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Author (up) During, G.; Picozzi, A.; Rica, S. pdf  doi
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  Title Breakdown of weak-turbulence and nonlinear wave condensation Type Journal Article
  Year 2009 Publication Physica D-Nonlinear Phenomena Abbreviated Journal Physica D  
  Volume 238 Issue 16 Pages 1524-1549  
  Keywords Weak-turbulence theory; Nonlinear Schrodinger equation; Wave condensation  
  Abstract The formation of a large-scale coherent structure (a condensate) as a result of the long time evolution of the initial value problem of a classical partial differential nonlinear wave equation is considered. We consider the nonintegrable and unforced defocusing NonLinear Schrodinger (NLS) equation as a representative model. In spite of the formal reversibility of the NLS equation, the nonlinear wave exhibits an irreversible evolution towards a thermodynamic equilibrium state. The equilibrium state is characterized by a homogeneous solution (condensate), with small-scale fluctuations superposed (uncondensed particles), which store the information necessary for “time reversal”. We analyze the evolution Of the cumulants of the random wave as originally formulated by DJ. Benney and P.G. Saffman [D.J. Bentley, P.G. Saffman, Proc. Roy. Soc. London A 289 (1966) 301] and A.C. Newell [A.C. Newell, Rev. Geophys. 6 (1968) 1]. This allows us to provide a self-consistent weak-turbulence theory of the condensation process, in which the nonequilibrium formation of the condensate is a natural consequence of the spontaneous regeneration of a non-vanishing first-order cumulant in the hierarchy of the cumulants' equations. More precisely, we show that in the presence of a small condensate amplitude, all relevant statistical information is contained in the off-diagonal second order cumulant, as described by the usual weak-turbulence theory. Conversely, in the presence of a high-amplitude condensate, the diagonal second-order cumulants no longer vanish in the long time limit, which signals a breakdown of the weak-turbulence theory. However, we show that all asymptotic closure of the hierarchy of the cumulants' equations is still possible provided one considers the Bogoliubov's basis rather than the standard Fourier's (free particle) basis. The nonequilibrium dynamics turns out to be governed by the Bogoliubov's off-diagonal second order cumulant, while the corresponding diagonal cumulants, as well as the higher order cumulants, are shown to vanish asymptotically. The numerical discretization of the NLS equation implicitly introduces an ultraviolet frequency cut-off. The simulations are in quantitative agreement with the weak turbulence theory without adjustable parameters, despite the fact that the theory is expected to breakdown nearby the transition to condensation. The fraction of condensed particles vs energy is characterized by two distinct regimes: For small energies (H << H-c) the Bogoliubov's regime is established, whereas for H less than or similar to H-c the small-amplitude condensate regime is described by the weak-turbulence theory. In both regimes we derive coupled kinetic equations that describe the coupled evolution of the condensate amplitude and the incoherent field component. The influence of finite size effects and of the dimensionality of the system are also considered. It is shown that, beyond the thermodynamic limit, wave condensation is reestablished in two spatial dimensions, in complete analogy with uniform and ideal 2D Bose gases. (C) 2009 Elsevier B.V. All rights reserved.  
  Address [Duering, Gustavo; Rica, Sergio] Univ Paris 06, Ecole Normale Super, Phys Stat Lab, F-75005 Paris, France, Email: rica@lps.ens.fr  
  Corporate Author Thesis  
  Publisher Elsevier Science Bv Place of Publication Editor  
  Language English Summary Language Original Title  
  Series Editor Series Title Abbreviated Series Title  
  Series Volume Series Issue Edition  
  ISSN 0167-2789 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000268621400003 Approved no  
  Call Number UAI @ eduardo.moreno @ Serial 61  
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