Léonie Canet

In this work, we investigate the Navier–Stokes equation in the presence of thermal noise, both at finite viscosity (revisiting the seminal work by Forster–Nelson–Stephen) and in the inviscid limit,which has not yet been explored. We determine the space-time velocity correlations in this dynamics, using functional renormalization group (RG) and direct numerical simulations. While spectrally truncated three-dimensional (3D) Euler flows reach a stationary equilibrium state, they exhibit non-trivial temporal correlations. We show that these non-trivial correlations persist for small but finite viscosity, yielding an emergent τ∼k-1 dynamical scaling, where τ is the decorrelation time. We characterize the crossover from the scaling τ∼1/(νk2), expected at large viscosity, to the scaling τ∼1/(urmsk) found in the inviscid limit. This article is part of the theme issue ‘Frontiers of turbulence and statistical physics’.

The Kardar–Parisi–Zhang (KPZ) equation is a celebrated non-linear stochastic dynamical equation yielding non-equilibrium universal scaling. It exhibits notorious non-perturbative aspects. The KPZ fixed point is strong-coupling, all the more in d > 1. Strikingly, another, even stronger-coupling fixed point of the KPZ equation, called inviscid Burgers fixed point, has been recently unveiled. These non-pertubative features can be theoretically accessed and studied in a controlled way in all dimensions using the functional renormalisation group. We propose an overview of the related results, which provide a unified picture of the fixed-point structure and associated scaling regimes of the KPZ equation in d = 1 and in higher dimensions.

The Kardar–Parisi–Zhang (KPZ) equation is a celebrated non-linear stochastic equation featuring non-equilibrium scaling. Although its statistical properties are well understood in one dimension, a new scaling regime has been reported in recent numerical simulations. This new regime is characterised by a dynamical exponent z = 1, markedly different from the expected one z = 3 / 2 for the KPZ universality class, and it emerges when approaching the inviscid limit. The origin of this scaling has been traced back to the existence of a new fixed point, termed the inviscid Burgers (IB) fixed point, which was uncovered using the functional renormalisation group (FRG). The FRG equations can be solved analytically in the asymptotic regime of vanishing viscosity and large momenta, showing that indeed z = 1 exactly at the IB fixed point. In this work, we set up an advanced method to numerically solve the full FRG flow equations in a certain approximation, which allows us to determine in a unified way the correlation function over the whole range of momenta, not restricted to some particular regime. We analyse the crossover between the different fixed points, and quantitatively determine the extent of the IB regime.

Many systems, classical or quantum, closed or open, exhibit universal statistical properties. Exciton–polariton condensates, being intrinsically driven-dissipative, offer a promising platform for observing nonequilibrium universal features. By conducting extensive numerical simulations of an incoherently pumped and interacting condensate coupled to an exciton reservoir we show that the effective nonlinearity of the condensate phase dynamics can be finely adjusted across a broad range, by varying the exciton–polariton interaction strength, allowing one to probe three main universal regimes with parameters accessible in current experiments: the weakly nonlinear Edwards-Wilkinson (EW) regime, where the phase fluctuations dominate, but the phase profile does not become rough, the strongly nonlinear Kardar-Parisi-Zhang regime, where the condensate phase fluctuations grow in a superdiffusive manner leading to roughening of the phase, and a vortex-dominated phase emerging at stronger interactions, where both density and phase dynamics play significant roles. Our results provide a unified picture of the phase diagram of two-dimensional exciton–polariton condensates under incoherent pumping, and shed light on recent experimental and numerical observations.

We investigate the correlation properties in the steady state of driven-dissipative interacting bosonic systems in the quantum regime, as for example non-linear photonic cavities. Specifically, we consider the Bose-Hubbard model on a periodic chain and with spatially homogeneous one-body loss and pump within the Markovian approximation. The steady state is non-thermal and is formally equivalent to an infinite-temperature state with finite chemical potential set by the dissipative parameters. While there is no effect of interactions on the steady state, we observe a nontrivial behaviour of the space-time two-point correlation function, obtained by exact diagonalisation. In particular, we find that the decay width of the propagator is not only renormalised at increasing interactions, as it is the case of a single non-linear resonator, but also at increasing hopping strength. Furthermore, we numerically predict at large interactions a plateau value of the decay rate which goes beyond perturbative results in the interaction strength. We then compute the full spectral function, finding that it contains both a dispersive free-particle like dispersion at low energy and a doublon branch at energy corresponding to the on-site interactions. We compare with the corresponding calculation for the ground state of a closed quantum system and show that the driven-dissipative nature – determining both the steady state and the dynamical evolution – changes the low-lying part of the spectrum, where noticeably, the dispersion is quadratic instead of linear at small wavevectors. Finally, we compare to a high temperature grand-canonical equilibrium state and show the difference with respect to the open system stemming from the additional degree of freedom of the dissipation that allows one to vary the width of the dispersion lines.

