Fluctuation dynamics in a relativistic fluid with a critical point

Abstract

To describe dynamics of bulk and fluctuations near the QCD critical point we develop general relativistic fluctuation formalism for a fluid carrying baryon charge. Feedback of fluctuations modifies hydrodynamic coefficients including bulk viscosity and conductivity and introduces nonlocal and noninstantaneous terms in constitutive equations. We perform necessary ultraviolet (short-distance) renormalization to obtain cutoffindependent deterministic equations suitable for numerical implementation. We use the equations to calculate the universal nonanalytic small-frequency dependence of transport coefficients due to fluctuations (long-time tails). Focusing on the critical mode we show how this general formalism matches existing Hydro+ description of fluctuations near the QCD critical point and nontrivially extends it inside and outside of the critical region.

Off-equilibrium non-Gaussian fluctuations near the QCD critical point: an effective field theory perspective

“…In a realistic situation, other refinements are needed. For example, the interpretation of t in those equations should be elaborated [18] and might be related to the affine parameter parametrizing the streamline in the background hydrodynamic solution. To make contact with experimental observations, a freezeout prescription for W n s has to be formulated [42] (see also refs.…”

Baryon transport and the QCD critical point

“…The feedback of fluctuating modes renormalizes the bare hydrodynamic variables and gives rise to a delayed response in the form of so-called "long-time tails". The hydro-kinetic approach can be implemented in the critical regime where the fluctuating modes relax to equilibrium on parametrically long time scales [34,40] (see also the reviews [46,80]).…”

“…Besides, the associated relaxation time for the bulk viscous pressure also diverges as ξ 3 [68]. In this case, the relaxation rate of the fluctuation modes contributing to bulk viscous pressure is much smaller than the typical hydrodynamic frequency, and they can no longer be treated hydrodynamically, requiring instead an extended framework such as Hydro+ [34,40]. In this work we focus on effects from baryon diffusion and neglect bulk and shear stress effects on the bulk evolution dynamics of the fireball; however, we use the same critical scaling laws for the transport coefficients in Model H as if they were restored (i.e., with non-vanishing shear viscosity η and bulk viscosity ζ).…”

“…Thus, to confirm or exclude the critical point via systematic model-data comparison, reliable dynamical simulations of off-equilibrium critical fluctuations and the associated final particle cumulants, on top of a wellconstrained comprehensive dynamical description of the bulk medium at various beam energies, are indispensable [28][29][30][31][32][33][34][35][36][37][38][39]. Recently, the Hydro+/++ framework [34,40] was developed for incorporating off-equilibrium fluctuations and critical slowing-down into hydrodynamic simulations, and some practical progress within simplified settings has since been made using this framework [21,22]. On the other hand, while a fully developed and calibrated (2+1)-dimensional multi-stage description of heavy-ion collisions (including initial conditions + pre-hydrodynamic dynamics + viscous hydrodynamics + hadronic afterburner) exists (see, e.g., the most recent versions described in [41][42][43][44]) and has met great phenomenological success at top RHIC and LHC energies [11,12,26], such a comprehensive and fully validated framework is still missing for collisions at the lower BES energies.…”

“…Compared to high-energy collisions at LHC and top RHIC energies, collisions at BES energies introduce a number of additional complications [45,46]. These include (i) a much more complex, intrinsically (3+1)dimensional and temporally extended nuclear interpenetration stage and its associated dynamical deposition of energy and baryon number [47][48][49][50][51], (ii) the need to account for and properly propagate conserved charge currents for baryon number and strangeness [51][52][53][54][55][56][57], (iii) a consistent treatment of singularities in the thermodynamic properties associated with the critical point [23,58], and (iv) the aforementioned off-equilibrium nature of critical dynamics [21,22,34,38,40,59]. The situation is made even more complicated by the backreaction of the non-equilibrium critical fluctuation dynamics on the bulk evolution of the medium.…”