We use the 2PI effective action of a relativistic scalar field theory to derive kinetic equations for a Bose-condensed system near the phase transition.We start from equations of motion derived within a 1/N-expansion at NLO. In taking the non-relativistic limit we obtain a generalized Gross-Pitaevskii equation for the condensate field. Within the Popov approximation we explicitly compute the collision term up to order g^2 using the Kadanoff-Baym formalism. For the sake of self-consistency we derive in the same way a Boltzmann equation for the non-condensate distribution function. The final results are in agreement to those previously obtained by Griffin, Nikuni and Zaremba and by Stoof.
We use the 2PI effective action of a relativistic scalar field theory to derive kinetic equations for a Bose-condensed system near the phase transition.We start from equations of motion derived within a 1/N-expansion at NLO. In taking the non-relativistic limit we obtain a generalized Gross-Pitaevskii equation for the condensate field. Within the Popov approximation we explicitly compute the collision term up to order g^2 using the Kadanoff-Baym formalism. For the sake of self-consistency we derive in the same way a Boltzmann equation for the non-condensate distribution function. The final results are in agreement to those previously obtained by Griffin, Nikuni and Zaremba and by Stoof.
Using the Kadanoff-Baym non-equilibrium Green's function formalism, we derive kinetic equations for the non-condensate atoms at finite temperatures which include the effect of binary collisions between atoms. The effect of collisions is included using the second-order self-energy given by the Beliaev (gapless) approximation. We limit our discussion to finite temperatures where we can use the single-particle Hartree-Fock spectrum for the excited atoms. In this limit, we can neglect the off-diagonal propagators (g~12 and g~21). As expected, this leads to the kinetic equations and collision integrals used in recent work by Zaremba, Nikuni, and Griffin (ZNG) [1]. We also derive a consistent equation of motion for the condensate wavefunction, involving a finite-temperature generalization of the well-known Gross-Pitaevskii equation which includes a dissipative term, as well as the mean field of the non-condensate.
We extend our earlier work on the nonequilibrium dynamics of a Bose-Einstein condensate initially loaded into a one-dimensional optical lattice. From the two-particle-irreducible (2PI) closed-time-path (CTP) effective action for the Bose-Hubbard Hamiltonian we derive causal equations of motion that treat mean-field effects and quantum fluctuations on an equal footing. We demonstrate that these equations reproduce well-known limits when simplifying approximations are introduced. For example, when the system dynamics admits two-time separation, we obtain the Kadanoff-Baym equations of quantum kinetic theory, and in the weakly interacting limit, we show that the local equilibrium solutions of our equations reproduce the second-order corrections to the self-energy of the type originally derived by Beliaev. The derivation of quantum kinetic equations from the 2PI-CTP effective action not only checks the viability of the formalism but also shows it to be a tractable framework for going beyond standard Boltzmann equations of motion.
We discuss the equivalence of two nonequilibrium kinetic theories that describe the evolution of a dilute, Bose-Einstein condensed atomic gas in a harmonic trap. The second-order kinetic equations of Walser et al. [Phys. Rev. A 63, 013607 (2001)] reduce to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the low- and high-temperature limits, respectively. These kinetic equations thus describe the system in equilibrium (finite temperature) as well as in nonequilibrium (real time). We have found this theory to be equivalent to the nonequilibrium Green's function approach originally proposed by Kadanoff and Baym and more recently applied to inhomogeneous trapped systems by Imamovi\ifmmode \acute{c}\else \'{c}\fi{}-Tomasovi\ifmmode \acute{c}\else \'{c}\fi{} and Griffin [in Progress in Nonequilibrium Green's Functions, edited by M. Bonitz (World Scientific, Singapore, 2000), p. 404].
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.
This paper extends an earlier quantum kinetics treatment for dilute, weakly interacting, partially Bose-Einstein condensed gases, presented by the author elsewhere (Proukakis N P and Burnett K 1996 J. Res. Natl Inst. Stand. Technol. 101 457), by consistently treating the dynamics of the uncondensed atoms to the same level of approximation as the condensed atoms. Our method is based on a hierarchy of coupled equations of motion for the condensate mean field and fluctuations around this mean field, truncated to second order in the (effective two-body) interatomic potential, and with suitable decoupling approximations for higher-order correlations. By applying perturbation theory in the Hartree-Fock-Bogoliubov basis, we re-derive the quantum kinetic theory of Walser et al (Walser R, Williams J, Cooper J and Holland M 1999 Phys. Rev. A 59 3878), which further indicates the consistency of our treatment with the Kadanoff-Baym non-equilibrium Green function formalism for trapped gases.
We present a detailed derivation of the quantum and quantum–thermal effective action for non-relativistic systems, starting from the single-particle case and extending to the Gross–Pitaevskii (GP) field theory for weakly interacting bosons. In the single-particle framework, we introduce the one-particle-irreducible (1PI) effective action formalism, taking explicitly into account the choice of the initial quantum state, its saddle-point plus Gaussian-fluctuation approximation, and its finite-temperature extension via Matsubara summation, yielding a clear physical interpretation in terms of zero-point and thermal contributions to the Helmholtz free energy. The formalism is then applied to the GP action, producing the 1PI effective potential at zero and finite temperature, including beyond-mean-field Lee–Huang–Yang and thermal corrections. We discuss the gapless and gapped Bogoliubov spectra, their relevance to equilibrium and non-equilibrium regimes, and the role of regularization. Applications include the inclusion of an external potential within the local density approximation, the derivation of finite-temperature Josephson equations, and the extension to D-dimensional systems, with particular attention to the zero-dimensional limit. This unified approach provides a transparent connection between microscopic quantum fluctuations and effective macroscopic equations of motion for Bose–Einstein condensates.