Kinetic evolution and correlation of fluctuations in an expanding quark gluon plasma.

Evolution of spatially anisotropic perturbation created in the system formed after Relativistic Heavy Ion Collisions has been studied. The microscopic evolution of the fluctuations has been examined within the ambit of Boltzmann Transport Equation (BTE) in a hydrodynamically expanding background. The expansion of the background composed of Quark Gluon Plasma (QGP) is treated within the framework of relativistic hydrodynamics. Spatial anisotropic fluctuations with different geometry have been evolved through Boltzmann equation. It is observed that the trace of such fluctuation survive the evolution. Within the relaxation time approximation analytical results have been obtained for the evolution of these anisotropies. Explicit relations between fluctuations and transport coefficients have been derived. The mixing of various Fourier (or $k$) modes of the perturbations during the evolution of the system has been explicitly demonstrated. This study is very useful in understanding the presumption that the measured anisotropies in the data from heavy ion collisions at relativistic energies imitate the initial state effects. The evolution of correlation function for the perturbation in pressure has been studied and shown that the initial correlation between two neighbouring points in real space evolves to a constant value at later time which gives rise to Dirac delta function for the correlation function in Fourier space. The power spectrum of the fluctuation in thermodynamic quantities (like temperature estimated in this work) can be connected to the fluctuation in transverse momentum of the thermal hadrons measured experimentally. The bulk viscous coefficient of the QGP has been estimated by using correlations of pressure fluctuation with the help of Green-Kubo relation. Angular power spectrum of the anisotropies has been estimated in the appendix.