Breakdown of weak-turbulence and nonlinear wave condensation
Introduction
In the present work we study the statistical initial value problem for the nonintegrable Hamiltonian defocusing NonLinear Schrödinger (NLS) equation or Gross–Pitaevskii equation in space dimensions higher or equal than 2. The random nonlinear wave is known to exhibit a thermalization process characterized by an irreversible evolution of the field towards an equilibrium state. In the defocusing regime of the NLS equation, the thermalization of the random field is characterized by a condensation process, i.e. by the spontaneous formation of a homogeneous solution in the long term evolution of the field [1], [2]. The condensation process manifests itself as a phase transition of the equilibrium state for sufficiently low energy density [3], [4], [5], [6]. In the framework of weak-turbulence theory [7], [8], the nonequilibrium formation of the condensate may be regarded as an inverse cascade of wave action from small scales to large scales and a direct cascade of energy towards small scale fluctuations, i.e. from small to large wavenumbers . We provide a self-consistent weak-turbulence theory of the condensation process, in which the spontaneous emergence of a non-vanishing average of the field () results from the natural asymptotic closure of the cumulant equations for the random field. The theory is confirmed by a direct numerical integration of the NLS equation, which implicitly introduces an ultraviolet frequency cut-off in the wave system, i.e. a finite number of degrees of freedom.
It is important to discuss the phenomenology of the condensation process into a more general perspective. The classical condensation process may be regarded as a self-organization process that occurs in a conservative and reversible wave system. Let us recall in this respect that, contrary to dissipative systems, a conservative Hamiltonian system cannot evolve towards a fully ordered state, because such an evolution would imply a loss of statistical information for the system that would violate its formal reversibility. However, in spite of its formal reversibility, a nonintegrable Hamiltonian system is expected to exhibit an irreversible evolution towards an equilibrium state, as a result of an irreversible process of diffusion in phase-space [9]. In this regard, an important achievement was accomplished when Zakharov and collaborators reported in Ref. [10] numerical simulations performed in the framework of the focusing nonintegrable NLS equation. This study revealed that the Hamiltonian system would evolve, as a general rule, towards the formation of a large-scale coherent localized structure, i.e., a solitary-wave, immersed in a sea of small-scale turbulent fluctuations. The solitary wave then plays the role of a “statistical attractor” for the Hamiltonian system, while the small-scale fluctuations contain, in principle, all the information necessary for time reversal. It is important to note, the solitary-wave solution corresponds to the solution that minimizes the energy (Hamiltonian), so that the system tends to relax towards the state of minimum energy, while the small-scale fluctuations compensate for the difference between the conserved energy and the energy of the coherent structure.
As was initially suggested in Refs. [10], a rigorous theoretical description of the long term evolution of the system would require a thermodynamic approach. It is only recently that statistical equilibrium models have been elaborated in the framework of statistical mechanics [11], [12], [13]. As a remarkable result, whenever the Hamiltonian system is constrained by an additional integral of motion (e.g., number of particles), the increase of entropy of small-scale turbulent fluctuations requires the formation of coherent structures [11], [13], so that it is thermodynamically advantageous for the system to approach the ground state which minimizes the energy [10]. More precisely, it is shown that a statistical equilibrium is reached, in which the energy not contained in the coherent structure is equally distributed among the modes of the small-scale fluctuations.
Let us now refers back to the condensation process that we shall discuss in the present paper. It is indeed important to note that the spontaneous formation of a homogeneous solution corroborates the general rule discussed above [10], [11], [13], because the homogenous solution realizes the minimum of the energy (Hamiltonian) in the defocusing case. This analogy between the focusing and defocusing regimes reveals that the formation of a condensate may be viewed as a consequence of the fact that the system naturally tends to increase its disorder (entropy). A simple explanation of this counterintuitive result may be given by recalling that the total energy of the field has a kinetic contribution and a nonlinear contribution. The kinetic energy being proportional to the gradient of the field, it provides a measure of the amount of fluctuations in the system. On the other hand, the nonlinear energy reaches its minimum value for a homogeneous solution. This merely explains why it is advantageous for the field to generate a condensate, because this permits the field to increase its disorder. In other terms, an increase of entropy in the field requires the generation of a homogenous solution (a plane-wave in optics terminology). According to this physical picture, there is a direct correspondence between the mechanisms underlying the spontaneous generation of a solitary-wave in the focusing regime and the condensation process in the defocusing regime. In both cases, the system tends to reach the most disordered state characterized by the presence of small-scale fluctuations in the field, which requires the generation of a large-scale coherent structure. The fact that the defocusing NLS equation exhibits a condensation process has been accurately confirmed in the context of thermal Bose fields, where numerical simulations of a “projected” NLS equation have been performed for both uniform [3], [4] and non-uniform (i.e. trapped) [14] configurations of the gas.
