Fig. 2. Typical behavior of the solutions to the simplest linear scale differential equation (see Sect. 3.1.1). One obtains an asymptotic fractal (power-law resolution-dependent) behavior at either large or small scales, and a transition to scale-independence toward the classical domain (intermediate scales). The transitions are given by the Compton-de Broglie scale in the microscopic case and by the typical static radius of objects (galaxy radii, cluster cores) in the macroscopic case. Note that the microscopic and macroscopic plots actually correspond to two different kinds of experiments: in the microscopic case, the "window" is kept constant while the "resolution" is changed, leading to an increase toward small scales, ; in the macroscopic case, the fractal behavior shows itself by increasing the window for a fixed resolution , this leading to an increase toward large scales, .