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Baudin, K., Fusaro, A., Garnier, J., Berti, N., Krupa, K., Carusotto, I.:, Rica, S., et al. (2021). Energy and waveaction flows underlying RayleighJeans thermalization of optical waves propagating in a multimode fiber((a)). EPL, 134(1), 14001.
Abstract: The wave turbulence theory predicts that a conservative system of nonlinear waves can exhibit a process of condensation, which originates in the singularity of the RayleighJeans equilibrium distribution of classical waves. Considering light propagation in a multimode fiber, we show that light condensation is driven by an energy flow toward the higherorder modes, and a bidirectional redistribution of the waveaction (or power) to the fundamental mode and to higherorder modes. The analysis of the nearfield intensity distribution provides experimental evidence of this mechanism. The kinetic equation also shows that the waveaction and energy flows can be inverted through a thermalization toward a negative temperature equilibrium state, in which the highorder modes are more populated than loworder modes. In addition, a Bogoliubov stability analysis reveals that the condensate state is stable.
Keywords: BOSEEINSTEIN CONDENSATION; TURBULENCE; TEMPERATURE; LIGHT

During, G., Picozzi, A., & Rica, S. (2009). Breakdown of weakturbulence and nonlinear wave condensation. Physica D, 238(16), 1524–1549.
Abstract: The formation of a largescale 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 smallscale 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 selfconsistent weakturbulence theory of the condensation process, in which the nonequilibrium formation of the condensate is a natural consequence of the spontaneous regeneration of a nonvanishing firstorder 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 offdiagonal second order cumulant, as described by the usual weakturbulence theory. Conversely, in the presence of a highamplitude condensate, the diagonal secondorder cumulants no longer vanish in the long time limit, which signals a breakdown of the weakturbulence 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 offdiagonal 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 cutoff. 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 << Hc) the Bogoliubov's regime is established, whereas for H less than or similar to Hc the smallamplitude condensate regime is described by the weakturbulence 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.

Kalyaan, A., Pinilla, P., Krijt, S., Banzatti, A., Rosotti, G., Mulders, G. D., et al. (2023). The Effect of Dust Evolution and Traps on Inner Disk Water Enrichment. Astrophys. J., 954(1), 66.
Abstract: Substructures in protoplanetary disks can act as dust traps that shape the radial distribution of pebbles. By blocking the passage of pebbles, the presence of gaps in disks may have a profound effect on pebble delivery into the inner disk, crucial for the formation of inner planets via pebble accretion. This process can also affect the delivery of volatiles (such as H2O) and their abundance within the water snow line region (within a few au). In this study, we aim to understand what effect the presence of gaps in the outer gas disk may have on water vapor enrichment in the inner disk. Building on previous work, we employ a volatileinclusive disk evolution model that considers an evolving icebearing drifting dust population, sensitive to dust traps, which loses its icy content to sublimation upon reaching the snow line. We find that the vapor abundance in the inner disk is strongly affected by the fragmentation velocity (v( f)) and turbulence, which control how intense vapor enrichment from pebble delivery is, if present, and how long it may last. Generally, for disks with low to moderate turbulence (a = 1 x 10(3)) and a range of v( f), radial locations and gap depths (especially those of the innermost gaps) can significantly alter enrichment. Shallow inner gaps may continuously leak material from beyond it, despite the presence of additional deep outer gaps. We finally find that for realistic v( f) (=10 m s(1)), the presence of gaps is more important than planetesimal formation beyond the snow line in regulating pebble and volatile delivery into the inner disk.

Picozzi, A., & Rica, S. (2012). Condensation of classical optical waves beyond the cubic nonlinear Schrodinger equation. Opt. Commun., 285(24), 5440–5448.
Abstract: A completely classical nonlinear wave is known to exhibit a process of condensation whose thermodynamic properties are analogous to those of the genuine BoseEinstein condensation. So far this phenomenon of wave condensation has been studied essentially in the framework of the nonlinear Schrodinger (NLS) equation with a pure cubic Kerr nonlinearity. We study wave condensation by considering two representative generalizations of the NLS equation that are relevant to the context of nonlinear optics, the nonlocal nonlinearity and the saturable nonlinearity. For both cases we derive analytical expressions of the condensate fraction in the weakly and the strongly nonlinear regime. The theory is found in quantitative agreement with the numerical simulations of the generalized NLS equations, without adjustable parameters. (C) 2012 Elsevier B.V. All rights reserved.
