Power law signatures of cosmic rays, star formation, and interstellar turbulence

Power laws are typical signatures of complex systems and examples in astrophysics are manifold. Understanding their origin is challenging and can be approached in very different ways. In this thesis, I first provide a mathematical treatment of power laws which starts from the defining property of scale-invariance. For physical systems this absence of scales holds only over a finite range. Power laws can be found as solutions to differential equations whereas their validity is usually constrained by initial and boundary conditions which I demonstrate with several examples. An almost physics-free interpretation for the origin of observed mass functions, dN/dM $\sim$ M${}^{-\alpha }$ with $\alpha$$\sim$ 2, is given by a simple hierarchical fragmentation model which I discuss next to a more physical model of star formation. Cosmic ray energy spectra are power laws over many orders of magnitude in contrast to exponentially decaying energy distribution functions which are typical for equilibrium statistical mechanichs. Superstatistics is a particular generalization of statistical mechanics for nonequilibrium systems. It generates power law distribution functions from a superposition of equilibrium distributions with variable temperatures. I carefully assess the physical motivation for this model in order to apply it to the observed energy spectra of cosmic rays. This requires a relation between the superstatistical distribution function and the observed differential intensity which has been treated inaccurately by previous studies of superstatistics. Hence, the provided derivation clarifies and improves the theoretical basis of superstatistical models applied to particle physics. I apply this model to recent AMS data for primary (He, C, O) and secondary (Li, Be, B) cosmic rays in order to determine the best fit parameters. Smolla et al. (2020) interpret the two observed universality classes of cosmic ray spectra as resulting from the characteristic energy scale $\sim$ 200 MeV in QCD scattering processes and two distinct types of superpositions of temperature fluctuations. In addition to presenting these results, I also provide a critical discussion for this novel interpretation and the superstatistical model in general. Interstellar turbulence, stellar initial mass function, star formation law, and far-infrared-radio correlation are examples for nearly universal power laws, in the sense that they are remarkably insensitive to variations in parameters of the respective physical system. For each case I review the available observational data and discuss models which account for their origin. Star formation turns out being an essential driver behind all these phenomena. I present a new schematic version of a galaxy model where thermal gas, turbulent gas, magnetic fields and cosmic rays all have comparable energy densities. The observed power laws characterize the interconnections between these four components and star formation. Due to the complexity and multi-scale nature of the galaxy, this model deliberately ignores many details. Its aim is to identify the self-regulating mechanisms which give rise to the observed power laws and the equipartition of energy.