Thermal noise in non-boost-invariant dissipative hydrodynamics

We study the effects of hydrodynamic fluctuations in non-boost-invariant longitudinal expansion of matter formed in relativistic heavy ion collisions. We formulate the theory of thermal noise within second-order viscous hydrodynamics treating noise as a perturbation on top of the non-boost-invariant flow. We develop a numerical simulation model to treat the (1+1)-dimension hydrodynamic evolution. The code is tested to reproduce the analytic results for the Riemann solver for expansion of matter in vacuum. For viscous hydrodynamic expansion, the initial energy density distribution are obtained by reproducing the measured charged hadron rapidity distribution at the RHIC energies. We show that the longitudinal rapidity correlations arising from space-time-dependent thermal noise and from an induced thermal perturbation have distinct structures. In general, the rapidity correlations are found to be dominated by temperature fluctuations at small rapidity separation and velocity fluctuations at large rapidities. We demonstrate that thermal noise produce ridgelike two-particle rapidity correlations which persist at moderately large rapidities. The magnitude and pattern of the correlations are quite sensitive to various second-order dissipative formalisms and to the underlying equations of state, especially at large rapidities. The short-range part of the rapidity correlation is found to be somewhat enhanced as compared to that in boost-invariant flow of matter.

Figure 1

Comparison between the Riemann analytical solutions (lines) and the numerical results (circles) for the rapidity dependence of energy density $\epsilon$, velocities $v^{\eta }$ and $v^z$ at proper times $\tau =3,4,5\mathrm{fm}/\mathrm{c}$ for the initial condition of Eq. (62) at time ${\tau }_0=1\mathrm{fm}/\mathrm{c}$.

Figure 2

Comparison between the analytical solutions (lines) and the numerical results (circles) for the rapidity dependence of energy density $\epsilon$ at proper times $\tau =510,600,700,1000\mathrm{fm}/\mathrm{c}$ for the initial condition of Eq. (64) at time ${\tau }_0=500\mathrm{fm}/\mathrm{c}$.

Figure 3

Rapidity distribution of ${\pi }^{\pm }$ and $K^{\pm }$ in $0–5\%$ central Au+Au collisions at $\sqrt{s_{\text{NN}}}=200\mathrm{GeV}$. The symbols represent the BRAHMS data [51], and the lines correspond to non-boost-invariant hydrodynamic calculations for the conformal fluid and lattice equation of state at the shear viscosity to entropy density ratio of ${\eta }_v/s=0$ and 0.08 in the MIS approach.

Figure 4

Space-time rapidity dependence of (a), (b) energy density, (c), (d) longitudinal velocity (scaled by time), and (e), (f) longitudinal to transverse pressure ratio $P_L/P_T$, in the hydrodynamic calculations at proper times of $\tau =5\mathrm{fm}$ (left panels) and 10 fm (right panels). The results are for ultrarelativistic gas (symbols) and lattice (lines) equations of state with ${\eta }_v/s=0,1/4\pi ,3/4\pi$ in the MIS theory. The black dotted line is the Bjorken scaling solution $\epsilon /{\epsilon }_0\sim {({\tau }_0/\tau )}^{4/3}$.

Figure 5

Equal-time longitudinal rapidity correlations $C_{\mathrm{\Delta }X,\mathrm{\Delta }Y}(\tau ,\mathrm{\Delta }\eta ;{\eta }_0)$ with $(\mathrm{\Delta }X,\mathrm{\Delta }Y)\equiv [\delta T/T_0,\tau \delta u^{\eta },\delta \pi /({\epsilon }_0+p_0)]$ computed as a function of space-time rapidity separation $\mathrm{\Delta }\eta$ at various later times from a thermal perturbation at an initial time of ${\tau }_0=0.4\mathrm{fm}/\mathrm{c}$ and at rapidity ${\eta }_0=0$. The perturbation is induced on top of non-boost-invariant background hydrodynamic flow in the MIS theory using an ultrarelativistic gas EoS with ${\eta }_v/s=1/4\pi$. The correlators in each panel are vertically scaled by a representative value.

Figure 6

Temperature-temperature and velocity-velocity rapidity correlations as a function of rapidity separation $\mathrm{\Delta }\eta$ arising from perturbations at various initial rapidities ${\eta }_0$ and computed at a freeze-out hypersurface $T_{\mathrm{dec}}=150\mathrm{MeV}$. The perturbations are induced on top of non-boost-invariant background hydrodynamic flow in the MIS theory for an ultrarelativistic gas EoS with ${\eta }_v/s=1/4\pi$. The correlators for large ${\eta }_0$ are scaled vertically by the values shown within braces.

Figure 7

Event averaged equal-time longitudinal rapidity correlations $\langle C_{X,Y}(\mathrm{\Delta }\eta ,\eta )\rangle$ with $(X,Y)\equiv (\delta T,\delta u^{\eta },\delta \pi )$ computed as a function of space-time rapidity separation $\mathrm{\Delta }\eta ={\eta }_1-{\eta }_2$ about ${\eta }_1=0$ at various times due to thermal noise perturbations on top of non-boost-invariant background flow. The results are in the MIS theory for ultrarelativistic gas EoS with ${\eta }_v/s=1/4\pi$. The correlators in each panel are scaled vertically by values given; the correlations at $\tau =1\mathrm{fm}/\mathrm{c}$ are further scaled by 0.25 for clarity.

Figure 8

Event averaged rapidity correlations $\langle C_{X,Y}(\mathrm{\Delta }\eta ,\eta )\rangle$ with $(X,Y)\equiv (\delta T,\delta u^{\eta },\delta \pi )$ at midrapidity from thermal noise at the freeze-out temperature $T_{\mathrm{dec}}=150\mathrm{MeV}$. The results are in the MIS formalism with ${\eta }_v/s=1/4\pi$ in the average and thermal noise evolution for $p=\epsilon /3$ EoS (red solid line), lattice EoS (green solid line) and an ideal background evolution with $p=\epsilon /3$ EoS (blue dashed line). The initial and final conditions for the background evolution are given in Table 1. All the correlations are scaled vertically by ${10}^6$.

Figure 9

Two-particle rapidity correlations for different fluctuations calculated for charged pions as a function of pion-rapidity separation $\mathrm{\Delta }y=y_1-y_2$ at rapidities $y_1=0,2,4$ in the MIS hydrodynamics. The results are for ideal gas EoS with ${\eta }_v/s=1/4\pi$ and the initial and final conditions are the same as in Fig. 7.

Figure 10

Correlation function of charged pions normalized with single-particle rapidity distribution in (1+1)D hydrodynamic expansion as a function of rapidity separation $\mathrm{\Delta }y=y_1-y_2$ at various rapidities $y_1$. The results are in the MIS theory for ideal gas EoS with ${\eta }_v/s=1/4\pi$ and the initial and final conditions are the same as in Fig. 7. The corresponding correlation in the Bjorken expansion but normalized by ${(dN/d{y}_1)}_{y_1=0}$ is shown by thick black dashed line.

Figure 11

Correlation function of charged pions normalized with single-particle rapidity distribution as a function of rapidity separation $\mathrm{\Delta }y=y_1-y_2$ at various rapidities $y_1$. The results are in the MIS and CE formalisms for thermal noise evolution and compared with the ideal background hydrodynamic evolution. An ideal gas EoS ($p=\epsilon /3$) is used and the initial and freeze-out conditions are given in Table 1.

Figure 12

Similar to Fig. 11 but with a lattice EoS. The initial and freeze-out conditions are given in Table 1.