The nonequilibrium statistical operator method is used to construct a two-velocity relativistic hydrodynamics of a quantum-field system with broken symmetry and complex scalar order parameter. The approach is based on model expansions of the operators (the energy-momentum tensor, the current density, etc.) used for statistical description of the system in the hydrodynamic stage. In the first part of the paper, these expansions are used to construct the statistical operator and the thermodynamics of a local-equilibriu m state. Below, we construct a quantum relativistic two-velocity hydrodynamics on the basis of first principles using Zubarev's nonequilibrium statistical operator method (11}. In Sec.2, we establish the connection between the 4-velocity of the motion of the ordered component with the gradient of the phase of the order parameter. In Sec.3 we note that the methods of (2-7) based on the use of unitary transformations in the Galileo group to obtain the equations of two-velocity hydrodynamics do not admit direct generalization to the relativistic domain. In this connection, we propose an approach based on model expansions of the energy-momentum tensor and the current density (and, possibly, other operators used for statistical des- cription of systems in the hydrodynamic stage). These expansions are then used in Sec.4 to construct the statistical operator and the thermodynamics of the local-equilibrium state. The transport relations and the hydrodynamic equations will be discussed in Part If. Below, we use a system of units in which c = l~ = k = 1. The signature of the metric tensor is
The nonequilibrium statistical operator method is used to construct a two-velocity relativistic hydrodynamics of a quantum-field system with broken symmetry and complex scalar order parameter. The approach is based on model expansions of the operators (the energy-momentum tensor, the current density, etc.) used for statistical description of the system in the hydrodynamic stage. In the first part of the paper, these expansions are used to construct the statistical operator and the thermodynamics of a local-equilibriu m state. Below, we construct a quantum relativistic two-velocity hydrodynamics on the basis of first principles using Zubarev's nonequilibrium statistical operator method (11}. In Sec.2, we establish the connection between the 4-velocity of the motion of the ordered component with the gradient of the phase of the order parameter. In Sec.3 we note that the methods of (2-7) based on the use of unitary transformations in the Galileo group to obtain the equations of two-velocity hydrodynamics do not admit direct generalization to the relativistic domain. In this connection, we propose an approach based on model expansions of the energy-momentum tensor and the current density (and, possibly, other operators used for statistical des- cription of systems in the hydrodynamic stage). These expansions are then used in Sec.4 to construct the statistical operator and the thermodynamics of the local-equilibrium state. The transport relations and the hydrodynamic equations will be discussed in Part If. Below, we use a system of units in which c = l~ = k = 1. The signature of the metric tensor is
Relativistic hydrodynamics is a low-energy e↵ective theory which universally describes macroscopic behaviors of relativistic many-body systems.Its application covers the broad branches of physics such as high-energy physics, astrophysics, and condensed matter physics.Nevertheless, its derivation from underlying microscopic theories, especially from quantum field theories, has not been clearly clarified.Furthermore, it is recently pointed out that novel transport phenomena such as the chiral magnetic e↵ect, which originates from the quantum anomaly in the underlying quantum field theories, take place in the macroscopic hydrodynamic regime.In this thesis, we derive relativistic hydrodynamics from quantum field theories on the basis of the recent development of nonequilibrium statistical mechanics.In order to derive the hydrodynamic equations we introduce an assumption that the density operator is given by a local Gibbs distribution at initial time, and decompose the energy-momentum tensor and charge current into nondissipative and dissipative parts.This leads to a generalization of the Gibbs ensemble method canonically employed in equilibrium statistical mechanics.Our formalism is also applicable to the situation in the presence of the quantum anomaly, and we can describe the anomaly-induced tranport phenomena.As a basic tool of our formalism, we first develop a path integral of the thermodynamic potential for locally thermalized systems.We show microscopically that the thermodynamic pontential, which is shown to be the generating functional of systems in local thermal equilibrium, is written in terms of the quantum field theory in the curved spacetime with one imaginary-time direction.The structure of this thermally emergent curved spacetime is determined by hydrodynamic variables such as the local temperature, and fluid-four velocity, and possesses notable symmetry properties: Kaluza-Klein gauge symmetry, spatial di↵eomorphism symmetry, and gauge symmetry.With the help of the symmetry argument, we can construct the nondissipative part of the hydrodynamic equations including the anomaly-induced transport phenomena.By the use of the perturbative calculation, we evaluate the anomalous transport coe cients at one-loop level.Furthermore, we also construct a solid basis to study dissipative corrections to hydrodynamic equations.In particular, by performing the derivative expansion, together with the result on nondissipative part of the constitutive relations, we derive the firstorder dissipative hydrodynamic equations, that is, the relativistic Navier-Stokes equation.Our formalism also provides the quantum field theoretical expression of the Green-Kubo formulas for transport coe cients.iii The formulation and whole works presented from Chapter 2 to Chapter 4 are based on our original work in collaboration with Yoshimasa Hidaka (RIKEN).The work presented in Chapter 3 is in collaboration with Yoshimasa Hidaka (RIKEN), Tomoya Hayata (RIKEN), and Toshifumi Noumi (Hong Kong University of Science and Technology) [1].1 There is close investigation of priority to the Navier-Stokes equation in Ref. [11].
