Pairing of Lyapunov Exponents for a Hard-Sphere Gas under Shear in the Thermodynamic Limit

We consider a dilute gas of hard spheres under shear. We use one of the predominant models to study this system, namely the SLLOD equations of motion, with an iso-kinetic Gaussian thermostat in between collisions, to get a stationary total peculiar kinetic energy. Based on the previously obtained result that in the non-equilibrium steady state and in case the number of particles $N$ becomes large, the coefficient of dynamical friction representing the iso-kinetic Gaussian thermostat for the SLLOD dynamics fluctuates with $1/\sqrt{N}$ fluctuations around a fixed value, we show on analytical grounds that for a hard sphere gas at small shear rate and with a large number of spheres, the conjugate pairing of the Lyapunov exponents is expected to be violated at the fourth power of the constant shear rate in the bulk.