Relativistic hydrodynamics and transport in strongly correlated systems

This thesis investigates second-order relativistic hydrodynamics and transport coefficients in strongly correlated systems. Our focus is mainly on the physical conditions relevant to heavy-ion collisions, as well as compact dense stellar objects at nonzero temperatures and in strong magnetic fields. Chapter 1 provides a brief introduction to the area of research covered by this thesis, specifically relativistic hydrodynamics and transport in hot and dense media, which occur in heavy-ion collisions and heated stellar matter. In Chapter 2 we give a new formulation of second-order dissipative hydrodynamics for relativistic systems using Zubarev's non-equilibrium statistical operator approach. We first solve the quantum Liouville equation with an infinitesimal source term to construct a non-equilibrium statistical operator which is a non-local functional of the thermodynamic parameters and their space-time gradients. Exploiting then the gradient expansion of the statistical operator we derive transport equations for the shear stress tensor, the bulk viscous pressure and the flavour diffusion currents up to the second order in hydrodynamic gradients. We show that the second-order corrections to the dissipative fluxes arise from (i) the quadratic terms of the Taylor expansion of the statistical operator; and (ii) the linear terms which are nonlocal in space and time. These non-local corrections generate finite relaxation time scales in the evolution of the dissipative quantities. We derive the most generic form of the transport equations which involve gradients of the dissipative fluxes, as well as products of two first-order quantities (i.e., either thermodynamic forces or dissipative fluxes). We then go on to express the first- and the second-order transport coefficients, which appear in these equations, via certain two- and three-point equilibrium correlation functions. Finally, we express the relaxation times for the dissipative fluxes via the frequency-derivatives of the corresponding first-order transport coefficients. In Chapter 3 we compute the transport coefficients of quark matter in the strong coupling regime within the two-flavor Nambu-Jona-Lasinio model. We apply the Kubo-Zubarev formalism to obtain the thermal and the electrical conductivities as well as the shear and the bulk viscosities by evaluating the corresponding equilibrium two-point correlation functions at the leading order in the 1/N_c expansion. In this approximation the conductivities and the shear viscosity are given by single-loop skeleton diagrams, whereas the bulk viscosity includes an infinite geometrical series of multi-loop diagrams. The dispersive effects that lead to nonzero transport coefficients arise from quark-meson fluctuations above the Mott transition temperature T_M, where meson decay into two on-mass-shell quarks is kinematically allowed. We find that the conductivities and the shear viscosity are decreasing functions of temperature and density above T_M. We also show that the Wiedemann-Franz law does not hold. The ratio of the shear viscosity to the entropy density is larger than unity close to the Mott temperature and approaches the AdS/CFT bound at higher temperatures. We conjecture on the basis of the uncertainty principle that the ratio of the thermal conductivity to the heat capacity per unit volume is bounded from below by 1/18. The case of the bulk viscosity turns out to be special, because the multi-loop contributions dominate the single-loop contribution close to the Mott line in the case where the chiral symmetry is explicitly broken. We find that in this case only at high temperatures the one-loop contribution becomes dominant. The resulting bulk viscosity exceeds the shear viscosity close to the Mott temperature by factors 5-20 when multi-loop contributions are included. In the high-temperature domain the bulk viscosity is negligible compared to the shear viscosity. For practical applications we provide simple, but accurate fits to the transport coefficients, which can facilitate the implementation of our results in hydrodynamics codes. In Chapter 4 we compute the electrical conductivity of finite temperature, strongly magnetized crust of a compact star which may be formed in the aftermath of a supernova explosion, binary neutron star merger, or during accretion processes in X-ray binaries. We focus on the temperature-density regime where plasma is in the liquid state and, therefore, the conductivity is dominated by the electron scattering off correlated nuclei. The dynamical screening of electron-ion interaction is implemented in terms of the polarization tensor computed in the hard-thermal-loop (HTL) effective field theory of QED plasma. The correlations of the background ionic component are accounted for via a structure factor derived from Monte Carlo simulations of one-component plasma. With this input we solve the Boltzmann kinetic equation in relaxation time approximation taking into account the anisotropy of transport due to the magnetic field. The electrical conductivity tensor is studied numerically as a function of temperature, density, magnetic field and the crust composition in a broad parameter range. We find that the conductivity as a function of temperature attains a minimum at the transition from the degenerate to the nondegenerate regime of electrons. We also provide accurate fit formulas to our numerical results for three components of the conductivity tensor. In addition, we provide supplemental tables which can be used in dissipative magneto-hydrodynamics(MHD) simulations of warm compact stars. We summarize our results and discuss the perspectives in Chapter