Nonlinear analysis of causality for heat flow in heavy-ion collisions: constraints from equation of state

TL;DR

The paper analyzes causality constraints in the Mueller-Israel-Stewart framework for heat-conducting relativistic fluids in nonlinear, one-dimensional flow. By solving the characteristic determinant in the rest frame, it maps how hyperbolicity depends on the EoS and the second-order coefficient $\beta_1$ (via $\lambda$), revealing a highly constrained causal domain that broadens with larger $\lambda$ and stiffer $c_s^2$. Incorporating a lattice-QCD EoS shows nontrivial modifications to the causal structure, including a narrowing of the causal window near the QCD crossover. Navier-Stokes estimates of heat flux with kinetic-theory values of $\kappa$ yield unrealistically large $|\mathbf{q}|/\varepsilon$ for RHIC-like conditions, implying possible breakdown of dissipative hydrodynamics in extreme regimes and highlighting the need for lattice-based transport coefficients. Overall, the work emphasizes the critical role of accurate transport coefficients and the equation of state in ensuring a well-posed, causal description of heat flow in heavy-ion collisions, and points to future work in higher dimensions and more realistic baryon-rich scenarios.

Abstract

We explore the causal parameter space of the Mueller-Israel-Stewart second-order theory for heat-conducting fluids in nonlinear regimes for one-dimensional fluid flow. We show that this parameter space is highly constrained and particularly sensitive to the equation of state and second-order transport coefficients. Through numerical analysis of the characteristic equations, we identify regions of strong hyperbolicity, weak hyperbolicity, and non-hyperbolicity, mapping the boundaries of causality violation as functions of the heat flux to energy density ratio $q/\varepsilon$ and relaxation parameters. We also explore the causality conditions using a realistic lattice QCD-based equation of state. Using the Navier-Stokes approximation, we estimate the heat flow magnitude to assess causality criteria for one-dimensional heat conduction in heavy-ion collisions. Our calculations reveal unrealistically large heat flux values ($|{\bf{q}}|/\varepsilon \sim 330-811$) for typical RHIC conditions when using thermal conductivity estimates from kinetic theory models, suggesting either significant overestimation of transport coefficients or breakdown of the fluid approximation in these extreme conditions. The pressure gradient corrections reduce the heat flow by approximately 15\% but do not resolve the causality concerns.

Figures (6)

• Figure 1: Characteristic velocities $v$ obtained from the numerical solution of Eq. \ref{['eq:fluid_charactersitic']} shown as a function of $q/\varepsilon$, for $\beta_1= \frac{5}{4p}$ and four different $c_s^2$=0.05, 0.10, 0.33, and 0.5. The shaded green region indicates the causal hyperbolic region.
• Figure 2: Characteristic velocities $v$ as functions of $\frac{q}{\varepsilon}$ for $\lambda = 10$ and four different values of $c_s^2$: $0.05, 0.1, 0.33, 0.5$. The causal regions, marked by the shaded green area, show a significant enhancement in the range of $\frac{q}{\varepsilon}$ compared to smaller $\lambda$ cases.
• Figure 3: Causality regions in the ($q/\varepsilon$, $c_s^2$) plane for different values of the relaxation parameter, $\lambda = 0.75, 1.0,$ and $10.0$. The green (Hyperbolic/ Causal), yellow (Hyperbolic/Acausal), and red (Non-hyperbolic/Acausal) regions denote the parameter space where the characteristic velocities are all real and subluminal, all real but at least one is superluminal, and not all real, respectively.
• Figure 4: Lattice QCD results for squared speed of sound $c_s^2$ as a function of energy density.
• Figure 5: Characteristic velocities $v$ as functions of $q/\varepsilon$ for three different values of $\lambda = 0.5, 1.0,$ and $20.0$, calculated using a lattice QCD equation of state. The plots show the results for positive $q/\varepsilon$, with the causal hyperbolic region highlighted by green rectangles.