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.
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].
Using the method of nonequilibrium statistical operator by Zubarev, an approach is proposed for the description of kinetics which takes into account the nonlinear hydrodynamic fluctuations for a quantum Bose system.Non-equilibrium statistical operator is presented which consistently describes both the kinetic and nonlinear hydrodynamic processes.Both a kinetic equation for the nonequilibrium one-particle distribution function and a generalized Fokker-Planck equation for nonequilibrium distribution function of hydrodynamic variables (densities of momentum, energy and particle number) are obtained.A structure function of hydrodynamic fluctuations in cumulant representation is calculated, which makes it possible to analyse the generalized Fokker-Planck equation in Gaussian and higher approximations of the dynamic correlations of hydrodynamic variables which is important in describing the quantum turbulent processes.
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.
By means of a quantum mechanical phase space distribution function
introduced by von Roos, the Schroedinger equation for a non-relativistic
system of N identical particles with scalar interactions is transformed
into a quantum mechanical generalization of the Liouville equation,
thereby formulating the problem in terms of a generalized density in
phase space, a quantity of primary interest in most treatments of the
corresponding classical system (or plasma). This transformation permits
a parallel development of the theories of classical and quantum plasmas
and thus allows the quantum many-body problem to be discussed virtually
completely in classical terms. In particular, a kinetic theory of
quantum plasmas is obtained by deriving the quantum analogue of the BBGKY
hierarchy, and applying thereto approximation techniques similar to those
of Rostocker and Rosenbluth, and Bogoliubov. The point of departure from
similar previous studies based on the Wigner distribution function is that
the proper exchange symmetry can be tractably introduced into the formalism. Attention is first focused on the Hartree and Hartree-Fock approximations,
in which case the quantum BBGKY system reduces to a simple quantum
generalization of the Vlasov equation. This equation is used to study
the response of spatially homogeneous systems to weak external forces,
and the associated problems of plasmon and spin-wave excitations. It is
also used to derive the quantum and exchange corrected equations of
inviscid hydrodynamical transport which are then applied to the problem
of sound propagation in the degenerate electron gas. The second part of the study is concerned with the theory of the
many-electron atom in the Hartree and Hartree-Fock approximations.
The relevant quantum Vlasov equations lead naturally to a statistical
theory of the atom which reduces to the Thomas-Fermi-Amaldi and Thomas Fermi
models (respectively) as ħ → 0. For ħ ≠ 0, the quantum and
exchange corrections to these models are simultaneously generated. The
quantum hydrodynamical theory developed earlier is used to determine
the influence of these corrections on the boundary conditions of the
model, and a theory of the compressed atom is consequently obtained.
Considered in somewhat less detail are the effects of non-zero temperature,
net orbital angular momentum, relativity and correlations, as well as
time dependent processes. The final part deals with the problem of the degenerate electron
gas with a uniform neutralizing background. Going beyond the Hartree-Fock approximation, the pair functions for particles with
parallel and anti-parallel spin are obtained by neglecting three particle
correlations. From these functions, a quantum-mechanical collision
integral is derived which differs from that obtained by Silin
and Guernsey and conjectured by Wyld and Pines in that dynamical exchange
effects are included. Also obtained from the pair function
is an expression for the correlation which reduces in the high
density limit to the result of Gell-Mann and Brueckner. At intermediate
densities an additional term appears in the energy due to the screening
of the exchange interaction by the dielectric properties of the medium.
It is evaluated in the high density limit and found to be -0.151 r_s ln r_s
Ryd/electron in marked disagreement with the corresponding value obtained
by DuBois.