We develop a combined hydro-kinetic approach which incorporates a hydrodynamical expansion of the systems formed in \textit{A}+\textit{A} collisions and their dynamical decoupling described by escape probabilities. The method corresponds to a generalized relaxation time ($τ_{\text{rel}}$) approximation for the Boltzmann equation applied to inhomogeneous expanding systems; at small $τ_{\text{rel}}$ it also allows one to catch the viscous effects in hadronic component - hadron-resonance gas. We demonstrate how the approximation of sudden freeze-out can be obtained within this dynamical picture of continuous emission and find that hypersurfaces, corresponding to a sharp freeze-out limit, are momentum dependent. The pion $m_{T}$ spectra are computed in the developed hydro-kinetic model, and compared with those obtained from ideal hydrodynamics with the Cooper-Frye isothermal prescription. Our results indicate that there does not exist a universal freeze-out temperature for pions with different momenta, and support an earlier decoupling of higher $p_{T}$ particles. By performing numerical simulations for various initial conditions and equations of state we identify several characteristic features of the bulk QCD matter evolution preferred in view of the current analysis of heavy ion collisions at RHIC energies.
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, which become dominant near the critical point. We use the hydro-kinetic equations to calculate the modifications of energy momentum tensor by thermal fluctuations from the earliest moments and at late times in Bjorken expansion. The solution to the kinetic equations precisely determines the coefficient of the first fractional power (…
We give a short review about the hydrodynamic model and its application to the elliptic flow phenomena and the pion interferometry in relativistic heavy ion collisions.
In this proceedings we discuss recent findings regarding the large order behavior of the Chapman-Enskog expansion in relativistic kinetic theory. It is shown that this series in powers of the Knudsen number has zero radius of convergence in the case of a Bjorken expanding fluid described by the Boltzmann equation in the relaxation time approximation. This divergence stems from the presence of non-hydrodynamic modes, which give non-perturbative contributions to the Knudsen series.
The transport coefficients of causal relativistic dissipative fluid dynamics are calculated both in a field-theoretical and a kinetic approach. We find that the results from the traditional kinetic calculation by Israel and Stewart are modified. The new expressions for the viscous transport coefficients agree with the results obtained in the field-theoretical approach when the contributions from pair creation and annihilation are neglected.
Understanding the production mechanism of light (anti-)nuclei in high-energy nuclear collisions and cosmic rays has been a long-standing problem in nuclear physics. In the present study, we develop a stochastic method to solve the relativistic kinetic equations for light nuclei production from many-body reactions with the inclusion of their finite sizes. The present approach gives an excellent description of the deuteron and helium-3 data from central Au+Au (Pb+Pb) collisions at $\sqrt{s_{\rm NN}}$ $=$ $200~\rm GeV$ ($2.76~\rm TeV$). It can also naturally explain their suppressed production in…
We use a macroscopic description of a system of relativistic particles based on adding a nonequilibrium tensor to the usual hydrodynamic variables. The nonequilibrium tensor is linked to relativistic kinetic theory through a nonlinear closure suggested by the Entropy Production Principle; the evolution equation is obtained by the method of moments, and together with energy-momentum conservation closes the system. Transport coefficients are chosen to reproduce second order fluid dynamics if gradients are small. We compare the resulting formalism to exact solutions of Boltzmann's equation in 0+1…
The relativistic kinetic equations for the two domains separated by the hypersurface with both space- and time-like parts are derived. The particle exchange between the domains separated by the time-like boundaries generates source terms and modifies the collision term of the kinetic equation. The correct hydrodynamic equations for the ``hydro+cascade'' models are obtained and their differences from existing freeze-out models of the hadronic matter are discussed.
