The following issues are discussed inspired by the recent paper of Kadanoff (arXiv: 1403:6162): (a) Construction of a generalized one-particle Wigner distribution (GWD) function (analog of the classical distribution function) from which the quantum kinetic equation due to Kadanoff and Baym (KB) is derived, often called the Quantum Boltzmann Equation (QBE); (b) The equation obeyed by this has a collision contribution in the form of a two-particle Green function. This term is manipulated to have Kinetic Entropy in parallel to its counterpart in the classical Boltzmann kinetic equation for the classical distribution function. This proved to be problematic in that unlike in the classical Boltzmann kinetic equation, the contribution from the kinetic entropy term was non-positive; (3) Kadanoff surmised that this situation could perhaps be related to quantum entanglement that may not have been included in his theory. It is shown that GWD is not positive everywhere (indicating dynamical quantumness) just like the commonly recognized property of the Wigner function (negative property indicating quantumness of the state). The issue of non-positive feature appearing in approximate evaluation of patently positive entities in many particle systems is here pointed to an early discussion of this issue (Phys. Rev. A10, 1852 (1974)) in terms of a theorem on truncation of cumulant expansion of a probability distribution function due to Marcinkeiwicz. The last issue of presence or absence of entanglement in an approximate evaluation of a many particle correlation poses a new problem; it is considered here in terms of fermionic entanglement theory in the light of density matrix and Green function theory of many-fermion systems. The clue comes from the fact that the Hartree-Fock approximation exhbits no entantanglement in two-particle fermion density matrix and hence also in two-particle Green function.
The following issues are discussed inspired by the recent paper of Kadanoff (arXiv: 1403:6162): (a) Construction of a generalized one-particle Wigner distribution (GWD) function (analog of the classical distribution function) from which the quantum kinetic equation due to Kadanoff and Baym (KB) is derived, often called the Quantum Boltzmann Equation (QBE); (b) The equation obeyed by this has a collision contribution in the form of a two-particle Green function. This term is manipulated to have Kinetic Entropy in parallel to its counterpart in the classical Boltzmann kinetic equation for the classical distribution function. This proved to be problematic in that unlike in the classical Boltzmann kinetic equation, the contribution from the kinetic entropy term was non-positive; (3) Kadanoff surmised that this situation could perhaps be related to quantum entanglement that may not have been included in his theory. It is shown that GWD is not positive everywhere (indicating dynamical quantumness) just like the commonly recognized property of the Wigner function (negative property indicating quantumness of the state). The issue of non-positive feature appearing in approximate evaluation of patently positive entities in many particle systems is here pointed to an early discussion of this issue (Phys. Rev. A10, 1852 (1974)) in terms of a theorem on truncation of cumulant expansion of a probability distribution function due to Marcinkeiwicz. The last issue of presence or absence of entanglement in an approximate evaluation of a many particle correlation poses a new problem; it is considered here in terms of fermionic entanglement theory in the light of density matrix and Green function theory of many-fermion systems. The clue comes from the fact that the Hartree-Fock approximation exhbits no entantanglement in two-particle fermion density matrix and hence also in two-particle Green function.
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.
A recently developed method [Semkat et al., Phys. Rev. E 59, 1557 (1999); Kremp et al., in Progress in Nonequilibrium Green’s Functions (World Scientific, Singapore, 2000), p. 34] for incorporating initial binary correlations into the Kadanoff–Baym equations (KBE) is used to derive a generalized T-matrix approximation for the self-energies. It is shown that the T-matrix obtains additional contributions arising from initial correlations. Using these results and taking the time-diagonal limit of the KBE, a generalized quantum kinetic equation in binary collision approximation is derived. This equation is a far-reaching generalization of Boltzmann-type kinetic equations: It self-consistently includes memory effects (retardation, off-shell T-matrices) as well as many-particle effects (damping, in-medium T-matrices) and spin-statistics effects (Pauli-blocking).
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.
The pair distribution function (PDF) is a key quantity for the analysis of correlation effects of a quantum system both in equilibrium and far from equilibrium. We derive an expression for the PDF in terms of the single-particle Green functions—the solutions of the Keldysh/Kadanoff-Baym equations in the two-time plane—for a one- or two-component system. The result includes initial correlations and generalizes previous density matrix expressions from single-time quantum kinetic theory. Explicit expressions for the PDF are obtained in second Born approximation.
In a previous publication [H. S. K\"ohler, Phys. Rev. C 51, 3232 (1995)] equilibration rates were studied for an extended system of nonequilibrium nuclear matter. The two-time Green's functions in the Kadanoff-Baym (KB) formalism were solved numerically. These quantum-mechanical calculations were compared with the Markovian (semi)classical Uehling-Uhlenbeck collision term often used in the analysis of collisions between heavy ions. The equilibration was found to be substantially slowed down by quantum effects. The theoretical as well as experimental study of nonequilibrium systems is of course of great interest in many areas of physics, e.g., within the realm of quantum transport in solid state devices. On the theoretical side various methods and approximations are the subject of intensive study. By the KB ansatz, for example, the equations can be simplified to a one-time kinetic equation while preserving the non-Markovian character. The generalized KB (GKB) ansatz of Lipavsky, Spicka, and Velicky [Phys. Rev. B 34, 6933 (1986)] is often preferred. In this paper approximations based upon the GKB ansatz are discussed in relation to the exact KB formalism. Numerical comparisons are made for both the strongly interacting nuclear medium as well as an electron plasma. A non-Markovian collision integral, including memory effects in a quasiparticle approximation, is often discussed in the literature. It is found that this approximation appreciably overestimates the collision rates even more than the classical (Boltzmann) approximation. The inclusion of a width in the spectral function, i.e., going beyond the quasiparticle approximation, gives considerable improvement. The competition between the memory and correlation effects is found to lead to an anomalous saturation of the relaxtion rate. \textcopyright{} 1996 The American Physical Society.
We propose a new method of calculating entanglement entropy of a many-body interacting Bosonic system (open or closed) in a field theoretic approach without replica methods. The Wigner function and Renyi entropy of a Bosonic system undergoing arbitrary non-equilibrium dynamics can be obtained from its Wigner characteristic function, which we identify with the Schwinger Keldysh partition function in presence of quantum sources turned on at the time of measurement. For non-interacting many body systems, starting from arbitrary density matrices, we provide exact analytic formulae for Wigner function and entanglement entropy in terms of the single particle Green's functions. For interacting systems, we relate the Wigner characteristic to the connected multi-particle correlators of the system. We use this formalism to study the evolution of an open quantum system from a Fock state with negative Wigner function and zero entropy, to a thermal state with positive Wigner function and finite entropy. The evolution of the Renyi entropy is non-monotonic in time for both Markovian and non-Markovian dynamics. The entropy is also found to be anti-correlated with negativity of the Wigner function of a 2 -mode open quantum system.