Kinetic regime of hydrodynamic fluctuations and long time tails for a Bjorken expansion

Abstract

We develop a set of kinetic equations for hydrodynamic fluctuations which are equivalent to nonlinear hydrodynamics with noise. The hydro-kinetic equations can be coupled to existing second order hydrodynamic codes to incorporate the physics of these fluctuations. We first show that the kinetic response precisely reproduces the renormalization of the shear viscosity and the fractional power (∝ ω 3/2 ) which characterizes equilibrium correlators of energy and momentum for a static fluid. Then we use the hydro-kinetic equations to analyze thermal fluctuations for a Bjorken expansion, evaluating the contribution of thermal noise from the earliest moments and at late times. In the Bjorken case, the solution to the kinetic equations determines the coefficient of the first fractional power of the gradient expansion (∝ 1/(τ T ) 3/2 ) for the expanding system. Numerically, we find that the contribution to the longitudinal pressure from hydrodynamic fluctuations is larger than second order hydrodynamics for typical medium parameters used to simulate heavy ion collisions.

Relativistic hydrodynamic fluctuations

“…Our equations can be directly simulated for any flow, not necessarily limited by Bjorken boost-invariance assumption (as in, e.g., Ref. [14]). In particular, the effects of vorticity, absent in the Bjorken flow, but important in heavy-ion collisions [17], can be studied.…”

“…Of course, the stochastic and the correlation function (deterministic) approaches are equivalent and complementary, in ways very similar to Langevin and Fokker-Plank description of stochastic processes, and the ultimate choice is to be made based on practicality, in particular, for numerical simulations. The deterministic approach (also referred to as hydro-kinetic approach due to the similarity of some of the additional equations to kinetic equations for phonons 1 ) has been recently discussed in a relativistic context [14][15][16] for a special case of Bjorken boost invariant solution where symmetries allow to reduce the effective dimensionality of the problem and simplify the analysis.…”

Rapidity decorrelation from hydrodynamic fluctuations

“…So far, almost all the hydrodynamic models applied to the analysis of experimental data are based on conventional viscous hydrodynamics without hydrodynamic fluctuations. Towards a thermodynamically consistent description of the event-by-event dynamics of thermodynamic fields, it is important to develop dynamical models for the high-energy nuclear collisions based on relativistic fluctuating hydrodynamics [39][40][41][42][43][44][45][46][47][48], which is the viscous hydrodynamics with hydrodynamic fluctuations. The fluctuation-dissipation relation (FDR) tells us the magnitude of the thermal fluctuations is determined by the corresponding dissipation rates.…”

Causal hydrodynamic fluctuations in non-static and inhomogeneous backgrounds

“…Instead of the Langevin type description ("event-by-event" description) of fluctuations, also the extension of the hydrodynamics (Hydro+) with slow modes and two-point functions is proposed for the critical dynamics [41]. Another important topic of the hydrodynamic fluctuations is the renormalization of hydrodynamics due to the nonlinear effects of the hydrodynamic fluctuations which are analyzed by various methods in various contexts [42][43][44][45][46][47][48][49][50]. The transport coefficients and the equation of state should be renormalized depending on the cutoff scales of the hydrodynamic fluctuations, and also there arises the long-time tail of the two-point correlations, which cannot be renormalized into ordinary transport coefficients.…”