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Technical Physics

Evolution of weakly correlated systems

I. N. Kosarev

Published Jun 1, 2014• 0 citations
Publisher

Abstract

A propagator with an effective action is constructed for a quantum many-particle system in the approximation taking into account only pair correlations between particles. The theory can easily be generalized to the case of two species of particles. In the case of a rarefied plasma and a gas, this propagator for a one-particle density matrix provides a solution to the kinetic problem over time intervals longer than the relaxation time.

DOI
10.1134/s1063784214060127

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A propagator with an effective action is constructed for a quantum many-particle system in the approximation taking into account only pair correlations between particles. The theory can easily be generalized to the case of two species of particles. In the case of a rarefied plasma and a gas, this propagator for a one-particle density matrix provides a solution to the kinetic problem over time intervals longer than the relaxation time.
In the paper we discuss possible approaches to the problem of the rigorous derivation of quantum kinetic equations from underlying many-particle dynamics. For the description of a many-particle evolution we construct solutions of the Cauchy problems of the BBGKY hierarchy and the dual BBGKY hierarchy in suitable Banach spaces. In the framework of the conventional approach to the description of kinetic evolution the mean-field asymptotics of the quantum BBGKY hierarchy solution is constructed. We develop also alternative approaches. One method is based on the construction of the solution asymptotics of the initial-value problem of the quantum dual BBGKY hierarchy. One more approach is based on the generalized quantum kinetic equation that is a consequence of the equivalence of the Cauchy problems of such evolution equation and the BBGKY hierarchy with initial data determined by the one-particle density operator.
The paper deals with the problem of the rigorous description of the evolution of states of large particle quantum systems by means of correlation operators. A nonperturbative solution of the Cauchy problem of the hierarchy of nonlinear evolution equations for a sequence of marginal correlation operators is constructed. For initial states specified in terms of a one-particle density operator and correlation operators we also develop an approach to the description of the processes of the creation and the propagation of correlations within the framework of a one-particle density operator. Moreover, the mean field asymptotic behavior of constructed marginal correlation operators is established.
This is the first of a series of papers which describe the functional-integral approach to the study of the statistical and kinetic properties of nonequilibrium quantum fields in flat and curved spacetimes. In this paper we treat a system of self-interacting bosons described by \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ scalar fields in flat space. We adopt the closed-time-path (CTP or ``in-in'') functional formalism and use a two-particle irreducible (2PI) representation for the effective action. These formalisms allow for a full account of the dynamics of quantum fields, and put the correlation functions on an equal footing with the mean fields. By assuming a thermal distribution we recover the real-time finite-temperature theory as a special case. By requiring the CTP effective action to be stationary with respect to variations of the correlation functions we obtain an infinite set of coupled equations which is the quantum-field-theoretical generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Truncation of this series leads to dissipative characteristics in the subsystem. In this context we discuss the nature of dissipation in interacting quantum fields.To one-loop order in a perturbative expansion of the CTP effective action, the 2PI formalism yields results equivalent to the leading 1/N expansion for an O(N)-symmetric scalar field. To higher-loop order we introduce a two-time approximation to separate the quantum-field effects of radiative correction and renormalization from the statistical-kinetic effects of collisions and relaxation. In the weak-coupling quasiuniform limit, the system of nonequilibrium quantum fields can subscribe to a kinetic theory description wherein the propagators are represented in terms of relativistic Wigner distribution functions. From a two-loop calculation we derive the Boltzmann equation for the distribution function and the gap equation for the effective mass of the quasiparticles. One can define an entropy function for the quantum gas of quasiparticles which satisfies the H theorem. We also calculate the limits to the validity of the binary collision approximation from a three-loop analysis. The theoretical framework established here can be generalized to nonconstant background fields and for curved spacetimes.
Universal dynamics of a dilute Bose gas is studied within the kinetic regime where the time evolution of the mode occupation number is governed by a wave-Boltzmann equation. The universality manifests itself in the form of the dynamical evolution which can be a cascade in the context of Kolmogorov-Zakharov wave turbulence or a self-similar shift in time and space. Which is the case depends on the relevant global conservation laws and on the particular range of scales in which the respective dynamics takes place. A nonperturbative kinetic equation is derived by applying the Schwinger-Keldysh closed-time functional integral and the 2-particle-irreducible formalism to an action of complex scalar Bose fields with quartic interaction. The resulting dynamic equation for two-point correlations are reduced to a wave-Boltzmann-type kinetic equation with an effective TTT-matrix which depends explicitly on the mode occupation numbers. This nonperturbative dependence on the solution occurs in the infrared regime where wave numbers are large and collective scattering of many particles prevails. Thus, the scaling analysis of wave turbulence theory can be applied. We explicitly calculate the effective TTT-matrix analytically taking into account an infrared cutoff for dealing with the infrared divergences. Our results show that the scaling behaviour of the TTT-matrix departs from the one predicted by naive dimensional counting due to the presence of the infrared cutoff. The Kolmogorov-Zakharov and the self-similar exponents are evaluated by power counting using our TTT-matrix and the results are confirmed by numerical integration of the wave-Boltzmann scattering integral. The scaling exponents governing the time evolution are determined by means of the global conservation laws and the kinetic equation. Depending on the scaling properties of the quasiparticle spectrum, the momentum cutoff scale in the infrared evolves critically slowed in time. The respective scaling exponent is universal in the nonperturbative regime regardless of whether the process is an inverse cascade or a self-similar shift towards the infrared. Thus, our results provide a general framework for classifying nonthermal fixed points in dilute ultracold Bose gases. They pave the way to a straightforward generalisation and application to trapped systems in different dimensionalities and to systems with more than one internal degree of freedom. Our results furthermore provide a possible interpretation of recent experimental results on wave turbulence in an ultracold Bose gas. %We studied the universal dynamics of diluted Bose gas within the kinetic regime where time evolution of occupation number is governed by the Boltzmann's scattering integral. %There are three possible scenarios, depending on the scaling behaviour of occupation number. They are the particle and energy cascade which are the steady state of turbulence transport and self-similar evolution where local occupation evolves in time. %We evaluated the scaling exponents in both momentum and time scales in each case under the constraints of global number and energy conservation. Furthermore, we can go beyond the perturbative regime by employing the closed-time-path contour and the 2-particle-irreducible formalism to the Gross-Pitaevskii action. The kinetic equation is modified by an effective T-matrix as a result of the resummation scheme. However, the structure of kinetic equation remains the same as in the perturbative case. Thus, all exponents in the nonperturbative regime depart from those in the perturbative regime because of the scaling of the effective T-matrix.
We develop a rigorous formalism for the description of the evolution of states of quantum many-particle systems in terms of a one-particle density operator. For initial states which are specified in terms of a one-particle density operator the equivalence of the description of the evolution of quantum many-particle states by the Cauchy problem of the quantum BBGKY hierarchy and by the Cauchy problem of the generalized quantum kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation is established in the space of trace class operators. The links of the specific quantum kinetic equations with the generalized quantum kinetic equation are discussed.