In a recent Letter [5] we formulated a thermodynamic description of this condensation process on the basis of the weak-turbulence theory. We showed that the thermodynamic properties of the condensation process are analogous to those of a Bose–Einstein transition, despite the fact that the considered wave system is inherently classical. More precisely, the theoretical analysis of the equilibrium properties of the condensation process were found to be in agreement with the numerical simulations of the three-dimensional (3D) NLS equation [5]. The present paper is aimed at providing a deeper insight into the weak-turbulence description of the classical wave condensation process, in which particular attention will be devoted to the kinetic dynamics of condensate formation.
The weak-turbulence theory is often referred to the so-called “random phase-approximation” approach [1], [7], [15], [16]. However, this approach should be rather considered as a convenient way of interpreting the results of the more rigorous technique based on the analysis of the cumulants of the nonlinear field, as originally formulated in Refs. [17], [18], [19]. This approach has been recently reviewed in Ref. [8], and studied in more detail through the analysis of the probability distribution function of the random field in Refs. [20]. The weak-turbulence theory describes the long time behavior of statistically homogeneous random waves under quite general assumptions [17], [18], [19]. It consists of a BBGKY-type of hierarchy equations for the cumulants of the random field, which is naturally closed because of the dispersive properties of the waves and the large separation between linear and nonlinear time scales [8], [17], [18], [19], [20]. The asymptotic closure of the hierarchy provides a kinetic equation governing the evolution of the spectrum of the random field, which actually refers to the off-diagonal second-order cumulant [] [8], [17], [18], [19]. The structure of the kinetic equation is analogous to that of the Boltzmann’s equation for classical dilute gases [21]. In particular, it exhibits a -theorem of entropy growth that describes an irreversible evolution of the field towards the Rayleigh–Jeans equilibrium distribution, , where and are called, by analogy with thermodynamics, the temperature and the chemical potential. Considering the system at equilibrium, it was shown in Ref. [5] that, in analogy with the standard Bose–Einstein condensation of an ideal quantum gas, wave condensation arises when the chemical potential reaches zero (from the negative side) for a non-vanishing critical temperature . Consequently, the number of particles was decomposed into a condensed part and an incoherent part, , where the sum excludes the fundamental mode . Let us underline that this decomposition is introduced artificially in the theory and, moreover, it does not consider as a dynamical variable and thus limits the theoretical description to the purely stationary equilibrium regime. Here we report a self-consistent weak-turbulence theory in which the nonequilibrium formation of the condensate, i.e., a non-vanishing average , is shown to naturally result from the spontaneous regeneration of the first-order cumulant of the field in the hierarchy of cumulants’ equations.
Moreover, the standard weak turbulence approach of the condensation process is not satisfactory whenever the equilibrium state is characterized by a significant fraction of condensed particles , i.e., for small values of the temperature, or equivalently for small values of the total energy of the field. Indeed, the nonlinear frequency renormalization indicates that a new dispersion relation that takes into account the nonlinear interaction should be introduced. It turns out that the Bogoliubov’s dispersion relation is the relevant dispersion relation that should be considered. A heuristic kinetic theory of this highly condensed regime (large condensate fraction ) was reported in Refs. [1], [5], [6] by adapting the Bogoliubov’s transformation to the classical wave problem. In particular, in a finite degree of freedom system, that is in a truncated non-linear Schödinger equation, it was shown in Ref. [5] that the nonlinear interaction changes the nature of the transition to condensation, i.e., the transition was shown to become subcritical (discontinuous). However, such a subcritical behavior was not identified in the numerical simulations [5]. Here we clarify this issue by showing that the transition to condensation of a classical wave system is actually a supercritical (continuous) transition.
In summary, we show that the phenomenon of wave condensation may be described by means of two distinct approaches. When the equilibrium state is characterized by a small fraction of condensed particles (i.e., high energy ), all the relevant statistical information is in the off-diagonal second order cumulant , so that the standard weak-turbulence closure is valid and the field relaxes to the usual Rayleigh–Jeans distribution. This approach describes well the critical energy for the transition to condensation (), as well as the small condensate regime. The corresponding condensation curve that expresses the fraction of condensed particles vs the energy , is given in Eq. (50). Conversely, for an equilibrium state characterized by a significant fraction of condensed particles (i.e., small energy ), we show that the diagonal second-order cumulants [and ] do not longer decrease to zero as in the usual weak-turbulence approach. This signals a breakdown of the standard weak-turbulence theory. The key idea behind the analysis is that the “normal variables” are no longer the Fourier’s modes, but instead the Bogoliubov’s modes. It is shown that in the framework of the Bogoliubov’s basis, an asymptotic closure of the hierarchy of the cumulants’ equations is still possible. The dynamics results in being 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 in the long time limit. The analysis reveals that the equilibrium spectrum turns out to be the Rayleigh–Jeans distribution with the Bogoluibov’s dispersion relation and a zero chemical potential. The corresponding expression of the condensation curve ( vs ) of this high-amplitude condensate regime is given by Eq. (52).
