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Zeitschrift für Naturforschung A

Die Vielteilchen-T-Matrix und ihre Anwendung in der Theorie realer Gase von mittlerer Dichte.

K. Baerwinkel

Published Jan 1, 1969• 21 citations
Publisher

Abstract

Abstract The formalism of the many-body T matrix suitable for application in nonequilibrium is presented. The corresponding approximation together with a density-expansion is applied to the general equation of motion due to Kadanoff-Baym . Correction terms to the Boltzmann-Landau equation are obtained.
DOI
10.1515/zna-1969-0105

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Die Vielteilchen-T-Matrix und ihre Anwendung in der Theorie realer Gase von mittlerer Dichte /

K. Baerwinkel

Abstract The improved Landau equation governing nonequilibrium behaviour of quantum gases of moderate density is shown to yield the exact second virial coefficient B(T). After simplifying the collisional operator the linearized stress tensor is evaluated to give the lowest order virial correction to the shear viscosity.

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Kadanoff–Baym derivation of the ZNG equations

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In Chapter 4, we introduced the Kadanoff–Baym equations of motion for the imaginary-time nonequilibrium Green's functions for a Bose gas, as given by (4.59) and (4.60). In this chapter, we will use the generalization of these equations of motion to find the equivalent equations of motion for the real-time Green's functions. These can be written in a natural way in the form of a kinetic equation. Using a simple Hartree–Fock approximation, we show how the coupled equations for the condensate and thermal cloud given in Chapter 3 emerge naturally from the Kadanoff–Baym (KB) formalism. This chapter is based on Imamović-Tomasović and Griffin (2001) and Imamović-Tomasović (2001), building on the pioneering work of Kane and Kadanoff (1965). In this chapter and Chapter 7 we review the KB formalism. However, we also encourage the reader to read the original account given by Kadanoff and Baym (1962). The goals and accomplishments of their seminal book are beautifully captured by the following quote from p. 138: Our rather elaborate Green's function arguments provide a means of describing transport phenomena in a self-contained way, starting from a dynamical approximation, i.e. an approximation for G 2 ( U ) in terms of G 1 ( U ). These calculations require no extra assumptions. The theory provides at the same time a description of the transport processes that occur and a determination of the quantities which appear in the transport equations. A closely related way of treating the nonequilibrium dynamics of a Bose-condensed gas is based on the two-particle irreducible (2PI) effective action together with the Schwinger–Keldysh closed-time path formalism. Berges (2004) gives a detailed review of this approach, which allows one to derive the nonequilibrium action on the basis of controllable approximations.

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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).

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Nonequilibrium Quantum-Field Dynamics and Off-Shell Transport

S. Juchem

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

•Jun 12, 2015•2 citations

Memory effects and virial corrections in nonequilibrium dense systems

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A dense interacting Bose or Fermi gas is considered. Within the Kadanoff-Baym real-time Green-function technique a generalized kinetic equation is derived in the nonequilibrium ladder approximation. As a Markovian limit the Boltzmann-Uehling-Uhlenbeck equation is recovered. The relevance of retardation effects in the kinetic equation of dense systems is shown. It turns out that the stationary solution of the generalized kinetic equation reproduces the second virial coefficient of the equation of state, including the Beth-Uhlenbeck quantum corrections. We establish a generalized kinetic equation describing correlation effects in a consistent approach to nonequilibrium for Bose and Fermi systems. In this way we give a generalization of the Beth-Uhlenbeck formula that applies to nonequilibrium and to finite systems. The derivation makes use of the time dependent T-matrix equation, which establishes a generalization of the Bethe-Goldstone equation. The correlated parts of the density and energy, which can be interpreted as virial corrections, are thermodynamically consistent. In this way global energy conservation is fulfilled.

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Kadanoff-Baym description of Hubbard clusters out of equilibrium: performance of many-body schemes, correlation-induced damping and multiple steady states

M. Puig von Friesen, Claudio Verdozzi, C.‐O. Almbladh

We present in detail a method we recently introduced (PRL. 103, 176404 (2009)) to describe finite systems in and out of equilibrium, where the evolution in time is performed via the Kadanoff-Baym Equations (KBE) within Many-Body Perturbation Theory (MBPT). The main property we analyze is the time-dependent density. We also study is the exchange-correlation potential of TDDFT, obtained via reverse engineering from the time-dependent density. Our systems consist of small, strongly correlated clusters, described by a Hubbard Hamiltonian within the Hartree-Fock, second Born, GW and T-matrix approximations. We compare the results from the KBE dynamics to those from exact numerical solutions. The outcome of our comparisons is that, among the many-body schemes considered, the T-matrix approximation is overall superior at all electron densities. Such comparisons permit a general assessment of the whole idea of applying MBPT, in the KBE sense, to finite systems. A striking outcome of our analysis is that when the system evolves under a strong external field, the KBE develop a steady-state solution as a consequence of a correlation-induced damping. This damping is present both in isolated (finite) systems, where it is purely artificial, as well as in clusters contacted to (infinite) macroscopic leads. To illustrate this point we present selected results for a system coupled to contacts within the T-matrix and second Born approximation. The extensive characterization we performed indicates that this behavior is present whenever approximate self energies, based upon infinite partial summations, are used. A second important result is that, for isolated clusters, the steady state reached is not unique but depends on how one switches on the external field. This may also true for clusters connected to leads.