We use the entropy production variational method to associate a one particle distribution function to the assumed known energy-momentum and entropy currents describing a relativistic conformal fluid. Assuming a simple form for the collision operator we find this one particle distribution function explicitly, and show that this method of linking the hydro and kinetic description is a non trivial generalization of Grad's ansatz. The resulting constitutive relations are the same as in the conformal dissipative type theories discussed in J. Peralta-Ramos and E. Calzetta, Phys. Rev. D {\bfseries 80…
In this article, we briefly review recent progress on hydrodynamic modeling and its implementations to relativistic heavy-ion collisions at RHIC and the LHC. The related topics include: 1) initial state fluctuations, final state correlations and event-by-event hydrodynamics, 2) extracting the QGP shear viscosity from flow data, 3) flow and hydrodynamics in 5.02 A TeV p+Pb collisions.
In this work, we start with chiral kinetic theory and construct the spin hydrodynamic framework for a chiral spinor system. Using the 14-moment expansion formalism, we obtain the equations of motion of second-order dissipative relativistic fluid dynamics with non-trivial spin polarization density. In a chiral spinor system, the spin alignment effect could be treated in the same framework as the Chiral Vortical Effect (CVE). However, the quantum corrections due to fluid vorticity induce not only CVE terms in the vector/axial charge currents but also corrections to the stress tensor. In this fra…
We re-derive the equations of motion of dissipative relativistic fluid dynamics from kinetic theory. In contrast to the derivation of Israel and Stewart, which considered the second moment of the Boltzmann equation to obtain equations of motion for the dissipative currents, we directly use the latter's definition. Although the equations of motion obtained via the two approaches are formally identical, the coefficients are different. We show that, for the one-dimensional scaling expansion, our method is in better agreement with the solution obtained from the Boltzmann equation.
We present an approach to derive a relativistic kinetic equation of the Vlasov type. Our approach is especially reliable for the description of quantum field systems with many internal degrees of freedom. The method is based on the Heisenberg picture and leads to a kinetic equation which fulfills the conservation laws. We apply the approach to the standard Walecka Lagrangian and an effective chiral Lagrangian.
We use the recently developed kinetic theory with Berry curvature to describe the fermions and antifermions of a chiral relativistic plasma. We check that this transport approach allows to reproduce the chiral anomaly equation of relativistic quantum field theory at finite temperature. We also check that it allows to describe the anomalous gauge polarization tensor that appears in the Hard Thermal (and/or Dense) effective field theory. We also construct an energy density associated to the gauge collective modes of the chiral relativistic plasma, valid in the case of small couplings or weak fie…
In this letter we show for the first time that the relativistic Chapman-Enskog series for a massless gas undergoing Bjorken expansion diverges. In order to fix this problem, we propose a novel type of expansion that includes non-perturbative contributions in the Knudsen number that are not considered in Chapman-Enskog theory. This approach is in good agreement with exact solutions of the Boltzmann equation for a wide range of values of Knudsen number and does not display the clear signs of divergence exhibited by the Chapman-Enskog series.
An approach is developed, which essentially simplifies derivation of kinetic equation for polarization in non-relativistic scattering. This approach is applicable for collision of projectile particles with a target for any spins of colliding particles. The most detailed consideration is performed for the case of spin 1/2 projectile particle interacting with spin 1/2 particle of a target. The solution of the kinetic equation for the case of zero initial polarization is presented.
We discuss some open problems in hydrodynamical approach to the relativistic heavy ion collisions. In particular, we propose a new, very simple alternative approach to the relativistic dissipative hydrodynamics of Israel and Stewart.
