Remarks on Kubo's formula and on Kadanoff-Baym's equations

Hannes Köhler, N. H. Kwong, Rolf H. Binder +2 more

Cian C. Reeves, Yuanran Zhu, Chao Yang +1 more

The generalized Kadanoff-Baym ansatz (GKBA) is an approximation to the Kadanoff-Baym equations (KBEs) that neglects certain memory effects that contribute to the Green's function at nonequal times. Here we present arguments and numerical results to demonstrate the practical insignificance of the quantities neglected when deriving the GKBA at conditions at which KBE and GKBA are appropriate. We provide a mathematical proof that places a scaling bound on the neglected terms, further reinforcing that these terms are typically small in comparison to terms that are kept in the GKBA. We perform calculations in a range of models, including different system sizes and filling fractions, as well as experimentally relevant nonequilibrium excitations. We find that both the GKBA and KBE capture the dynamics of interacting systems with moderate and even strong interactions well. We explicitly compute terms neglected in the GKBA approximation and show, in the scenarios tested here, that they are orders of magnitude smaller than the terms that are accounted for, i.e., they offer only a small correction when included in the full KBEs.

Cian C. Reeves, Yuanran Zhu, Chao Tung Yang +1 more

The generalized Kadanoff-Baym ansatz (GKBA) is an approximation to the Kadanoff-Baym equations (KBE), that neglects certain memory effects that contribute to the Green's function at non-equal times. Here we present arguments and numerical results to demonstrate the practical insignificance of the quantities neglected when deriving the GKBA at conditions at which KBE and GKBA are appropriate. We provide a mathematical proof that places a scaling bound on the neglected terms, further reinforcing that these terms are typically small in comparison to terms that are kept in the GKBA. We perform calculations in a range of models, including different system sizes and filling fractions, as well as experimentally relevant non-equilibrium excitations. We find that both the GKBA and KBE capture the dynamics of interacting systems with moderate and even strong interactions well. We explicitly compute terms neglected in the GKBA approximation and show, in the scenarios tested here, that they are orders of magnitude smaller than the terms that are accounted for, i.e., they offer only a small correction when included in the full Kadanoff-Baym equations.

Markus Michael Müller

(Classical) Boltzmann equations suffer from several shortcomings as compared to their quantum mechanical generalizations, the so-called Kadanoff-Baym equations. Nevertheless, in practical calculations approximations to Boltzmann equations are widely used. A prominent example is the computation of the baryon asymmetry of the universe. In this work, we present a detailed comparison between the Kadanoff-Baym and Boltzmann equations in the framework of a scalar phi 4 quantum field theory. We solve the Kadanoff-Baym and the Boltzmann equations numerically in 3 + 1 dimensions and compare their predictions on the evolution of systems out of thermal equilibrium for various initial conditions. The obtained numerical solutions reveal significant discrepancies between both types of equations. Most notably, the universality respected by Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations.

John M. Blatt