טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentEvyatar Sabag
SubjectHolographic Turbulence in a Large Number of Dimensions
DepartmentDepartment of Physics
Supervisor Professor Yarom Amos
Full Thesis textFull thesis text - English Version


Abstract

We consider relativistic hydrodynamics in the limit where the number of spatial dimensions is very large. We first show that under certain restrictions, the resulting equations of motion simplify significantly. Holographic theories in a large number of dimensions satisfy the aforementioned restrictions and their dynamics are captured by hydrodynamics with a naturally truncated derivative expansion. The resulting hydrodynamic equations of motion in the aforementioned limit bare a striking resemblance to the compressible Navier-Stokes equations. This resemblance allows for a comparison of the former equations to known results from classical hydrodynamics, especially in the subsonic regime, where we expect to find the Kolmogorov and Kraichnan scaling laws, in addition to the direct and inverse cascades.

The resulting equations in a large number of dimensions are non-linear as the Navier-Stokes equations, and are hard to tackle analytically. Thus, we turn to numerical techniques to analyze two and three-dimensional turbulent flow of such fluids in various regimes. The Kolmogorov and Kraichnan scaling laws are not found in the simulations of two dimensional flows, probably due to insufficient lattice size. Inverse cascade-like behavior, on the other hand is apparent, and remains visible even when the Mach number is not small. The three-dimensional turbulent flow shows a transient effect of the Kolmogorov scaling law as expected for decaying turbulence, in addition to a direct-cascade like behavior. Remarkably, these two observations remain true even for high Mach numbers. Finally, using our improved analytic control in the limit of large number of dimensions, we studied a proposed relation between the horizon curvature power spectrum and the hydrodynamic energy power spectrum. We found these quantities to be linearly related by analytic and numeric analysis for low Mach numbers. However, the relation becomes unclear as we increase the Mach number.