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Progress of Theoretical Physics

The Quantum-statistical Theory of Transport Phenomena, III

Syû Ono

Published Aug 1, 1954• 30 citations
Publisher

Abstract

In order to introduce the coarse-grained measurement of coordinates in quantum mechanics, the whole space is divided into cubic cells, and the coordinate of the center of cell is regarded as the coarse-grained coordinate of every point inside the cell. The function which vanishes outside a certain cell and which agrees with plane wave within the same cell is called a cell function. If one imposes a certain type of boundary condition upon such cell functions, the set of these functions is complete and orthonormal, and gives a generalized type of the Wigner distribution functions, from which the coarse-grained distribution functions are derived. The change of the density matrices with time is calculated by means of the method analogous to the time-dependent perturbation theory. From this result the collision terms of the Uehling-Uhlenbeck equation are derived on the basis of the equation of motion for the generalized phase-space distribution function, under the assumption that the maximum range of the intermolecular force is sufficiently short compared with the edge length of elementary cell. Owing to the cell funcions employed instead of the plane waves, the distribution functions retain dependence on the spatial coordinates irrespective of the postulate of random a priori phases, which plays an important role in derivation of the collision terms. In the present article, this postulate is necessary for derivation of the stresming terms, in which the differentiation operators with respect to the coarse-grained coordinates and the coarse-grained distribution functions appear. The domain of validity of the Uehling-Uhlenbeck equation is discussed, and, in conclusion, it seems doubtful that the Uehling-Uhlenbeck equation is valid even for the transport phenomena in the degenerate Bose gas, on account of the singularity of the distribution function in momentum space.