We present the complete first order relativistic quantum kinetic theory with spin for massive fermions derived from the Wigner function formalism in a concise form that shows explicitly how the 32 Wigner equations reduce to 4 independent transport equations. We solve modified on-shell conditions to obtain the general solution and present the corresponding transport equations in three different forms that are suitable for different purposes. We demonstrate how different spin effects arise from the kinetic theory by calculating the chiral separation effect with mass correction, the chiral anomal…
We present an analytical derivation of the transport coefficients of a relativistic gas in (2+1) dimensions for both Chapman-Enskog (CE) asymptotics and Grad's expansion methods. Moreover, we develop a systematic calibration method, connecting the relaxation time of relativistic kinetic theory to the transport parameters of the associated dissipative hydrodynamic equations. Comparison between the analytical results and numerical simulations, shows that the CE method correctly captures dissipative effects, while Grad's method does not. The resulting calibration procedure based on the CE method …
Hydrodynamics can be formulated in terms of a perturbative series in derivatives of the temperature, chemical potential, and flow velocity around an equilibrium state. Different formulations for this series have been proposed over the years, which consequently led to the development of various hydrodynamic theories. In this work, we discuss the relativistic generalizations of the perturbative expansions put forward by Chapman and Enskog, and Hilbert, using general matching conditions in kinetic theory. This allows us to describe, in a comprehensive way, how different out-of-equilibrium definit…
We demonstrate by simple mathematical considerations that a power-law tailed distribution in the kinetic energy of relativistic particles can be a limiting distribution seen in relativistic heavy ion experiments. We prove that the infinite repetition of an arbitrary composition rule on an infinitesimal amount leads to a rule with a formal logarithm. As a consequence the stationary distribution of energy in the thermodynamical limit follows the composed function of the Boltzmann-Gibbs exponential with this formal logarithm. In particular, interactions described as solely functions of the relati…
The hydro-kinetic formalism has been used as a complementary approach to solving the Stochastic Differential Equations (SDE) corresponding to noisy hydrodynamics. The hydro-kinetic formalism consists of a deterministic set of relaxation type equations that tracks the evolution of 2-point correlation functions of stochastic hydrodynamic quantities. Hence they are comparatively easier to solve than the SDEs, which are computationally intensive and need to deal with arbitrarily large gradients. This work compares the two approaches for the propagation and diffusion of conserved charge fluctuation…
In this paper, we study the Cauchy problem of the linearized kinetic equations for the models of Marle and Anderson-Witting, and compare these dispersion relations with the 14-moment theory. First, we propose a modification of the Marle model to improve the resultant transport coefficients in accord with those obtained by the full Boltzmann equation. Using the modified Marle model and Anderson-Witting model, we calculate dispersion relations that are kinetically correct within the validity of the BGK approximation. The 14-moment theory that includes the time derivative of dissipation currents …
In this work, the causality and stability of a first-order relativistic dissipative hydrodynamic theory, that redefines the hydrodynamic fields from a first principle microscopic estimation, have been analyzed. A generic approach of gradient expansion for solving the relativistic transport equation has been adopted using the Chapman-Enskog iterative method. Next, the momentum dependent relaxation time approximation (MDRTA) has been employed to quantify the collision term for analytical estimation of the field correction coefficients from kinetic theory. At linear regime, in local rest frame th…
The Giessen Boltzmann-Uehling-Uhlenbeck transport model with relativistic mean fields is used to simulate $\bar p$-nucleus collisions. Antiproton absorption cross sections and momentum distributions of annihilation products are calculated by varying the $\bar p$ coupling strength to the mean meson fields. Parameters of the antiproton-nucleus optical potential are extracted from the comparison of the model calculations with experimental data.
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 ($\propto ω^{3/2}$) 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, ev…
Based on the chiral kinetic approach using initial conditions from a multiphase transport model, we study the spin polarizations of quarks and antiquarks in non-central heavy ion collisions at the Relativistic Heavy Ion Collider. Because of the non-vanishing vorticity field in these collisions, quarks and antiquarks are found to acquire appreciable spin polarizations in the direction perpendicular to the reaction plane. Converting quarks and antiquarks to hadrons via the coalescence model, we further calculate the spin polarizations of Lambda and anti-Lambda hyperons and find their values comp…
Hydro-kinetic theory of thermal fluctuations is applied to a non-conformal relativistic fluid. Solving the hydro-kinetic equations for an isotropically expanding background we find that hydrodynamic fluctuations give ultraviolet divergent contributions to the energy-momentum tensor. After shifting the temperature to account for the energy of non-equilibrium modes, the remaining divergences are renormalized into local parameters, e.g. pressure and bulk viscosity. We also confirm that the renormalization of the pressure and bulk viscosity is universal by computing them for a Bjorken expansion. T…
Within the $σ-ω$ model of coupled nucleon-meson systems, a generalized relativistic Lenard--Balescu--equation is presented resulting from a relativistic random phase approximation (RRPA). This provides a systematic derivation of relativistic transport equations in the frame of nonequilibrium Green's function technique including medium effects as well as fluctuation effects. It contains all possible processes due to one meson exchange and special attention is kept to the off--shell character of the particles. As a new feature of many particle effects, processes are possible which can be interpr…
Relativistic hydrodynamics is essential to our current understanding of nucleus-nucleus collisions at ultrarelativistic energies (current experiments at the Relativistic Heavy Ion Collider, forthcoming experiments at the CERN Large Hadron Collider). This is an introduction to relativistic hydrodynamics for graduate students. It includes a detailed derivation of the equations, and a description of the hydrodynamical evolution of a heavy-ion collisions. Some knowledge of thermodynamics and special relativity is assumed.
