Renormalized Statistical Cumulants in Stochastic Surface Growth and Fluid Turbulence

Abstract

Fluctuating geometries occur in a variety of macroscopic non-equilibrium phenomena in Nature. Examples start from trajectories of Brownian particles, structure of sea shores, landscapes, mountains, islands, rivers, sediments, and even the surface geometry of thin films grown in the laboratory. Dynamics of these non-equilibrium phenomena being stochastic in nature, their statistical properties are of interest in modern trends of research. In this thesis, we focus mainly on surface growth dynamics and fluid turbulence, which are of great value due to their practical importance. In particular, we study a few normalized cumulants of distribution functions (height distribution for surface growth and distribution of velocity derivative for fluid turbulence), namely, skewness, kurtosis, hyper-skewness. The non-zero value of those cumulants confirm that the corresponding probability distributions are non-Gaussian in nature. There are various types of surface growth, which can be described by different continuum equations. The most generic continuum nonlinear equation for surface growth is Kardar-Parisi-Zhang (KPZ) equation driven by a Gaussian white noise. The equation does not respect the up-down symmetry of the surface height because of the nonlinear term in the equation. First, we study one-dimensional KPZ equation by employing a perturbative renormalization scheme. We calculate the second and third order cumulants of the height-distribution from the renormalized Feynman loop diagrams obtaining the skewness value S = 0.3237 which is independent of any model parameter.