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Chinese Physics C

NON-RELATIVISTIC BUU EQUATION

Liang Ge, Zhuo Ye

Published Jan 1, 1989• 0 citations
OpenAlex

Abstract

The Boltzmann-Uehlig-Uhlenbeck (BUU) equation, which is the time evolution of the wigner function of the single particle Green's function, is derived by using the closed-time Green's function approach. The quantum mechanical approximation in deriving the BUU equation is discussed.

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The Boltzmann equation is the traditional framework in which one extends the time-dependent mean field classical description of a many-body system to include the effect of particle-particle collisions in an approximate manner. A semiclassical extension of this approach to quantum many-body systems was suggested by Uehling and Uhlenbeck in 1933 for both Fermi and Bose statistics, and many further generalization of this approach are known as the Boltzmann-Uehling-Uhlenbeck (BUU) equations. Here I suggest a pure quantum version of the BUU type of equations, which is mathematically equivalent to a generalized Time-Dependent Density Functional Theory extended to superfluid systems.
The Boltzmann equation is the traditional framework in which one extends the time-dependent mean field classical description of a many-body system to include the effect of particle-particle collisions in an approximate manner. A semiclassical extension of this approach to quantum many-body systems was suggested by Uehling and Uhlenbeck in 1933 for both Fermi and Bose statistics, and many further developments of this approach are known as the Boltzmann-Uehling-Uhlenbeck (BUU) equations. Here I introduce a pure quantum version of the BUU type of equations, which is mathematically equivalent to a generalized time-dependent density functional theory extended to superfluid systems. As expected, during nonequilibrium processes the quantum Boltzmann one-body entropy increases during evolution.
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 the framework of real-time Green's functions, the general kinetic equations are investigated in a first-order gradient expansion. Within this approximation, the problem of the reconstruction of the two-time correlation functions from the one-time Wigner function was solved. For the Wigner function, a cluster expansion is found in terms of a quasiparticle distribution function. In equilibrium, this expansion leads to the well-known generalized Beth-Uhlenbeck expression of the second virial coefficient. As a special case, the T-matrix approximation for the self-energy is investigated. The quantum kinetic equation derived thus has, besides the (Markovian) Boltzmann collision integral, additional terms due to the retardation expansion which reflect memory effects. Special interest is paid to the case that bound states exist in the system. It is shown that the bound state contribution, which can be introduced via a bilinear expansion of the two-particle T matrix, follows from the first-order retardation term in the general kinetic equation. The full Wigner function is now a sum of one function describing the unbound particles and another one for the bound state contribution. The latter two functions have to be determined from a coupled set of kinetic equations. In contrast to the quantum Boltzmann equation, energy and density of a nonideal system are conserved. \textcopyright{} 1996 The American Physical Society.
The inversion method to derive the quantum kinetic equation is extended to the case of an inhomogeneous system. Within the closed-time-path formalism, the source is introduced to probe the (Fourier transform of the) Wigner distribution function and the equation of motion for the Wigner function is derived perturbatively by the inversion formulae. The equation of motion has the form of the generalized Boltzmann equation, which is non-local in space and time. For the obtained kinetic equation, the local conservation laws of number, energy and momentum are confirmed.