Mode-by-mode hydrodynamics: Ideas and concepts

Abstract

The main ideas, technical concepts and perspectives for a mode resolved description of the hydrodynamical regime of relativistic heavy ion collisions are discussed. A background-fluctuation splitting and a Bessel-Fourier expansion for the fluctuating part of the hydrodynamical fields allows for a complete characterization of initial conditions, the fluid dynamical propagation of single modes, the study of interaction effects between modes, the determination of the associated particle spectra and the generalization of the whole program to event-by-event correlations and probability distributions.In recent years it has become apparent that what one may call the "fluid dynamic standard model of heavy ion collisions" according to which initial density anisotropies are evolved to final momentum anisotropies by almost ideal hydrodynamics works surprisingly well, see [1,2,3,4] for recent reviews. Some of the remaining puzzles are about the nature of the initial state directly after the collision and the early out-of-equilibrium dynamics that drives it towards local equilibrium on a rather short time scale, see [5] for an overview over recent literature on that question. On a more quantitative level one would like to better understand how thermodynamic and transport properties govern the hydrodynamic regime in order to allow for precise comparison between theoretical calculations and experimental measurements of these material properties of strongly interacting quantum field theory.Interesting new insights into both the properties of the initial state and the material properties of QCD in the hydrodynamic regime may come from the study of fluctuations in the hydrodynamic fields. More specific, from many models or on general grounds one expects event-by-event fluctuations around the average in the hydrodynamical fields such as energy density , fluid velocity u µ , shear stress π µν (and more general also baryon number density n B , electric charge density, electromagnetic fields and so on) at the initialization time τ 0 where the hydrodynamical description becomes (approximately) valid. These fluctuations should contain interesting information from early times, both from the initial state and the early non-equilibrium dynamics. Their dynamical evolution is governed by universal fluid dynamic equations, depending only on the thermodynamic and transport properties. Since one can distinguish fluctuations of different characteristic spatial size by their wave numbers, the information content is much richer than for averaged quantities only.There is an interesting analogy to cosmology. Indeed, the cosmic microwave background also contains interesting information from early times and the time evolution is sensitive to the history of the universe. The spectrum of fluctuations involves many numbers (or whole functions) that can be compared between theory and experiment. Historically, the detailed study of this spectrum has lead to quite a detailed quantitative understanding of cosmology and one may hope that a similar

Heavy ion collisions and cosmology

“…Besides the linear formalism, there is also a diagrammatic formalism to treat non-linear mode-mode couplings. In principle one could use a very similar theoretical approach to describe the evolution of perturbations in the fluid dynamic description of heavy ion collisions [1,9]. The somewhat smaller degree of global symmetries increases somewhat the technical effort.…”