The macroscopic hydrodynamic equations are derived for many-body systems in the local-equilibrium approach, using the Schrödinger picture of quantum mechanics. In this approach, statistical operators are defined in terms of microscopic densities associated with the fundamentally conserved quantities and other slow modes possibly emerging from continuous symmetry breaking, as well as macrofields conjugated to these densities. Functional identities can be deduced, allowing us to identify the reversible and dissipative parts of the mean current densities, to obtain general equations for the time evolution of the conjugate macrofields, and to establish the relationship to projection-operator methods. The entropy production is shown to be nonnegative by applying the Peierls-Bogoliubov inequality to a quantum integral fluctuation theorem. Using the expansion in the gradients of the conjugate macrofields, the transport coefficients are given by Green-Kubo formulas and the entropy production rate can be expressed in terms of quantum Einstein-Helfand formulas, implying its nonnegativity in agreement with the second law of thermodynamics. The results apply to multicomponent fluids and can be extended to condensed matter phases with broken continuous symmetries.
We derive relativistic hydrodynamics from quantum field theories by assuming that the density operator is given by a local Gibbs distribution at initial time. We decompose the energy-momentum tensor and particle current into nondissipative and dissipative parts, and analyze their time evolution in detail. Performing the path-integral formulation of the local Gibbs distribution, we microscopically derive the generating functional for the nondissipative hydrodynamics. We also construct a basis to study dissipative corrections. In particular, we derive the first-order dissipative hydrodynamic equations without a choice of frame such as the Landau-Lifshitz or Eckart frame.
In this thesis, we present and discuss the quantum statistical foundations of relativistic hydrodynamics with special emphasis on the entropy current. We show that it is possible to provide a rigorous definition for this quantity in the framework of relativistic quantum statistical mechanics and we put forward a general method to calculate it at local thermodynamic equilibrium. The method is based on the proof of existence of the thermodynamic potential current, a usually tacitly understood hypothesis. We then apply the method for the entropy current to the study of two different systems. The first system is a relativistic quantum fluid at global thermodynamic equilibrium, phenomenologically related to the quark-gluon plasma and, from a quantum field theoretical standpoint, to the Unruh effect. The second systems is a relativistic quantum fluid with boost invariance, also related to the quark-gluon plasma and of great interest as the first solvable system out of equilibrium. The thermal expectation value of the energy-momentum tensor as well as the entropy current are calculated for both systems, and renormalization is discussed.
Differential conservation equations are derived for the mass-, momentum-, and energy-density operators for a 1-component simple fluid of Bose or Fermi particles with arbitrary pairwise interactions. These equations are used in a statistical mechanical derivation of exact equations of motion for the expectations of these operators. The equations of motion are coupled to equations relating these expectations to the local temperature, chemical potential, and fluid velocity. The coupled equations are closed in the sense that the expectations and their thermodynamic conjugates listed above are the only unknowns, although some of the dependence in the equations on the conjugates is expressed only implicitly. The equations of motion are memory-retaining nonlocal generalizations of the classical hydrodynamic equations and apply to a normal fluid arbitrarily far from equilibrium. The formalism is not carried as far as has the corresponding classical formalism because the local equilibrium expectation of the momentum density here does not equal the fluid velocity times the expectation of the mass density as is true in classical statistical mechanics.
Using the method of non-equilibrium statistical Zubarev operator an approach to the description of kinetics taking into account the nonlinear hydrodynamic fluctuations for quantum Bose system is proposed. The non-equilibrium statistical operator that consistently describes both the kinetic and nonlinear hydrodynamic fluctuations in quantum liquid is calculated. A kinetic equation for the non-equilibrium one-particle distribution function and generalized Fokker-Planck equation for non-equilibrium distribution function of hydrodynamic variables: densities of momentum, energy and particle numbers are obtained. A structural function of hydrodynamic fluctuations in cumulant representation is calculated. It provides a possibility to analyzing the generalized Fokker-Planck equation in Gaussian and higher approximations for dynamic correlations of hydrodynamic variables that is important to describe the quantum turbulent processes. The generalized Gross-Pitaevkii equation for the order parameter which takes into account nonlinear hydrodynamic fluctuations is obtained.