## 1 Introduction  
Inter-particle collision of small particles immersed in a turbulent flow is a common micro-process found in both industrial and natural environments. Examples of physical phenomena where the effects of inter-particle interactions are revealed as relevant are: models of planet formation from dust particle collisions and coalescence in turbulent protoplanetary nebulae [1; 2], rain drops' formation in warm clouds [3; 4], chemical reaction rates enhancement due to active suspended particles in fluidised beds [5; 6; 7], particle transport and reduction of wall erosion in pneumatic conveying [8; 9; 10; 11] and turbomachinery [12], or sand storm evolution and dune transport in deserts [13; 14]. Fundamental ingredients of inter-particle collisions are the role of particles' inertia and their interaction with the carrier turbulent flow structures, which lead to a non-linear increase of collision rates [15; 16; 17; 18]. As a result of the efforts to understand and quantify the enhancement of collision rates induced by inertia, two main mechanisms have been identified: on the one hand, preferential concentration, i.e., particle clustering, increases the probability for two particles to be at a colliding distance, and, on the other hand, detachment from fluid trajectories may enhance the relative velocity between two approaching particles. Two limiting cases can be devised: in the case of vanishingly small inertia, particles behave almost as tracers so that their relative velocity is given by the fluid velocity shear; therefore, the deviation of a uniform concentration distribution is responsible for an enhancement of collisions. In the other limit, of very large inertia, particle motion is essentially ballistic, and they distribute uniformly but they may reach the same position with very different velocities. Moreover, following [PERSON] et al. [19], interactions between the dispersed and continuous phases not only structure the particle number density, but also lead to correlation of the velocities of neighbouring particles, which is an alternative point of view of the description of enhancement of collision rates by inertia. For small-inertia particles, neighbouring velocities will be spatially correlated through the interactions with the same local fluid flow as basically both particles experience similar fluid velocities during long enough times. In contrast, for large-inertia particles with response times that are long compared to the fluid turbulence macroscales, neighbouring particle velocities are uncorrelated since these particles maintain a stronger connection (memory) to their interactions, with very distant, and independent, turbulent eddies. In the large inertia limit, statistics of the particle velocity distribution will satisfy the assumption of molecular chaos and can be described using kinetic theory (e.g., see Reference [20]). An important consequence is that in the large-inertia limit, the particle velocity distribution cannot be assumed to correspond to a spatially continuous velocity field. The random nature of the particle motion will lead to a crossing of individual trajectories, inducing a collision in physical systems. The above small- and large-inertia limit cases illustrate that substantially different spatial features in the particle velocity are possible with these differences depending on particle inertia. In the tracer limit, the spatial correlation function between any two particle velocities should be accurately modelled by a decaying exponential, analogous to the correlation describing the fluid turbulence. On the other hand, in the limit of very inertial particles, motion becomes stochastically equivalent to a Brownian motion with independent random velocities, even for those particles passing close to each other, also called random uncorrelated motion [19]. For intermediate values of the particle response time, the particle velocity distribution should be expected to exhibit spatial features with characteristics representative of these two limiting regimes.