DOI
10.1143/ptp.12.113

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In order to introduce the coarse-grained measurement of coordinates in quantum mechanics, the whole space is divided into cubic cells, and the coordinate of the center of cell is regarded as the coarse-grained coordinate of every point inside the cell. The function which vanishes outside a certain cell and which agrees with plane wave within the same cell is called a cell function. If one imposes a certain type of boundary condition upon such cell functions, the set of these functions is complete and orthonormal, and gives a generalized type of the Wigner distribution functions, from which the coarse-grained distribution functions are derived. The change of the density matrices with time is calculated by means of the method analogous to the time-dependent perturbation theory. From this result the collision terms of the Uehling-Uhlenbeck equation are derived on the basis of the equation of motion for the generalized phase-space distribution function, under the assumption that the maximum range of the intermolecular force is sufficiently short compared with the edge length of elementary cell. Owing to the cell funcions employed instead of the plane waves, the distribution functions retain dependence on the spatial coordinates irrespective of the postulate of random a priori phases, which plays an important role in derivation of the collision terms. In the present article, this postulate is necessary for derivation of the stresming terms, in which the differentiation operators with respect to the coarse-grained coordinates and the coarse-grained distribution functions appear. The domain of validity of the Uehling-Uhlenbeck equation is discussed, and, in conclusion, it seems doubtful that the Uehling-Uhlenbeck equation is valid even for the transport phenomena in the degenerate Bose gas, on account of the singularity of the distribution function in momentum space.
A derivation of the quantum-mechanical analog of the Maxwell-Boltzmann equation of transport for gases at low density is presented. The analysis requires: the definition of a time differential which is necessary for the selection of only secular variations in the time dependence of the distribution functions (coarse-graining in time); a binary collision approximation; the connection between the phase-space transformation function, relating the Wigner distribution function at different times, and the transition matrix in the theory of scattering of two particles; and a coarse graining in configuration space leading to a kinetic equation for spatially nonuniform systems. A consequence of the binary collision approximation is the exclusion of the effects of symmetrization (Bose-Einstein or Fermi-Dirac statistics) resulting in density-dependent terms.
Close to equilibrium, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. The general expression derived for the kernel is evaluated to second order in the interparticle potential. The result is a wavevector- and frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born approximation cross section. The theory is formulated in terms of second-quantized phase space operators whose equilibrium averages are the n-particle Wigner distribution functions. Convenient expressions for the commutators and anticommutators of the phase space operators are obtained. The two-particle equilibrium distribution function is analyzed in terms of momentum-dependent quantum generalizations of the classical pair distribution function h(k) and direct correlation function c(k). The kinetic equation is presented as the equation of motion of a two -particle correlation function, the phase space density-density anticommutator, and is derived by a formal closure of the quantum BBGKY hierarchy. An alternative derivation using a projection operator is also given. It is shown that the method used for approximating the kernel by a second order expansion preserves all the sum rules to the same order, and that the second-order kernel satisfies the appropriate positivity and symmetry conditions.
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.
The theory of gases in nonstationary states, given by Lorentz and Enskog, is generalized for the quantum statistics to give the hydrodynamical equations and the distribution function in first and second approximation, and formal expressions for the viscosity and heat conductivity coefficients. These results are valid for all statistics and for all degrees of degeneration. Two essential points contribute to this generality: (a) Exact expressions independent of statistics and degeneration can be given for the coordinate and time derivatives of the coefficient AAA of the equilibrium distribution function in terms of the pressure and temperature gradients and time derivatives, though a closed expression for this coefficient as a function of vvv and TTT is known only in limiting cases; (b) The function WWW of the general equation of state for ideal gases in all statistics pv=(RTM)W(v23T)pv=(\frac{\mathrm{RT}}{M})W({v}^{\frac{2}{3}}T)pv=(MRT​)W(v32​T) is adiabatically invariant. Numerical values of the viscosity and heat conductivity coefficients, which should come out of the formal theory on the introduction of special assumptions about the molecular forces, have not yet been obtained. It is our hope that these results, when found, may furnish an experimental test of the existence of Einstein-Bose statistics in real gases, as is required by theory.
The quantum Boltzmann equation (QBE), originally formulated as the quantum extension of Boltzmann’s classical transport theory by Uehling and Uhlenbeck and subsequently developed within the nonequilibrium Green-function formalisms of Kadanoff–Baym and Keldysh, provides the central kinetic framework for describing relaxation and transport in weakly interacting quantum many-body systems. Although its formal structure is well established, conventional derivations often obscure the microscopic equivalence between operator and nonequilibrium Green-function approaches. Wick factorization is typically applied without explicitly enumerating real-time contractions, while Green-function treatments encode the same physics in Born-level self-energies that conceal their operator origin. In this paper, we present a pedagogical microscopic derivation of the quantum Boltzmann equation that makes this equivalence fully explicit. Starting from the ladder-operator formulation, we enumerate all fifteen Wick pairings of the second-order six-field correlator, identifying three Hartree–Fock contributions and twelve real-time Born scattering channels generated by a quartic, number-conserving interaction Hamiltonian. We then establish a one-to-one correspondence between these twelve operator contractions and the Keldysh components of the second-order Born self-energy, demonstrating explicitly how the Uehling–Uhlenbeck collision integral emerges after Wigner transformation and quasiparticle projection. Under controlled Born–Markov and quasiparticle approximations, this work offers a transparent synthesis of operator and Green-function formulations, and clarifies how kinetic fluctuations can be understood as emergent under coarse-graining.
In this paper, we derive the equations characterizing the boundary layer which describes the transition of the distribution function of a gas of weakly interacting bosons to the distribution function of the gas in the presence of a Bose–Einstein condensate. To this end, we first rederive the classical Uehling–Uhlenbeck equation very briefly, taking as a starting point the dynamics of a system of many weakly interacting quantum particles. The solutions of the Uehling–Uhlenbeck equation yield blow up in finite time. Near the blow-up time, the approximations used to derive the Uehling–Uhlenbeck equation break down. We derive the set of equations that describe the building of correlations and the onset of quantum interference effects for the many-particle Hamiltonian system under the assumption that the blow-up for the Uehling–Uhlenbeck equation takes place in a self-similar form.