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

TL;DR

The paper analyzes how momentum-dependent relaxation in the relativistic Boltzmann equation under Bjorken flow affects the evolution of an infinite set of moments $\rho_{n,l}$, revealing that MDRTA introduces couplings across both energy exponents $n$ and angular indices $l$. By formulating the collision term to conserve invariants, and employing Grad-based moment closure, the authors stabilize the truncated hierarchy and isolate the role of the MDRTA parameter $\Lambda$ in shaping the dynamics. They show that larger $\Lambda$ enhances momentum anisotropy, slows isotropisation, and prolongs the intermediate cooling phase, especially at low $\eta/s$, with higher-order moments being particularly sensitive. The work provides a framework to incorporate microscopic momentum transfer into kinetic-theory-based moment analyses, offering improved insights for systems with low shear viscosity such as the quark-gluon plasma. It also identifies computational challenges and proposes a robust closure scheme, paving the way for extensions to massive particles and reduced symmetries.

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.

Figures (5)

• Figure 1: The ratio of absolute values of alternate moments at late time for $n= 2$ for a free streaming system. The ratio saturates near 1.
• Figure 2: The evolution of temperature for $\Lambda = 0$ and $\Lambda = 0.125$. We see a growth in temperature when $\tau/\tau_R > 1$ .
• Figure 3: Evolution of the scaled moments and their dependence on $\Lambda$ and initial specific viscosity $\eta/s$.
• Figure 4: Dependence of temperature and $\mu/T$ evolution on $\Lambda$ and initial specific viscosity $\eta/s$.
• Figure 5: log plot of the minimum eigenvalue of $M(1,\Lambda)$ for $N=5$ as a function of $\Lambda$