  We consider particle acceleration by large-scale incompressible turbulence
with a lengthscale larger than the particle mean free path. We derive an
ensemble-averaged transport equation of energetic charged particles from an
extended transport equation which contains the shear acceleration. The
ensemble-averaged transport equation describes particle acceleration by
incompressible turbulence (turbulent shear acceleration). We find that for
Kolmogorov turbulence, the turbulent shear acceleration becomes important in
small scale. Moreover, by Monte Carlo simulations, we confirm that the
ensemble-averaged transport equation describes the turbulent shear
acceleration.
