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Abstract: My PhD thesis is concerned with Functional Renormalization Group (FRG) approach to turbulence. In this talk I will cover the results that we obtained for the Burgers equation, which was introduced as a simplified model of turbulence (it has same nonlinear term as the Navier-Stokes equation). It maps exactly to the Kardar-Parisi-Zhang (KPZ) equation (which describes completely different physics – randomly growing surfaces) and is a very famous model of non-equilibrium statistical physics. Within the FRG approach, we can obtain a “coarse-grained” effective description in order to access statistical properties of the model, such as scaling regimes, distinguished by a dynamical exponent z, which tells how the correlation time t is related to momentum k, t~k^(-z). In fact, different scaling regimes correspond to different fixed points of the FRG equation. The motivation of our study was that a new scaling regime characterized by a z=1 dynamical critical exponent has been reported in several numerical simulations of the one-dimensional KPZ and noisy Burgers equations \1-2\. This scaling was found to emerge in the tensionless limit for the interface (KPZ) and in the inviscid limit for the fluid (Burgers). Based on FRG, the origin of this scaling has been elucidated \3\. It was shown to be controlled by a yet unpredicted fixed point of the one-dimensional Burgers-KPZ equation, termed inviscid Burgers fixed point. The associated universal properties, including the scaling function, were calculated. In this talk, I will show how this analysis can be generalized to the multi-dimensional Burgers-KPZ equation \4\. I will show that the inviscid Burgers fixed point exists in all dimensions d, 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.
References
\1\ C. Cartes, E. Tirapegui, R. Pandit, M. Brachet, The Galerkin-truncated Burgers equation: crossover from inviscid-thermalized to Kardar-Parisi-Zhang scaling, Phil. Trans. Roy. Soc. A 380, 20210090 (2022)
\2\ E. Rodriguez-Fernandez, S. N. Santalla, M. Castro, R. Cuerno, Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation, Phys. Rev. E 106, 024802 (2022)
\3\ C. Fontaine, F. Vercesi, M. Brachet, L. Canet, Unpredicted scaling of the one-dimensional Kardar-Parisi-Zhang equation, Phys. Rev. Lett. 131, 247101 (2023)
\4\ L. Gosteva, M. Tarpin, N. Wschebor, L. Canet, Inviscid fixed point of the multidimensional Burgers–Kardar-Parisi-Zhang equation, Phys. Rev. E 110, 054118 (2024)