Abstract
This chapter presents a re-consideration of the second Born approximation that was studied in Chap. 6 and discusses its relation to the familiar SBA for the selfenergy of nonequilibrium Green functions theory (NEGF) that are subject of Chap. 13. The key difference lies in additional renormalization effects characteristic for NEGF. In this chapter we demonstrate how the SBA of NEGF can be recovered in density operator theory and give the appropriate approximation for the two- and three-particle correlation operators. We derive the solution for the renormalized pair correlation operator and the quantum Landau equation with selfenergy effects included. The direct correspondence between density operators and NEGF is established in Sect. 7.4 where we, for the first time in this book, encounter the generalized Kadanoff-Baym ansatz (GKBA). A second subject of this chapter is the connection to Markovian kinetic equations of the Boltzmann type. This was discussed already in Chap. 6, but here we present a more detailed analysis that allows to include memory effects approximately via a retardation expansion.
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Notes
- 1.
See Table 5.1.
- 2.
Notice that the r.h.s. of (7.1) is not permutation invariant in the particle indices. This is because the symmetric terms containing \(g_{13}\) and \(g_{23}\) have been dropped since they do not contribute to the renormalization of the Hamiltonian \(H_{12}\) in the equation for \(g_{12}\). These terms are generalized polarization terms which are beyond the static Born approximation. This asymmetry indicates that there could be problems with the energy conservation of this approximation. We will return to this question in Sect. 7.3.2.
- 3.
The original form of the third hierarchy equation (2.16) contains the \(H^0_{123}=H^0_1+H^0_2+H^0_3\) (the three-particle ladder terms are beyond the Born approximation). This is one possible choice, however, a consistent treatment will require a renormalization \(H^0_{123}\rightarrow H_{123}^{0 \mathrm{eff}}\), which we derive below. We further mention that an analogous approach for the simpler case of carrier-phonon scattering has been given by Kuhn, Schilp and Mahler [128, 129].
- 4.
They do not have this property and, therefore, do not contribute to selfenergy diagrams, although they are of the same order in the interaction and the density.
- 5.
Since \({\bar{H}}_{1}\) is non-hermitean, it does not lead to a commutator on the l.h.s.
- 6.
Details of the derivation are given in Appendix D.
- 7.
- 8.
Strictly speaking, we have to write \(\gamma _{12} \rightarrow {\tilde{\gamma }}_{12}= \gamma ({\tilde{p}}_1,{\tilde{t}})+ \gamma ({\tilde{p}}_2,{\tilde{t}})\), where \({\tilde{p}}_{1,2}\) are some average momentum values, but we will see below (Fig. 7.6) that the momentum dependence is rather weak. Also, \(\gamma \) depends on time. But it turns out that for times larger than the correlation time, there is only small further change.
- 9.
- 10.
For a detailed discussion, see Chap. 8.
- 11.
For a discussion of Boltzmann’s H-theorem, see Chap. 8.
- 12.
In a multi-component plasma, despite repulsive interaction of identical charges, the net interaction energy is negative, assuring stability. This applies also to a “one-component” plasma, where one has to account for the neutralizing background of oppositely charged carriers.
- 13.
Otherwise, this term can be calculated using e.g. the numerically obtained solution f(t).
- 14.
- 15.
We mention that the numerical calculation of the correlation energy in a quantum plasma is very complicated due to the large number of arguments of \(g_{12}\). A much simpler method is given by formula (13.84) of the Green functions approach, which is discussed in Chap. 13, and which was used to compute the data of Fig. 7.4.
- 16.
Moving \(t_0\) to \(-\infty \), the problem changes qualitatively, it is transformed from an initial value to a boundary (on the time arrow) value problem.
- 17.
As mentioned in Chap. 5, exceptions are long-living correlations, such as bound states. In that case, \(g_{12}\) is split in a short-living and a long-living contribution, and only the first is required to decay, \(\lim _{(t-t_0)\rightarrow -\infty }\,g(t_0)=g^{long}(t_0)\).
- 18.
It has been introduced into the density operator approach by Bogolyubov [76] and has been generalized to real time Greeb functions by Kremp et al. [264]. While these and related works postulated this principle, in our approach it follows directly, since the dynamical evolution of the correlations (i.e. non-Markovian behavior) is included.
- 19.
If inserted into (7.39).
- 20.
Plus possibly higher orders, but, in most cases, the series converges very fast.
- 21.
Recall that the result (7.11) was obtained from the second and third hierarchy equations with the closure (7.1) without any approximation with respect to the time dependencies. Based on the discussion in Sect. 2.2.1, we, therefore, conclude that the time-reversibility of the full hierarchy is retained.
- 22.
We briefly outline the main steps in the derivation of approximation (7.60): the selfenergy terms are small corrections to the free two-particle dynamics (in Born approximation, they are of the order \(V^2\), constituting a small perturbation of the free energy). Therefore, one may apply perturbation theory to treat \(\epsilon \) under the time integral. In particular, the renormalized energy may often be approximated by the zeroth order retardation term \(\epsilon (t-\tau )\approx \epsilon (t)\). Also, under the time integral, selfenergy corrections may often be neglected compared to the free energy \(E^0\). Using these arguments, one can approximate under the time integral g by its Markovian limit \(g^M_0\), (7.48). As a result, g can be taken out of the integral, turning \({\tilde{\Sigma }}\) into a function instead of an operator. Furthermore, the delta function in \(g^{M_0}\) leads to cancellation in the difference of six energies under the integral which becomes \({\bar{E}}^0_{1}+{\bar{E}}^0_{3}-E^0_{1}-E^0_{3}\approx {\bar{\epsilon }}^0_{1}+{\bar{\epsilon }}^0_{3}-\epsilon ^0_{1}-\epsilon ^0_{3}\). Neglecting further the retardation in \(\epsilon \), we arrive at (7.60) .
- 23.
Although (7.60) defines \(\Delta _1(t)\) and \(\gamma _1(t)\) in a complicated way by an integral equation, this poses no problem for a numerical solution, and these integrals may be treated in the same way as the collision integrals of kinetic equations, cf. Appendix F.
- 24.
- 25.
In general, it has to be expected that \(\Sigma \) contains two-particle contributions. How they have to be introduced in the BBGKY-hierarchy is still an open problem.
- 26.
Strictly speaking, the last term in the selfenergy expressions, i.e. the term \(\sim g_{13}\) is missing in the Green functions results. In the BBGKY-hierarchy, it appears straightforwardly from the (anti-)symmetrization of the third hierarchy equation, cf. Sect. 3.4. A possible approach is to iteratively replace \(g_{13}\) by the solution of the second hierarchy equation. Most likely, the resulting terms can be “absorbed” by a generalization of the relations (7.81), which should agree with the additional terms (beyond the generalized Kadanoff-Baym ansatz) in the exact reconstruction formula of [131], see (13.119).
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Bonitz, M. (2016). Non-Markovian Kinetic Equations with Selfenergy. In: Quantum Kinetic Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-24121-0_7
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