Relaxation-time approximation and relativistic third-order viscous hydrodynamics from kinetic theory

Using the iterative solution of Boltzmann equation in the relaxation-time approximation, the derivation of a third-order evolution equation for shear stress tensor is presented. To this end we first derive the expression for viscous corrections to the phase-space distribution function, $f(x,p)$, up to second-order in derivative expansion. The expression for $\delta f(x,p)$ obtained in this method does not lead to violation of the experimentally observed $1/\sqrt{m_T}$ scaling of the femtoscopic radii, as opposed to the widely used Grad's 14-moment approximation. Subsequently, we present the derivation of a third-order viscous evolution equation and demonstrate the significance of this derivation within one-dimensional scaling expansion. We show that results obtained using third-order evolution equations are in excellent accordance with the exact solution of Boltzmann equation as well as with transport results.