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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.
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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) internal energy and of (generalised) work done on, or extracted from, the system. We aim to explicitly quantify changes in these two quantities in a system of simulated self-propelled particles, and contrast them with changes in the systems configuration entropy. In doing so, we adapt a thermodynamic formulation of the curvatures of the internal energy and the work, with respect to two parameters that control the particles alignment. This allows us to systematically investigate the behaviour of the system by varying the two control parameters to drive the system across a kinetic phase transition. Our results identify critical regimes and show that during the phase transition, where the configuration entropy of the system decreases, the rates of change of the work and of the internal energy also decrease, while their curvatures diverge. Importantly, the reduction of entropy achieved through expenditure of work is shown to peak at criticality. We relate this both to a thermodynamic efficiency and the significance of the increased order with respect to a computational path. Additionally, this study provides an information-geometric interpretation of the curvature of the internal energy as the difference between two curvatures: the curvature of the free entropy, captured by the Fisher information, and the curvature of the configuration entropy.
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 spin and velocity of the same particle inspired by the coupling observed in self-propelled hard discs. Because of this coupling Galilean invariance is broken and the conserved linear momentum associated to translation invariance is not proportional to the velocity of the center of mass. Also, the dynamics is not invariant under a global rotation of the spins alone. This, in principle, leaves room for collective motion and thus raises the question whether collective motion can arise in Hamiltonian systems. We study the statistical mechanics of such a system, and show that, in the fully connected (or mean-field) case, a transition to collective motion does exist in spite of momentum conservation. Interestingly, the velocity of the center of mass, which in the absence of Galilean invariance, is a relevant variable, also feeds back on the magnetization properties, as it acts as an external magnetic field that smoothens the transition. Molecular dynamics simulations of finite size systems indeed reveal a rich phase diagram, with a transition from a disordered to a homogeneous polar phase, but also more complex inhomogeneous phases with local order interrupted by topological defects.
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 ways. Specifically, constituents (or clusters of constituents) may aggregate to form clusters (or clusters of clusters) that serve as building blocks for later stages of assembly. Here we evaluate the character and consequences of such collective motion in a set of prototypical assembly processes. We do so using computer simulations in which a systems capacity for hierarchical dynamics can be controlled systematically. By explicitly allowing or suppressing collective motion, we quantify its effects. We find that coarsening within a two dimensional attractive lattice gas (and an analogous off-lattice model in three dimensions) is naturally dominated by collective motion over a broad range of temperatures and densities. Under such circumstances, cluster mobility inhibits the development of uniform coexisting phases, especially when macroscopic segregation is strongly favored by thermodynamics. By contrast, the assembly of model viral capsids is not frustrated but is instead facilitated by collective moves, which promote the orderly binding of intermediates consisting of several monomers.
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 $alpha$ means that the closer neighbors have stronger influence on the moving direction. We find that there exists an optimal value of $alpha$, leading to the highest degree of direction consensus. The value of optimal $alpha$ increases as the system size increases, while it decreases as the absolute velocity, the sensing radius and the noise amplitude increase.
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 mean field level in a parent paper. Here, we first obtain its finite size ground states, as a function of the spin-velocity coupling intensity and system size, with numerical techniques. These ground states, namely a collectively moving polar state of aligned spins, and two non moving states embedded with topological defects, are recovered from the analysis of the continuum limit theory and simple energetic arguments that allow us to predict their domains of existence in the space of control parameters. Next, the finite temperature regime is investigated numerically. In some specific range of the control parameters, the magnetisation presents a maximum at a finite temperature. This peculiar behaviour, akin to an order-by-disorder transition, is explained by the examination of the free energy of the system and the metastability of the states of minimal energy. The robustness of our results is checked against the geometry of the boundary conditions and the dimensionality of space.
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