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.
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.
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 Kadanoff-Baym (KB) equations are solved numerically for infinite nuclear matter. In particular we calculate correlation energies and correlation times. Approximating the Green's functions in the KB collision kernel by the free Green's functions the Levinson equation is obtained. This approximation is valid for weak interactions and/or low densities. It relates to the extended quasiclassical approximation for the spectral function. Comparing the Levinson, Born, and KB calculations allows for an estimate of higher order spectral corrections to the correlations. A decrease in binding energy is reported due to spectral correlations and off-shell parts in the reduced density matrix.
Abstract Correlated classical and quantum many‐particle systems out of equilibrium are of high interest in many fields, including dense plasmas, correlated solids, and ultracold atoms. Accurate theoretical description of these systems is challenging both, conceptionally and with respect to computational resources. While for classical systems, in principle, exact simulations are possible via molecular dynamics, this is not the case for quantum systems. Alternatively, one can use many‐particle approaches such as hydrodynamics, kinetic theory, or nonequilibrium Green functions (NEGF). However, NEGF exhibit a very unfavorable cubic scaling of the CPU time with the number of time steps. An alternative is the G1–G2 scheme [N. Schlünzen et al., Phys. Rev. Lett. 124, 076601 (2020)] which allows for NEGF simulations with time linear scaling, however, at the cost of large memory consumption. The reason is the need to store the two‐particle correlation function. This problem can be overcome for a number of approximations by reformulating the kinetic equations in terms of fluctuations – an approach that was developed, for classical systems, by Yu.L. Klimontovich [JETP 33, 982 (1957)]. Here, we present an overview of his ideas and extend them to quantum systems. In particular, we demonstrate that this quantum fluctuations approach can reproduce the nonequilibrium GW approximation [E. Schroedter et al., Cond. Matt. Phys. 25, 23401 (2022)] promising high accuracy at low computational cost which arises from an effective semiclassical stochastic sampling procedure. We also demonstrate how to extend the approach to the two‐time exchange‐correlation functions and the density response properties. [E. Schroedter et al., Phys. Rev. B 108, 205109 (2023)]. The results are equivalent to the Bethe–Salpeter equation for the two‐time exchange‐correlation function when the generalized Kadanoff‐Baym ansatz with Hartree‐Fock propagators is applied [E. Schroedter and M. Bonitz, phys. stat. sol. (b) 2024, 2300564].
We solve the Kadanoff-Baym equations for nonequilibrium initial configurations of the φ 4-theory in 2+1 dimensions and compare to explicit solutions of generalized transport equations for the same theory. The latter transport equations are derived from the Kadanoff-Baym equation in a first order gradient expansion in phase space and explicitly retain the off-shell dynamics as inherent in the timedependent spectral functions. The solutions of these equations compare very well with the exact solutions of the full Kadanoff-Baym equations with respect to the occupation numbers of the individual modes, the spectral evolution as well as the chemical equilibration process. Furthermore, the proper equilibrium off-shell distribution is reached for large times contrary to the quasiparticle Boltzmann limit. We additionally present a direct comparison of the solution of the generalized transport equations in the Kadanoff-Baym and Botermans-Malfliet form; both solutions are found to agree very well with each other. PACS: 06.60.+w; 05.70.Ln; 11.10.Wx; 24.10.-i; 25.75.-q Keywords: Many-body theory; finite temperature field theory; out-of-equilibrium quantum field theory; nonequilibrium statistical physics; relativistic heavy-ion collisions Part of the PhD thesis of S. Juchem
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 method of nonequilibrium Green functions (NEGFs) constitutes a solid framework to describe closed and open quantum systems. It captures situations far from equilibrium including arbitrary perturbations and correlations - effects higher than first order in the binary interactions. On the contrary, in the absence of external fields, it reduces to the imaginary-time Matsubara formalism. The abilities (and successes) of the method originate from its self-consistency and the conserving character of many-body approximations that are introduced by diagram technique. On this account, NEGFs have, now for a long time, brought insight into the dynamics of nuclear matter, non-ideal quantum gases and (semiconductor) plasmas. However, applications to localized, finite and strongly interacting systems have emerged only within the recent decade. The dynamical properties follow from quantum kinetic equations - the two-time Kadanoff-Baym equations for the one-particle NEGF - the numerical solution of which is the central topic of this thesis. The non-Markovian structure of these equations thereby inhibits an unlimited, gradual propagation of the nonequilibrium Green function: Any self-consistent, future dynamics of the system is tightly linked to the (complete) past by non-local self-energies. Further, the success in obtaining solutions is being diminished if the system under investigation is not isotropic or inhomogeneous and, in addition, the spatial topology (e.g., the charge carrier or spin density) needs to be resolved in time. On top of this, small isolated systems are generally not candidates showing fast decoherence which offers little hope for smooth and well-damped integration kernels. Despite these difficulties, the thesis at hand is devoted to the computation of the two-time NEGF for finite, closed systems of which ensembles of electrons bound by nuclei forming atoms or molecules are the most natural representatives. Here, in the spirit of the well-studied one-dimensional helium atom, the electron dynamics in model atoms and molecules is addressed in ab initio fashion. To render calculations possible in the first place, (i) a novel hybrid representation of the NEGF is introduced based on the conception of finite elements (FE) and the discrete variable representation (DVR), and (ii) a parallel algorithm is developed that allows for an efficient, well scalable distributed-memory computation of the full two-time Green function. The first point attributes plenty of flexibility and makes the method readily adjustable to different Hamiltonians avoiding the need to, by hand or in intricate numerical manner, calculate the whole bunch of matrix elements of one- and two-particle operators. Also, in consequence, it drastically simplifies the evaluation of self-energies. The second point makes the NEGF approach ready for large-scale, high-performance computing facilities, the resources (of computer power: mainly memory!) of which are indispensable for (some of) the calculations presented. The applicability of the developed methods is demonstrated inter alia on the basis of helium, beryllium, hydrogen and lithium hydride modeled in one spatial dimension whereby electron-electron correlations are treated in the second(-order) Born approximation. After a thorough discussion on the self-consistent ground states, the thesis focuses on the electron dynamics in the linear response regime as well as in the presence of strong external fields. Overall, to assess the the performance of the second Born approximation, the results are compared to the Hartree-Fock approximation, on the one hand, and to the exact solution of the time-(in)dependent Schrödinger equation, on the other hand. In addition, aiming at the possibility of extended time-dependent calculations, the behavior of the generalized Kadanoff-Baym ansatz is discussed.