arXiv:1004.3662•Apr 21, 2010•0 citations
In Chapter 4, we introduced the Kadanoff–Baym equations of motion for the imaginary-time nonequilibrium Green's functions for a Bose gas, as given by (4.59) and (4.60). In this chapter, we will use the generalization of these equations of motion to find the equivalent equations of motion for the real-time Green's functions. These can be written in a natural way in the form of a kinetic equation. Using a simple Hartree–Fock approximation, we show how the coupled equations for the condensate and thermal cloud given in Chapter 3 emerge naturally from the Kadanoff–Baym (KB) formalism. This chapter is based on Imamović-Tomasović and Griffin (2001) and Imamović-Tomasović (2001), building on the pioneering work of Kane and Kadanoff (1965). In this chapter and Chapter 7 we review the KB formalism. However, we also encourage the reader to read the original account given by Kadanoff and Baym (1962). The goals and accomplishments of their seminal book are beautifully captured by the following quote from p. 138: Our rather elaborate Green's function arguments provide a means of describing transport phenomena in a self-contained way, starting from a dynamical approximation, i.e. an approximation for G 2 ( U ) in terms of G 1 ( U ). These calculations require no extra assumptions. The theory provides at the same time a description of the transport processes that occur and a determination of the quantities which appear in the transport equations. A closely related way of treating the nonequilibrium dynamics of a Bose-condensed gas is based on the two-particle irreducible (2PI) effective action together with the Schwinger–Keldysh closed-time path formalism. Berges (2004) gives a detailed review of this approach, which allows one to derive the nonequilibrium action on the basis of controllable approximations.
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).
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
A dense interacting Bose or Fermi gas is considered. Within the Kadanoff-Baym real-time Green-function technique a generalized kinetic equation is derived in the nonequilibrium ladder approximation. As a Markovian limit the Boltzmann-Uehling-Uhlenbeck equation is recovered. The relevance of retardation effects in the kinetic equation of dense systems is shown. It turns out that the stationary solution of the generalized kinetic equation reproduces the second virial coefficient of the equation of state, including the Beth-Uhlenbeck quantum corrections. We establish a generalized kinetic equation describing correlation effects in a consistent approach to nonequilibrium for Bose and Fermi systems. In this way we give a generalization of the Beth-Uhlenbeck formula that applies to nonequilibrium and to finite systems. The derivation makes use of the time dependent T-matrix equation, which establishes a generalization of the Bethe-Goldstone equation. The correlated parts of the density and energy, which can be interpreted as virial corrections, are thermodynamically consistent. In this way global energy conservation is fulfilled.
We present in detail a method we recently introduced (PRL. 103, 176404 (2009)) to describe finite systems in and out of equilibrium, where the evolution in time is performed via the Kadanoff-Baym Equations (KBE) within Many-Body Perturbation Theory (MBPT). The main property we analyze is the time-dependent density. We also study is the exchange-correlation potential of TDDFT, obtained via reverse engineering from the time-dependent density. Our systems consist of small, strongly correlated clusters, described by a Hubbard Hamiltonian within the Hartree-Fock, second Born, GW and T-matrix approximations. We compare the results from the KBE dynamics to those from exact numerical solutions. The outcome of our comparisons is that, among the many-body schemes considered, the T-matrix approximation is overall superior at all electron densities. Such comparisons permit a general assessment of the whole idea of applying MBPT, in the KBE sense, to finite systems. A striking outcome of our analysis is that when the system evolves under a strong external field, the KBE develop a steady-state solution as a consequence of a correlation-induced damping. This damping is present both in isolated (finite) systems, where it is purely artificial, as well as in clusters contacted to (infinite) macroscopic leads. To illustrate this point we present selected results for a system coupled to contacts within the T-matrix and second Born approximation. The extensive characterization we performed indicates that this behavior is present whenever approximate self energies, based upon infinite partial summations, are used. A second important result is that, for isolated clusters, the steady state reached is not unique but depends on how one switches on the external field. This may also true for clusters connected to leads.