Let us remark that, as was pointed out in Ref. [5], wave condensation does not occur in two spatial dimensions (2D) in the thermodynamic limit. Nevertheless, we shall see that for situations of practical interest in which a finite number of particles and a finite size of the system are considered, wave condensation is reestablished in 2D. This corroborates the numerical observations reported, e.g., in Ref. [22]. Because the system is not in the thermodynamic limit, the chemical potential does not vanish exactly, and the condensation curve is given by means of two parametrically coupled equations for the condensate amplitude and the energy [see Eqs. (56), (57)]. Actually, the critical energy for the transition to condensation tends to zero in the thermodynamic limit, because of the logarithmic infrared divergence of the equilibrium distribution, in complete analogy with uniform and ideal 2D Bose gases.
We finally note that, although we provide a consistent wave turbulence theory with a condensate, which is in quantitative agreement with the numerical simulations at equilibrium, the non-equilibrium dynamics of the random field still leaves open many questions. As many times stressed by A. Newell and collaborators [8], weak turbulence still does not constitute a complete and fully-satisfactory theory. Indeed, there are always some spatial scales (large or small) at which the theory breaks down [8], [23], [24]. In the considered example of wave condensation, we show that the theory breaks down near in the thermodynamic limit for dimensions , and nonlinear processes come into play, such as, e.g., a new dispersion relation, or the three-wave resonant interactions. A complete picture of wave condensation should also include the interaction with localized coherent structures, such as, e.g., vortices or solitary waves as the Jones–Roberts rarefaction pulses [25]. In this respect, a complete statistical description of large scale coherent structures and their interactions with weakly nonlinear turbulent fluctuations is still lacking and will be the subject of future investigations.
The paper is organized as follows. In Section 2 we introduce the NLS model equation and recall the usual weak turbulence description, its properties and the mechanism underlying the dynamical formation of a condensate. In Section 3 we follow the weak-turbulence approach based on the analysis of the cumulants of the random field [8], [17], [18], [19]. This approach will be shown to be more appropriate for the description of the spontaneous formation of a condensate, which turns out to be associated to the emergence of a non-vanishing first-order cumulant of the field, i.e., a non-vanishing average (). Moreover, this approach naturally decomposes the random field into its coherent part (condensate) and its incoherent part, whereas such a decomposition is introduced in an artificial way in the usual treatment of wave condensation. In Section 4 we develop the weak-turbulence theory in the Bogoliubov’s basis, which is relevant for the description of the turbulent regime in the presence of a high-amplitude condensate. In this treatment the standard result of vanishing second-order diagonal cumulants does not hold. Finally, in Section 5 we analyze the role of finite size effects and of the dimensionality of the system, and we compare the predictions of weak turbulence theory with direct numerical simulations of the NLS equation.
Section snippets
NLS model equation
Let us consider the normalized defocusing NLS equation for the complex wave-function [26]: where refers to the Laplacian operator in spatial dimension while the constant allow us to trace the role of the nonlinearity in the theoretical analysis. This equation describes the evolution of defocusing interacting waves through the cubic nonlinear term. The dynamics conserves the number of particles (mass) the linear momentum and the total energy or
Cumulants’ hierarchy
To provide a statistical description of the evolution of the random field , we shall make use of the cumulants of the field rather than its moments [8], [17], [18], [19] (see the Appendix A.1 for the definitions of the cumulants). Indeed, given the homogenous character of the statistics, the advantage of the cumulants with respect to the moments relies on the fact that the cumulants decrease to zero as the spatial coordinates involved in the average tend to infinity. For instance, for a field
Weak-turbulence theory with a high amplitude condensate: The Bogoluibov’s regime
In the presence of a small energy density, the weak-turbulence theory presented above only describes a transient evolution of the field, because in the long-time limit the condensate fraction at equilibrium may not be small, i.e., . In this regime of high-condensate amplitude, the diagonal terms of the second-order cumulant (, ) no longer vanish, a peculiar feature which signals a breakdown of the usual weak-turbulence theory. Note that this breakdown seems to be of different
Numerical simulations of wave turbulence and wave condensation
To numerically integrate the NLS equation (1) one has to resort to a spatial discretization of the equation. The discretized field then exhibits a finite number of degrees of freedom, and the Rayleigh–Jeans distribution (25) may be reached without the trouble of the ultraviolet divergence discussed above. In the following we shall see that a good agreement is obtained between the numerical simulations and the wave turbulence theory reported in Sections 3 Weak turbulence theory with a small
Conclusion
In summary, we analyzed the classical wave condensation process by means of the defocusing NLS equation. We extended the standard weak-turbulence theory by analyzing the asymptotic closure of the hierarchy of the cumulants’ equations induced by the dispersive properties of the waves. This allowed us to formulate a self-consistent wave turbulence theory, in which the condensation process appears as a natural consequence of the emergence of a non-vanishing first-order cumulant. The condensation
Acknowledgements
The authors acknowledge Christophe Josserand for stimulating discussions and contribution at the early stage of this work. This work has been supported by the grant ANR-08-SYSC-004 of the Agence National de la Recherche, France (ANR). G.D. also thanks a Conicyt fellowship.
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