing rate before saturating in the QS
regime. A scaling theory for the diffusion is derived using the QS avalanche
rate distribution and generalized to the finite strain rate regime. This theory
is used to collapse curves for different system sizes and confirm $\beta/\nu$.
