On the Approach Towards Equilibrium Through Momentum-Dependent Relaxation:Insights from Evolution of the Moments in Kinetic Theory

Abstract: We investigate the impact of momentum-dependent relaxation time approximation in the Boltzmann equation within the Bjorken flow framework by analyzing the moments of the single-particle distribution function. The moment equations, which form an infinite hierarchy, provide important insights about the system dynamics and the approach towards equilibrium for systems far from equilibrium. The momentum-dependent collision kernel couples moments through both the energy exponents and the angular dependence via various-order Legendre polynomials, resulting in an intricate system of infinitely coupled equations. A naive truncation of the coupled equations results in diverging moments at late times. We outline strategies for solving the coupled system, including a novel approach for managing the divergences and non-integer moments. We show a significant influence of momentum dependent relaxation time on the time evolution of the moments, particularly for higher-order moments and system with smaller shear viscosity over entropy density, emphasizing the importance of incorporating such dependence for a more accurate description of the system dynamics with low shear viscosity such as the quark-gluon-plasma produced in high-energy heavy-ion collisions.