  We extend a recent theory for the rheology of frictionless granular materials
[K. Suzuki and H. Hayakawa, Phys. Rev. Lett. 2015, 115, 098001] to the case of
frictional disks in two dimensions. Employing a frictional contact model for
molecular dynamics simulations, we derive difference equations of the shear
stress, the granular temperature, and the spin temperature from the generalized
Green-Kubo formula, where all the terms are given by microscopic expressions.
The numerical solutions of the difference equations not only describe the flow
curve, but also reproduce the hysteresis of shear stress, which can be the
signature of discontinuous shear thickening of frictional disks.
