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Contributions to Plasma Physics

Nonequilibrium Two‐Particle Dyson Equation

Th. Bornath, M. Schlanges, D. Kremp

Published Jan 1, 1999• 1 citation
Publisher

Abstract

Abstract The two‐particle problem within a nonequilibrium many‐particle system is investigated in the framework of real‐time Green's functions. A nonequilibrium Dyson equation is given for the two‐time two‐particle Green's function. The two‐particle self‐energy in this equation consists of two qualitatively different parts: one part is local in time, whereas the other part is a two‐time quantity having thus a dynamical character. The latter part of the self‐energy can be expressed in terms of multi‐particle Green's functions.

DOI
10.1002/ctpp.2150390108

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Abstract The two‐particle problem within a nonequilibrium many‐particle system is investigated in the framework of real‐time Green's functions. A nonequilibrium Dyson equation is given for the two‐time two‐particle Green's function. The two‐particle self‐energy in this equation consists of two qualitatively different parts: one part is local in time, whereas the other part is a two‐time quantity having thus a dynamical character. The latter part of the self‐energy can be expressed in terms of multi‐particle Green's functions.
The two-particle problem within a nonequilibrium many-particle system is investigated in the framework of real-time Green's functions. Starting from the dynamically screened ladder approximation of the nonequlibrium Bethe-Salpeter equation, a nonequilibrium Dyson equation is given for two-time two-particle Green's functions. Thereby the well-known Kadanoff-Baym equations are generalized to the case of two-particle functions. The two-time structure of the equations is achieved in an exact way avoiding the so-called Shindo approximation. For the case of thermodynamic equilibrium, the differences to former results obtained for the effective two-particle hamiltonian are discussed.
The two-particle problem within a nonequilibrium many-particle system is investigated in the framework of real-time Green's functions. Starting from the nonequilibrium Bethe-Salpeter equation on the Keldysh contour, a Dyson equation is given for two-time two-particle Green's functions. Thereby the well-known Kadanoff-Baym equations are generalized to the case of two-particle functions. The two-time structure of the equations is achieved in an exact way using the semigroup property of the free-particle propagators. The frequently used Shindo approximation is thus avoided. It turns out that results obtained earlier are valid only in limiting cases of a nondegenerate system or a static interaction, respectively. For the case of thermodynamic equilibrium, the differences to former results obtained for the effective two-particle Hamiltonian are discussed.
A numerical method to solve the set of Dyson‐like equations in the framework of non‐equilibrium Green's functions is presented. The technique is based on the self‐consistent solution of the Dyson equations for the interacting single‐particle Green's function once a choice for the self‐energy, functional of the single‐particle Green's function itself, is made. The authors briefly review the theory of the non‐equilibrium Green's functions in order to highlight the main point useful in discussing the proposed technique. Then, the numerical implementationis presented and discussed, which is based on the distribution of the Keldysh components of the Green's function and the self‐energy on a grid of processes. It is discussed how the structure of the considered self‐energy approximations influences the distribution of the matrices in order to minimize the communication time among processes and which should be considered in the case of other approximations. The authors give an example of the application of this technique to the case of quenches in ultracold gases and to the single impurity Anderson model, also discussing the convergence and the stability features of the approach.
An approximate solution of a Dyson equation satisfied by a single particle Green's function in many-impurity problems in solids is obtained for an extended impurity perturbation potential. We assume that there is no overlapping of any two impurity potentials and that the range of the impurity potential is arbitrary. The coordinate representation of the Dyson equation is solved by transforming an iterative solution of the equation into a form with no repetitive summation indices appearing in any term of given order. By neglecting fluctuations of the impurity distribution, a set of equations are derived to determine the Green's function and the self-energy in a self-consistent manner. A group-theoretic consideration of the self-energy and some sum rules are given.
Entropy in nonequilibrium statistical mechanics is investigated theoretically so as to extend the well-established equilibrium framework to open nonequilibrium systems. We first derive a microscopic expression of nonequilibrium entropy for an assembly of identical bosons/fermions interacting via a two-body potential. This is performed by starting from the Dyson equation on the Keldysh contour and following closely the procedure of Ivanov, Knoll and Voskresensky [Nucl. Phys. A {\bf 672} (2000) 313]. The obtained expression is identical in form with an exact expression of equilibrium entropy and obeys an equation of motion which satisfies the HHH-theorem in a limiting case. Thus, entropy can be defined unambiguously in nonequilibrium systems so as to embrace equilibrium statistical mechanics. This expression, however, differs from the one obtained by Ivanov {\em et al}., and we show explicitly that their ``memory corrections'' are not necessary. Based on our expression of nonequilibrium entropy, we then propose the following principle of maximum entropy for nonequilibrium steady states: ``The state which is realized most probably among possible steady states without time evolution is the one that makes entropy maximum as a function of mechanical variables, such as the total particle number, energy, momentum, energy flux, etc.'' During the course of the study, we also develop a compact real-time perturbation expansion in terms of the matrix Keldysh Green's function.
The Dyson equation represents a well-defined connection between the one-particle Green's function and the self-energy. By means of the Dyson equation, the evaluation of the one-particle Green's function can be reduced to the evaluation of the self-energy. As it is well known, the one-particle Green's function is composed of an advanced and a retarded part, which can be represented in the (N+1)- and (N-1)-particle configuration spaces, respectively. The Dyson equation combines these two spaces into a matrix in the union space. This is difficult to understand in configuration-interaction language, where the Hamiltonian matrices for systems containing different numbers of particles are naturally separated. Starting with the usual matrix representations of the Hamiltonian in the (N+1)- and (N-1)-particle spaces, it is shown that it is possible to define a class of unitary transformations that mixes the two parts and gives rise to an effective Hamiltonian matrix in the union space. This effective Hamiltonian leads to all possible representations of the Dyson equation. The transformations used arise from suitable combinations of the matrices of residues of the advanced and retarded Green's functions. Several specific members of the above class of unitary transformations are discussed in some detail also in connection with possible approximation schemes for the Green's function. Special attention is paid to those unitary transformations that depend on the ground state of the N-particle system as the only unknown quantity. This investigation has given rise to some general conclusions and to an alternative approach to the one-particle Green's function.