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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. Schl\"unzen et al., Phys. Rev. Lett. \textbf{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 \textbf{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 \textit{et al.}, Cond. Matt. Phys. \textbf{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 \textit{et al.}, Phys. Rev. B \textbf{108}, 205109 (2023)].

Correlated quantum many-particle systems out of equilibrium are of high interest in many fields, including correlated solids, ultracold atoms or dense plasmas. Accurate theoretical description of these systems is challenging both, conceptionally and with respect to computational resources. We have recently presented a quantum fluctuations approach which is equivalent to the nonequilibrium $GW$ approximation [E. Schroedter \textit{et al.}, Cond. Matt. Phys. \textbf{25}, 23401 (2022)] that promises high accuracy at low computational cost. In a second publication [E. Schroedter \textit{et al.}, Phys. Rev. B \textbf{108}, 205109 (2023)], this approach was extended to the two-time exchange-correlation functions and the density response properties. Here, we analyze the properties of this approach in more detail. We demonstrate that the method is 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.

The theory of Nonequilibrium Green functions (NEGF) has seen a rapid development over the recent three decades. Applications include diverse correlated many-body systems in and out of equilibrium. Very good agreement with experiments and available exact theoretical results could be demonstrated if the proper selfenergy approximations were used. However, full two-time NEGF simulations are computationally costly, as they suffer from a cubic scaling of the computation time with the simulation duration. Recently we have introduced the G1-G2 scheme that exactly reformulates the Kadanoff-Baym ansatz with Hartree-Fock propagators (HF-GKBA) into time-local equations, allowing for a dramatic reduction of the scaling to time-linear scaling [Schluenzen et al., Phys. Rev. Lett. \textbf{124}, 076601 (2020)]. Remarkably, this scaling is achieved quickly, and also for high-level selfenergies, including the nonequilibrium $GW$ and $T$-matrix approximations [Joost et al., Phys. Rev. B \textbf{101}, 245101 (2020)]. Even the dynamically screened ladder approximation is now feasible [Joost et al., Phys. Rev. B \textbf{105}, 165155 (2022)], and also applications to electron-boson systems were demonstrated. Here we present an overview on recent results that were achieved with the G1--G2 scheme. We discuss problems and open questions and present further ideas how to overcome the current limitations of the scheme.We illustrate the G1--G2 scheme by presenting applying it to the excitation dynamics of Hubbard clusters, to optical excitation of graphene, and to charge transfer during stopping of ions by correlated materials.

The quantum dynamics of correlated fermionic or bosonic many-body systems following external excitation can be successfully studied using nonequilibrium Green functions (NEGF) or reduced density matrix methods. Approximations are introduced via a proper choice of the many-particle selfenergy or decoupling of the BBGKY-hierarchy, respectively. These approximations are based on Feynman's diagram approaches or on cluster expansions into single-particle and correlation operators. In a recent paper [E. Schroedter, J.-P. Joost, and M. Bonitz, Cond. Matt. Phys. \textbf{25}, 23401 (2022)] we have presented a different approach where, instead of equations of motion for the many-particle NEGF (or density operators), equations for the correlation functions of fluctuations are analyzed. In particular, we derived the stochastic GW and polarization approximations that are closely related to the nonequilibrium GW approximation. Here, we extend this approach to the computation of two-time observables depending on the specific ordering of the underlying operators. In particular, we apply this extension to the calculation of the density correlation function and dynamic structure factor of correlated Hubbard clusters in and out of equilbrium.

The quantum dynamics of fermionic or bosonic many-body systems following external excitation can be successfully studied using two-time nonequilibrium Green's functions (NEGF) or single-time reduced density matrix methods. Approximations are introduced via a proper choice of the many-particle self-energy or decoupling of the BBGKY hierarchy. These approximations are based on Feynman's diagram approaches or on cluster expansions into single-particle and correlation operators. Here, we develop a different approach where, instead of equations of motion for the many-particle NEGF (or density operators), single-time equations for the correlation functions of fluctuations are analyzed. We present a derivation of the first two equations of the alternative hierarchy of fluctuations and discuss possible decoupling approximations. In particular, we derive the polarization approximation (PA) which is shown to be equivalent to the single-time version [following by applying the generalized Kadanoff-Baym ansatz (GKBA)] of the nonequilibrium $GW$ approximation with exchange effects of NEGF theory, for weak coupling. The main advantage of the quantum fluctuations approach is that the standard ensemble average can be replaced by a semiclassical average over different initial realizations, as was demonstrated before by Lacroix and co-workers [see e.g. D. Lacroix et al., Phys. Rev. B, 2014, 90, 125112]. Here, we introduce the stochastic $GW$ (SGW) approximation and the stochastic polarization approximation (SPA) which are demonstrated to be equivalent to the single-time $GW$ approximation without and with exchange, respectively, in the weak coupling limit. Our numerical tests confirm that our approach has the same favorable linear scaling with the computation time as the recently developed G1-G2 scheme [Schluenzen et al., Phys. Rev. Lett., 2020, 124, 076601]., Comment: 36 pages, 14 figures