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This work considers the distribution of inertial particles in turbulenceusing the point-particle approximation. We demonstrate that the random pointprocess formed by the positions of particles in space is a Poisson pointprocess with log-normal random intensity ("log Gaussian Cox process" or LGCP).The probability of having a finite number of particles in a small volume isgiven in terms of the characteristic function of a log-normal distribution.Corrections due to discreteness of the number of particles to the previouslyderived statistics of particle concentration in the continuum limit areprovided. These are relevant for dealing with experimental or numerical data.The probability of having regions without particles, i.e. voids, is larger forinertial particles than for tracer particles where voids are distributedaccording to Poisson processes. Further, the probability of having large voidsdecays only log-normally with size. This shows that particles cluster, leavingvoids behind. At scales where there is no clustering there can still be anincrease of the void probability so that turbulent voiding is stronger thanclustering. The demonstrated double stochasticity of the distributionoriginates in the two-step formation of fluctuations. First, turbulence bringsthe particles randomly close together which happens with Poisson-typeprobability. Then, turbulence compresses the particles' volume in theobservation volume. We confirm the theory of the statistics of the number ofparticles in small volumes by numerical observations of inertial particlemotion in a chaotic ABC flow. The improved understanding of clusteringprocesses can be applied to predict the long-time survival probability ofreacting particles. Our work implies that the particle distribution in weaklycompressible flow with finite time correlations is a LGCP, independently of thedetails of the flow statistics.

Inertial particles distribute in turbulence as Poissonian points with random intensity inducing clustering and supervoiding