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We study the statistics of fluid (gas) density and concentration of passivetracer particles (dust) in compressible turbulence. We raise the question ofwhether the fluid density which is an active field that reacts back on thetransporting flow and the passive concentration of tracers must coincide in thesteady state, which we demonstrate to be crucial both theoretically andexperimentally. The fields' coincidence is provable at small Mach numbers,however at finite Mach numbers the assumption of mixing is needed, not evidentdue to the possibility of self-organization. Irrespective of whether the fieldscoincide we obtain a number of rigorous conclusions on both fields. As Maincreases the fields in the inertial range go through a phase transition from afinite continuous smooth to a singular multifractal distribution. We propose away to calculate fractal dimensions from numerical or experimental data. Wederive a simple expression for the spectrum of fractal dimensions of isothermalturbulence and describe limitations of lognormality. The expression depends ona single parameter: the scaling exponent of the density spectrum. We propose amechanism for the phase transition of concentration to multifractality. Wedemonstrate that the pair-correlation function is invariant under the action ofthe probability density function of the inter-pair distance that has the Markovproperty implying applicability of the Kraichnan turbulence model. We use themodel to derive an explicit expression for the tracers pair correlation thatdemonstrates their smooth transition to multifractality and confirms thetransition's mechanism. Our results are of potentially important implicationson astrophysical problems such as star formation as well as on technologicalapplications such as supersonic combustion. As an example we demonstrate strongincrease of planetesimals formation rate at the transition.

Density and tracer statistics in compressible turbulence: phase transition to multifractality