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arXiv:1510.04552 · Physical review. E

Kinetic approach to a relativistic Bose-Einstein condensate

Alex Meistrenko, Hendrik van Hees, Kai Zhou, Carsten Greiner

Published Mar 16, 2016• 7 citations
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Abstract

We apply a Boltzmann approach to the kinetic regime of a relativistic Bose-Einstein condensate of scalar bosons by decomposing the one-particle distribution function in a condensate part and a nonzero momentum part of excited modes, leading to a coupled set of evolution equations which are then solved efficiently with an adaptive higher order Runge-Kutta scheme. We compare our results to the partonic cascade Monte Carlo simulation BAMPS for a critical but far from equilibrium case of massless bosons. Motivated by the color glass condensate initial conditions in QCD with a strongly overpopulated initial glasma state, we also discuss the time evolution starting from an overpopulated initial distribution function of massive scalar bosons. In this system a self-similar evolution of the particle cascade with a nonrelativistic turbulent scaling in the infrared sector is observed as well as a relativistic exponent for the direct energy cascade, confirming a weak wave turbulence in the ultraviolet region.

DOI
10.1103/physreve.93.032131

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We apply a Boltzmann approach to the kinetic regime of a relativistic Bose-Einstein condensate of scalar bosons by decomposing the one-particle distribution function in a condensate part and a nonzero momentum part of excited modes, leading to a coupled set of evolution equations which are then solved efficiently with an adaptive higher order Runge-Kutta scheme. We compare our results to the partonic cascade Monte Carlo simulation BAMPS for a critical but far from equilibrium case of massless bosons. Motivated by the color glass condensate initial conditions in QCD with a strongly overpopulated initial glasma state, we also discuss the time evolution starting from an overpopulated initial distribution function of massive scalar bosons. In this system a self-similar evolution of the particle cascade with a nonrelativistic turbulent scaling in the infrared sector is observed as well as a relativistic exponent for the direct energy cascade, confirming a weak wave turbulence in the ultraviolet region.
The formation of Bose condensates far from equilibrium can play an important role in our understanding of collision experiments of heavy nuclei or for the evolution of the early Universe. In the relativistic quantum world particle number changing processes can counteract Bose condensation, and there is a considerable debate about the relevance of this phenomenon in this context. We show that the involved question of Bose condensation from initial overpopulation can be answered for the example of scalar field theories. Condensate formation occurs as a consequence of an inverse particle cascade with a universal power-law spectrum. This particle transport towards low momenta is part of a dual cascade, in which energy is also transferred by weak wave turbulence towards higher momenta. To highlight the importance of number changing processes for the subsequent decay of the condensate, we also compare to nonrelativistic theories with exact number conservation. We discuss the relevance of these results for non-Abelian gauge theories.
We outline a kinetic theory of nonthermal fixed points for the example of a dilute Bose gas, partially reviewing results obtained earlier, thereby extending, complementing, generalizing, and straightening them out. We study universal dynamics after a cooling quench, focusing on situations where the time evolution represents a pure rescaling of spatial correlations, with time defining the scale parameter. The nonequilibrium initial condition set by the quench induces a redistribution of particles in momentum space. Depending on conservation laws, this can take the form of a wave-turbulent flux or of a more general self-similar evolution, signaling the critically slowed approach to a nonthermal fixed point. We identify such fixed points using a nonperturbative kinetic theory of collective scattering between highly occupied long-wavelength modes. In contrast, a wave-turbulent flux, possible in the perturbative Boltzmann regime, builds up in a critically accelerated self-similar manner. A key result is the simple analytical universal scaling form of the nonperturbative many-body scattering matrix, for which we lay out the concrete conditions under which it applies. We derive the scaling exponents for the time evolution as well as for the power-law tail of the momentum distribution function, for a general dynamical critical exponent zzz and an anomalous scaling dimension $\ensuremath{\eta}$. The approach of the nonthermal fixed point is, in particular, found to involve a rescaling of momenta in time ttt by ${t}^{\phantom{\rule{0.16em}{0ex}}\ensuremath{\beta}}$, with $\ensuremath{\beta}=1/z$, within our kinetic approach independent of $\ensuremath{\eta}$. We confirm our analytical predictions by numerically evaluating the kinetic scattering integral as well as the nonperturbative many-body coupling function. As a side result, we obtain a possible finite-size interpretation of wave-turbulent scaling recently measured by Navon et al.
We map the infrared dynamics of a relativistic single-component (N=1N=1N=1) interacting scalar field theory to that of nonrelativistic complex scalar fields. The Gross-Pitaevskii (GP) equation, describing the real-time dynamics of single-component ultracold Bose gases, is obtained at first nontrivial order in an expansion proportional to the powers of $\ensuremath{\lambda}{\ensuremath{\phi}}^{2}/{m}^{2}$ where $\ensuremath{\lambda}$, $\ensuremath{\phi}$, and mmm are the coupling constant, the scalar field, and the particle mass respectively. Our analytical studies are corroborated by numerical simulations of the spatial and momentum structure of overoccupied scalar fields in (2+1)-dimensions. Universal scaling of infrared modes, vortex-antivortex superfluid dynamics, and the off-equilibrium formation of a Bose-Einstein condensate are observed. Our results for the universal scaling exponents are in agreement with those extracted in the numerical simulations of the GP equation. As in these simulations, we observe coarsening phase kinetics in the Bose superfluid with strongly anomalous scaling exponents relative to that of vertex resummed kinetic theory. Our relativistic field theory framework further allows one to study more closely the coupling between superfluid and normal fluid modes, specifically the turbulent momentum and spatial structure of the coupling between a quasiparticle cascade to the infrared and an energy cascade to the ultraviolet. We outline possible applications of the formalism to the dynamics of vortex-antivortex formation and to the off-equilibrium dynamics of the strongly interacting matter formed in heavy-ion collisions.
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 investigate formation of Bose-Einstein condensates under nonequilibrium conditions using numerical simulations of the three-dimensional Gross-Pitaevskii equation. For this, we set initial random weakly nonlinear excitations and the forcing at high wave numbers and study propagation of the turbulent spectrum toward the low wave numbers. Our primary goal is to compare the results for the evolving spectrum with the previous results obtained for the kinetic equation of weak wave turbulence. We demonstrate existence of a regime for which good agreement with the wave turbulence results is found in terms of the main features of the previously discussed self-similar solution. In particular, we find a reasonable agreement with the low-frequency and the high-frequency power-law asymptotics of the evolving solution, including the anomalous power-law exponent ${x}^{*}\ensuremath{\approx}1.24$ for the three-dimensional wave action spectrum. We also study the regimes of very weak turbulence when the evolution is affected by the discreteness of the Fourier space, and the strong turbulence regime when the emerging condensate modifies the wave dynamics and leads to formation of strongly nonlinear filamentary vortices.
We study the bosonic Boltzmann-Nordheim kinetic equation, which describes the kinetic regime of weakly interacting bosons with s-wave scattering only. We consider a spatially homogeneous fluid with an isotropic momentum distribution. The issue of the dynamical formation of a Bose-Einstein condensate has been studied extensively. We supply here the completed equations of motion for the coupled system, the energy density distribution of the normal fluid and the density of the condensate. With this information the post-nucleation self-similar solution is investigated in more detail than before.