Molecular dynamics approach to dissipative relativistic hydrodynamics: Propagation of fluctuations

Relativistic generalization of hydrodynamic theory has attracted much attention from a theoretical point of view. However, it has many important practical applications in high energy as well as astrophysical contexts. Despite various attempts to formulate relativistic hydrodynamics, no definitive consensus has been achieved. In this work, we propose to test the predictions of four types of first-order hydrodynamic theories for nonperfect fluids in the light of numerically exact molecular dynamics simulations of a fully relativistic particle system in the low density regime. In this regard, we study the propagation of density, velocity, and heat fluctuations in a wide range of temperatures using extensive simulations and compare them to the corresponding analytic expressions we obtain for each of the proposed theories. As expected, in the low temperature classical regime all theories give the same results, consistent with the numerics. In the high temperature extremely relativistic regime, not all considered theories are distinguishable from one another. However, in the intermediate regime, a meaningful distinction exists in the predictions of various theories considered here. We find that the predictions of the recent formulation due to Tsumura, Kunihiro, and Ohnishi are more consistent with our numerical results than the traditional theories: the Meixner, modified Eckart, and modified Marle-Stewart theories.

Figure 1

Analytical correlation functions up to zeroth order in terms of amplitudes (solid black lines) for (a) density, (b) velocity, and (c) heat fluctuations are compared to numerical results (symbols) for a system with $\rho =0.04, T=0.01$, and $\text{MFT}=33.8$ at $t=600$, 1400, and 2800 in rescaled time units. (d)–(f) show first-order correlation profiles (dashed red lines) compared to their zeroth-order counterparts (solid gray lines) at $t=600$. Note that the correlation functions in all theories coincide in the low-temperature limit and we have chosen to plot TKO's theory as a representative.

Figure 2

Analytical correlation profile of density fluctuations obtained from TKO, MMS, ME, and MX theories (colored lines) are compared to simulation results (black dots) for a system with $\rho =0.04, T=3, \text{MFT}=5.79$, and $t=500$. Note that the difference between ME, MMS, and TKO theories is almost negligible at $T=3$ and cannot be observed in this figure.

Figure 3

Height of the Brillouin peak in the density correlation profile is plotted as a function of temperature for different formalisms for a system with $\rho =0.04$, at two different times.

Figure 4

Analytical density correlation profiles obtained from TKO, MMS, ME, and MX theories (colored lines) are compared to simulation results (black dots) for a system with $\rho =0.04, T=0.2$, and $\text{MFT}=8.99$, observed at $t=700$ in rescaled time units. Inset shows details.

Figure 5

The heights of the Brillouin peak in density correlation profiles of first-order TKO and ME formalisms are compared to the second-order ME formalism for three different ratios $\tau /t=0.01,0.02,0.03$, in a system with $\rho =0.04$ and $t=700$. Note that the correction due to finite relaxation time ($\tau \ne 0$) does not alter ME ($\tau =0$) results in such a way to make it comparable with TKO results.