Relaxation Time Approximation for a multi-species relativistic gas

TL;DR

This work extends the relativistic relaxation-time framework to multi-species systems by introducing counter-terms that restore energy–momentum conservation for momentum-dependent relaxation times. It provides a first-order Chapman–Enskog solution and proves entropy production is non-negative, maintaining the second law. The formalism yields explicit expressions for shear and bulk transport coefficients that reduce to the Anderson–Wilkins results in the momentum-independent limit and remain consistent with Landau matching. Applied to a hadron–resonance gas with a momentum-dependent relaxation time $\tau_{R\mathbf p,i} \propto (E_{\mathbf p,i}/T)^{\gamma}$, the method reveals a novel bulk contribution proportional to $E_{\mathbf p,i}/T$ and a $\gamma$-dependent behavior of both bulk and shear sectors, with AW recovered as $\gamma\to0$. This approach improves the physical fidelity of particlization models in heavy-ion simulations by preserving fundamental conservation laws while enabling momentum-dependent dissipative dynamics.

Abstract

We generalize a recent prescription for the relaxation time approximation for the relativistic Boltzmann equation for systems with multiple particle species at finite temperature. This is performed by adding counter-terms to the traditional Anderson-Witting ansatz for each particle species. Our approach allows for the use of momentum-dependent relaxation times and the obedience of local conservation laws regardless of the definition of the local equilibrium state. As an application, we derive the first order Chapman-Enskog corrections to the equilibrium distribution and display results for the hadron-resonance gas. We also demonstrate that our collision term ansatz obeys the second law of thermodynamics.

Figures (1)

• Figure 1: Normalized coefficients as a function of temperature for the hadron-resonance gas computed numerically