A new scaling regime characterized by a z = 1 dynamical critical exponent has been reported in several numerical simulations of the one-dimensional Kardar-Parisi-Zhang and noisy Burgers equations. In these works, this scaling, differing from the well-known KPZ one z = 3/2, was found to emerge in the tensionless limit for the interface and in the inviscid limit for the fluid. Based on functional renormalization group, the origin of this scaling has been elucidated. It was shown to be controlled by a yet unpredicted fixed point of the one-dimensional Burgers-KPZ equation, termed inviscid Burgers (IB) fixed point. The associated universal properties, including the scaling function, were calculated. All these findings were restricted to d = 1, and it raises the intriguing question of the fate of this new IB fixed point in higher dimensions. In this work, we address this issue and analyze the multidimensional Burgers-KPZ equation using functional renormalization group. We show that the IB fixed point exists in all dimensions d 0, and that it controls the large momentum behavior of the correlation functions in the inviscid limit. It turns out that it yields in all d the same super-universal value z = 1 for the dynamical exponent.

In the absence of vortices or phase slips, the phase dynamics of exciton-polariton condensates was shown to map onto the Kardar-Parisi-Zhang (KPZ) equation, which describes the stochastic growth of a classical interface. This implies that the coherence of such nonequilibrium quasicondensates decays in space and time following stretched exponentials, characterized by KPZ universal critical exponents. In this work, we focus on the time evolution of the average phase of a one-dimensional exciton-polariton condensate in the KPZ regime and determine the frequency of its evolution, which is given by the blueshift, i.e., the nonequilibrium analog of the chemical potential. We determine the stochastic corrections to the blueshift within Bogoliubov linearized theory and find that while this correction physically originates from short scale effects, and depends both on density and phase fluctuations, it can still be related to the effective large-scale KPZ parameters. Using numerical simulations of the full dynamics, we investigate the dependence of these blueshift corrections on both noise and interaction strength, and compare the results to the Bogoliubov prediction. Our finding contributes both to the close comparison between equilibrium and nonequilibrium condensates, and to the theoretical understanding of the KPZ mapping.

Shell models are simplified models of hydrodynamic turbulence, retaining only some essential features of the original equations, such as the non-linearity, symmetries and quadratic invariants. Yet, they were shown to reproduce the most salient properties of developed turbulence, in particular universal statistics and multi-scaling. We set up the functional renormalisation group (FRG) formalism to study generic shell models. In particular, we formulate an inverse RG flow, which consists in integrating out fluctuation modes from the large scales (small wavenumbers) to the small scales (large wavenumbers), which is physically grounded and has long been advocated in the context of turbulence. Focusing on the Sabra shell model, we study the effect of both a large-scale forcing, and a power-law forcing exerted at all scales. We show that these two types of forcing yield different fixed points, and thus correspond to distinct universality classes, characterised by different scaling exponents. We find that the power-law forcing leads to dimensional (K41-like) scaling, while the large-scale forcing entails anomalous scaling.

Shell models are simplified models of hydrodynamic turbulence, retaining only some essential features of the original equations, such as the non-linearity, symmetries and quadratic invariants. Yet, they were shown to reproduce the most salient properties of developed turbulence, in particular universal statistics and multi-scaling. We set up the functional renormalisation group (RG) formalism to study generic shell models. In particular, we formulate an inverse RG flow, which consists in integrating out fluctuation modes from the large scales (small wavenumbers) to the small scales (large wavenumbers), which is physically grounded and has long been advocated in the context of turbulence. Focusing on the Sabra shell model, we study the effect of both a large-scale forcing, and a power-law forcing exerted at all scales. We show that these two types of forcing yield different fixed points, and thus correspond to distinct universality classes, characterised by different scaling exponents. We find that the power-law forcing leads to dimensional (K41-like) scaling, while the large-scale forcing entails anomalous scaling.