  We provide a first-principles derivation of the Langevin equation with shear
flow and its corresponding fluctuation-dissipation theorems. Shear flow of
simple fluids has been widely investigated by numerical simulations. Most
studies postulate a Markovian Langevin equation with a simple shear drag term
\`a la Stokes. However, this choice has never been justified from first
principles. We start from a particle-bath system described by a classical
Caldeira-Leggett Hamiltonian modified by adding a term proportional to the
strain-rate tensor according to Hoover's DOLLS method, and we derive a
generalized Langevin equation for the sheared system. We then compute,
analytically, the noise time-correlation functions in different regimes. Based
on the intensity of the shear-rate, we can distinguish between
close-to-equilibrium and far-from-equilibrium states. According to the results
presented here, the standard, simple and Markovian form of the Langevin
equation with shear flow postulated in the literature is valid only in the
limit of extremely weak shear rates compared to the effective vibrational
temperature of the bath. For even marginally higher shear rates, the
(generalized) Langevin equation is strongly non-Markovian and non-trivial
fluctuation-dissipation theorems are derived.
