Thermally fluctuating second-order viscous hydrodynamics and heavy-ion collisions

The fluctuation-dissipation theorem requires the presence of thermal noise in viscous fluids. The time and length scales of heavy-ion collisions are small enough so that the thermal noise can have a measurable effect on observables. Thermal noise is included in numerical simulations of high-energy lead-lead collisions, increasing average values of the momentum eccentricity and contributing to its event-by-event fluctuations.

Figure 1

The cell averages of autocorrelations $\langle \delta u^i\delta u^i/3\rangle$ and $\langle \delta p\delta p\rangle$ (in ${\mathrm{fm}}^{-8}$) as a function of energy density.

Figure 2

Fluctuations in energy density in the transverse plane as a function of time in a typical Pb+Pb collision for $b=5.99$ fm at LHC energies.

Figure 3

Transverse momentum distribution for pions in linear and log scales showing that the single-particle distribution is unaffected by noise.

Figure 4

The evolution in time of ${\epsilon }_p$ from the calculations of two $b=5.99$ fm collisions with thermal noise as well as from a calculation without noise. Notice both the rapidly decorrelating variations in ${\epsilon }_p$, coming from $\delta W^{\prime \mu \nu }$, as well as the variations over longer timescales.

Figure 5

Top panel ($b=5.99$ fm): Inclusion of noise increases the average ${\epsilon }_p$ and broadens its distribution. Bottom panel ($b=0$ fm): Even for exactly central collisions there is a nonzero average $v_2$ due to noise.