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We review the observations and the basic laws describing the essential aspects of collective motion -- being one of the most common and spectacular manifestation of coordinated behavior. Our aim is to provide a balanced discussion of the various facets of this highly multidisciplinary field, including experiments, mathematical methods and models for simulations, so that readers with a variety of background could get both the basics and a broader, more detailed picture of the field. The observations we report on include systems consisting of units ranging from macromolecules through metallic rods and robots to groups of animals and people. Some emphasis is put on models that are simple and realistic enough to reproduce the numerous related observations and are useful for developing concepts for a better understanding of the complexity of systems consisting of many simultaneously moving entities. As such, these models allow the establishing of a few fundamental principles of flocking. In particular, it is demonstrated, that in spite of considerable differences, a number of deep analogies exist between equilibrium statistical physics systems and those made of self-propelled (in most cases living) units. In both cases only a few well defined macroscopic/collective states occur and the transitions between these states follow a similar scenario, involving discontinuity and algebraic divergences.
We study self-organisation of collective motion as a thermodynamic phenomenon, in the context of the first law of thermodynamics. It is expected that the coherent ordered motion typically self-organises in the presence of changes in the (generalised)
We propose a conservative two-dimensional particle model in which particles carry a continuous and classical spin. The model includes standard ferromagnetic interactions between spins of two different particles, and a nonstandard coupling between spi
The simplest prescription for building a patterned structure from its constituents is to add particles, one at a time, to an appropriate template. However, self-organizing molecular and colloidal systems in nature can evolve in much more hierarchical
We propose a dynamic model for a system consisting of self-propelled agents in which the influence of an agent on another agent is weighted by geographical distance. A parameter $alpha$ is introduced to adjust the influence: the smaller value of $alp
Using analytic and numerical methods, we study a $2d$ Hamiltonian model of interacting particles carrying ferro-magnetically coupled continuous spins which are also locally coupled to their own velocities. This model has been characterised at the mea