A detailed analysis of the coupled relativistic kinetic equations for two domains separated by a hypersurface having both space- and time-like parts is presented. Integrating the derived set of transport equations, we obtain the correct system of the hydro+cascade equations to model the relativistic nuclear collision process. Remarkably, the conservation laws on the boundary between domains conserve separately both the incoming and outgoing components of energy, momentum and baryonic charge. Thus, the relativistic kinetic theory generates twice the number of conservation laws compared to tradi…
We derive relativistic dissipative spin hydrodynamics from kinetic theory featuring a nonlocal collision term using the method of moments. In this framework, the components of the spin tensor are dynamical variables which obey relaxation-type equations. We find that the corresponding relaxation times are determined by the local part of the collision term, while the nonlocal part contributes to the Navier-Stokes terms in these equations of motion. The spin relaxation time scales are comparable to those of the usual dissipative currents. Finally, the Navier-Stokes limit of the Pauli-Lubanski vec…
Within the $σ-ω$ model of coupled nucleon-meson systems, a generalized relativistic Lenard--Balescu--equation is presented resulting from a relativistic random phase approximation (RRPA). This provides a systematic derivation of relativistic transport equations in the frame of nonequilibrium Green's function technique including medium effects as well as flucuation effects. It contains all possible processes due to one meson exchange and special attention is kept to the off--shell character of the particles. As a new feature of many particle effects, processes are possible which can be interpre…
Simulations of the viscous hydrodynamic model for relativistic heavy-ion collisions at RHIC and LHC energies are presented. Results for spectra, femtoscopy radii, and transverse momentum fluctuations are favorably compared to the experimental data. Effects of the local charge conservation on correlation observables are also studied.
The current status of the application of hydrodynamics to ultrarelativistic heavy ion collisions is reviewed. We elaborate on the arguments for strong transverse flow and rapid thermalization and discuss future applications and trends in hydrodynamics.
Freeze-out of particles in relativistic hydrodynamics is considered across a 3-dimensional space-time hypersurface. The conservation laws for time-like parts of the freeze-out hypersurface require different values of temperature, baryonic chemical potential and flow velocity in the fluid and in the final particle spectra. We analyze this freeze-out discontinuity and its connection to the shock-wave phenomena in relativistic hydrodynamics.
The chiral kinetic theory of Weyl fermions with collisions in the presence of weak electric and magnetic fields is derived from quantum field theories. It is found that the side-jump terms in the perturbative solution of Wigner functions play a significant role for the derivation. Moreover, such terms manifest the breaking of Lorentz symmetry for distribution functions. The Lorentz covariance of Wigner functions thus leads to modified Lorentz transformation associated with side-jump phenomena further influenced by background fields and collisions.
We describe the details of 3+1 dimensional relativistic hydrodynamic code for the simulations of quark-gluon/hadron matter expansion in ultra-relativistic heavy ion collisions. The code solves the equations of relativistic viscous hydrodynamics in the Israel-Stewart framework. With the help of ideal-viscous splitting, we keep the ability to solve the equations of ideal hydrodynamics in the limit of zero viscosities using a Godunov-type algorithm. Milne coordinates are used to treat the predominant expansion in longitudinal (beam) direction effectively. The results are successfully tested again…
