Methods to extract information from the tracking of mobile objects/particles have broad interest in biological and physical sciences. Techniques based on simple criteria of proximity in timeconsecutive snapshots are useful to identify the trajectories of the particles. However, they become problematic as the motility and/or the density of the particles increases due to uncertainties on the trajectories that particles followed during the images' acquisition time. Here, we report an efficient method for learning parameters of the dynamics of the particles from their positions in time-consecutive images. Our algorithm belongs to the class of message-passing algorithms, known in computer science, information theory, and statistical physics as belief propagation (BP). The algorithm is distributed, thus allowing parallel implementation suitable for computations on multiple machines without significant intermachine overhead. We test our method on the model example of particle tracking in turbulent flows, which is particularly challenging due to the strong transport that those flows produce. Our numerical experiments show that the BP algorithm compares in quality with exact Markov Chain Monte Carlo algorithms, yet BP is far superior in speed. We also suggest and analyze a random distance model that provides theoretical justification for BP accuracy. Methods developed here systematically formulate the problem of particle tracking and provide fast and reliable tools for the model's extensive range of applications.belief propagation | message passing | statistical inference | turbulence | particle image velocimetry T racking of mobile objects is widespread in the natural sciences, with numerous applications both for living and inert "particles." Trajectories of the particles are to be obtained from successive images, acquired sequentially in time at a suitable rate. Examples of living "particles" include birds in flocks (1) and motile cells (2). Among inert objects, nanoparticles (3) and particles advected by turbulent fluid flow (4-6) provide two important examples. The general goal of tracking particles is to extract clues about their dynamics and to make inferences about the laws of motion and/or unknown modeling parameters.Ideal cases for tracking are those where the density and the mobility of particles is low and the acquisition rate of images is high. The nondimensional parameter governing the stiffness of the problem is the ratio Λ ¼ ℓρ 1∕d of the typical distance ℓ traveled by the particles during the time between images and the average interparticle distance 1∕ρ 1∕d . Here, ρ is the number density of particles and d is the space dimensionality. Tracking is rather straightforward if Λ is small: the positions of each particle in two successive images will be relatively far from those of all other particles. Trajectories are thus defined without ambiguity. Such a situation is encountered for instances of the tracking of nanoparticles (7). More generally, effective methods are available to identify the assignment (de...

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We propose a general framework for solving inverse self-assembly problems, i.e. designing interactions between elementary units such that they assemble spontaneously into a predetermined structure. Our approach uses patchy particles as building blocks, where the different units bind at specific interaction sites (the patches), and we exploit the possibility of having mixtures with several components. The interaction rules between the patches is determined by transforming the combinatorial problem into a Boolean satisfiability problem (SAT) which searches for solutions where all bonds are formed in the target structure. Additional conditions, such as the non-satisfiability of competing structures (e.g. metastable states) can be imposed, allowing to effectively design the assembly path in order to avoid kinetic traps. We demonstrate this approach by designing and numerically simulating a cubic diamond structure from four particle species that assembles without competition from other polymorphs, including the hexagonal structure.