A question of scale

If you search for 'collective behaviour' with your web browser most of the texts popping up will be about group activities of humans, including riots, fashion and mass panic. Nevertheless, collective behaviour is also considered to be an important aspect of observed phenomena in atoms and molecules, for example, during spontaneous magnetization. In your web search, you might also find articles on collectively migrating bacteria, insects or birds; or phenomena where groups of organisms or non- living objects synchronize their signals or motion (think of fireflies flashing in unison or people clapping in phase during rhythmic applause).Comment: Concepts essay, published in Nature http://www.nature.com/nature/journal/v411/n6836/full/411421a0.htm

Synchronization of oscillators with long range interaction: phase transition and anomalous finite size effects

Synchronization in a lattice of a finite population of phase oscillators with algebraically decaying, non-normalized coupling is studied by numerical simulations. A critical level of decay is found, below which full locking takes place if the population contains a sufficiently large number of elements. For large number of oscillators and small coupling constant, numerical simulations and analytical arguments indicate that a phase transition separating synchronization from incoherence appears at a decay exponent value equal to the number of dimensions of the lattice. In contrast with earlier results on similar systems with normalized coupling, we have indication that for the decay exponent less than the dimensions of the lattice and for large populations, synchronization is possible even if the coupling is arbitarily weak. This finding suggests that in organisms interacting through slowly decaying signals like light or sound, collective oscillations can always be established if the population is sufficiently large.Comment: 15 pages, 12 figures, submitted to Phys. Rev. E; Text slightly changed; References added; Fig. 9 update

Aggregation of magnetic holes in a rotating magnetic field

We have experimentally investigated field induced aggregation of nonmagnetic particles confined in a magnetic fluid layer when rotating magnetic fields were applied. After application of a magnetic field rotating in the plane of the fluid layer, the single particles start to form two-dimensional (2D) clusters, like doublets, triangels, and more complex structures. These clusters aggregated again and again to form bigger clusters. During this nonequilibrium process, a broad range of cluster sizes was formed, and the scaling exponents, $z$ and $z'$, of the number of clusters $N(t)\sim t^{z'}$and average cluster size $S(t)\sim t^{z}$ were calculated. The process could be characterized as diffusion limited cluster-cluster aggregation. We have found that all sizes of clusters that occured during an experiment, fall on a single curve as the dynamic scaling theory predicts. Hovewer, the characteristic scaling exponents $z',\: z$ and crossover exponents $\Delta$ were not universal. A particle tracking method was used to find the dependence of the diffusion coefficients $D_{s}$ on cluster size $s$. The cluster motions show features of \textit{\emph{Brownian}} motion. The average diffusion coefficients $$ depend on the cluster sizes $s$ as a power law $\propto s^{\gamma}$ where values of $\gamma$ as different as $\gamma=-0.62\pm0.19$ and $\gamma=-2.08\pm0. were found in two of the experiments

Multifractal Network Generator

We introduce a new approach to constructing networks with realistic features. Our method, in spite of its conceptual simplicity (it has only two parameters) is capable of generating a wide variety of network types with prescribed statistical properties, e.g., with degree- or clustering coefficient distributions of various, very different forms. In turn, these graphs can be used to test hypotheses, or, as models of actual data. The method is based on a mapping between suitably chosen singular measures defined on the unit square and sparse infinite networks. Such a mapping has the great potential of allowing for graph theoretical results for a variety of network topologies. The main idea of our approach is to go to the infinite limit of the singular measure and the size of the corresponding graph simultaneously. A very unique feature of this construction is that the complexity of the generated network is increasing with the size. We present analytic expressions derived from the parameters of the -- to be iterated-- initial generating measure for such major characteristics of graphs as their degree, clustering coefficient and assortativity coefficient distributions. The optimal parameters of the generating measure are determined from a simple simulated annealing process. Thus, the present work provides a tool for researchers from a variety of fields (such as biology, computer science, biology, or complex systems) enabling them to create a versatile model of their network data.Comment: Preprint. Final version appeared in PNAS

Staggered and extreme localization of electron states in fractal space

We present exact analytical results revealing the existence of a countable infinity of unusual single particle states, which are localized with a multitude of localization lengths in a Vicsek fractal network with diamond shaped loops as the 'unit cells'. The family of localized states form clusters of increasing size, much in the sense of Aharonov-Bohm cages [J. Vidal et al., Phys. Rev. Lett. 81, 5888 (1998)], but now without a magnetic field. The length scale at which the localization effect for each of these states sets in can be uniquely predicted following a well defined prescription developed within the framework of real space renormalization group. The scheme allows an exact evaluation of the energy eigenvalue for every such state which is ensured to remain in the spectrum of the system even in the thermodynamic limit. In addition, we discuss the existence of a perfectly conducting state at the band center of this geometry and the influence of a uniform magnetic field threading each elementary plaquette of the lattice on its spectral properties. Of particular interest is the case of extreme localization of single particle states when the magnetic flux equals half the fundamental flux quantum.Comment: 9 pages, 8 figure