Mean-field theory of collective motion due to velocity alignment

We introduce a system of self-propelled agents (active Brownian particles) with velocity alignment in two spatial dimensions and derive a mean-field theory from the microscopic dynamics via a nonlinear Fokker-Planck equation and a moment expansion of the probability distribution function. We analyze the stationary solutions corresponding to macroscopic collective motion with finite center of mass velocity (ordered state) and the disordered solution with no collective motion in the spatially homogeneous system. In particular, we discuss the impact of two different propulsion functions governing the individual dynamics. Our results predict a strong impact of the individual dynamics on the mean field onset of collective motion (continuous vs discontinuous). In addition to the macroscopic density and velocity field we consider explicitly the dynamics of an effective temperature of the agent system, representing a measure of velocity fluctuations around the mean velocity. We show that the temperature decreases strongly with increasing level of collective motion despite constant fluctuations on individual level, which suggests that extreme caution should be taken in deducing individual behavior, such as, state-dependent individual fluctuations from mean-field measurements [Yates {\em et al.}, PNAS, 106 (14), 2009].Comment: corrected version, Ecological Complexity (2011) in pres

Swarming and Pattern Formation due to Selective Attraction and Repulsion

We discuss the collective dynamics of self-propelled particles with selective attraction and repulsion interactions. Each particle, or individual, may respond differently to its neighbors depending on the sign of their relative velocity. Thus, it is able to distinguish approaching (coming closer) and moving away individuals. This differentiation of the social response is motivated by the response to looming visual stimuli and may be seen as a generalization of the previously proposed, biologically motivated, escape and pursuit interactions. The model can account for different types of behavior such as pure attraction, pure repulsion, or escape and pursuit depending on the values (signs) of the different response strengths, and provides, in the light of recent experimental results, an interesting alternative to previously proposed models of collective motion with an explicit velocity-alignment interaction. We show the onset of large scale collective motion in a subregion of the parameter space, which corresponds to an effective escape and/or pursuit response. Furthermore, we discuss the observed spatial patterns and show how kinetic description of the dynamics can be derived from the individual based model.Comment: Preprint, 24 pages, submitted to Interface Focu

Self-propelled particles with selective attraction-repulsion interaction - From microscopic dynamics to coarse-grained theories

In this work we derive and analyze coarse-grained descriptions of self-propelled particles with selective attraction-repulsion interaction, where individuals may respond differently to their neighbours depending on their relative state of motion (approach versus movement away). Based on the formulation of a nonlinear Fokker-Planck equation, we derive a kinetic description of the system dynamics in terms of equations for the Fourier modes of a one-particle density function. This approach allows effective numerical investigation of the stability of possible solutions of the system. The detailed analysis of the interaction integrals entering the equations demonstrates that divergences at small wavelengths can appear at arbitrary expansion orders. Further on, we also derive a hydrodynamic theory by performing a closure at the level of the second Fourier mode of the one-particle density function. We show that the general form of equations is in agreement with the theory formulated by Toner and Tu. Finally, we compare our analytical predictions on the stability of the disordered homogeneous solution with results of individual-based simulations. They show good agreement for sufficiently large densities and non-negligible short-ranged repulsion. Disagreements of numerical results and the hydrodynamic theory for weak short-ranged repulsion reveal the existence of a previously unknown phase of the model consisting of dense, nematically aligned filaments, which cannot be accounted for by the present Toner and Tu type theory of polar active matter.Comment: revised version, 37pages, 11 figure

Implementation of a Special Education Parent Advisory Committee: A Mixed Methods Investigation into the Members’ Experience of Parental Involvement with the School System

This research was intended as a mixed methods case study of the initial effectiveness of one school system’s Special Education Parent Advisory Committee (PAC). It was not until well into the study that it became clear the phenomenon at the root of the this research was actually the broader one of special education parental involvement in the schools, with the Rush County (a pseudonym) School System’s Department of Special Education as the case study. Although phenomenological inquiry is primary, the mixed methods research design employed included both thematic development and verification based on data obtained by both qualitative and quantitative means. Quantitative data were collected annually from 2002 through 2005, using a state- developed surve y instrument sent each spring to half of the families with children receiving special education services. The primary qualitative data were collected from nine individual interviews of PAC charter members. Observational notes, the researcher’s field log, and archival documents from the PAC were also examined. The main quantitative findings were that the parents of special education students in Rush County return consistently positive responses when asked yes/no type questions about their children’s educational programs. The only areas in which negative responses were more than 20 to 30% concern the parents’ own participation in school system activities. The quantitative finding that special education parental involvement in the school system is limited was also one of the qualitative findings. These are the four phenomenological themes developed: “It’s all about the kids” (the parent as primary advocate), “Our own little group” (parents’ focus on special education), “One person can’t get it done” (being helped or hindered by a range of others), and “Get them involved, and then we’ll make them care” (the range of parental involvement in the school system). These findings were verified using member checking, peer examination and debriefing, and commentary from a group of university instructors and graduate students who regularly read transcripts with the goal of understanding the essence of each experience described. The main outcome of analyzing these themes was the realization that in public education (particularly special education), as others decrease in proximity to the child, their impact on that child also decreases. The PAC has become more than an advisory committee for the special education director; it is a support and advocacy group for special education parents as well. The discussion of findings explored the possibility that information sharing (support) is taking a primary role because the PAC investigated is still in its early years. The discussion also pointed out that the support, advisory, and advocacy functions of the PAC were all written into its charter from the start. To relate the main result of this research to theory and practice in public education: the parents provide the most support, then the child’s teacher to a lesser degree. The parents’ view is that the school system and community have very little to do with the day-to-day help the child receives, other than keeping a structure in place for education to occur. Parental involvement is a spectrum and the school system has to have methods in place (especially during students’ transitions from one school to another) that allow parents to get involved to the levels with which they are comfortable. One way to do this is for school systems that do not already have special education PACs to organize them. A lesson learned from this study is that the PAC will need years to grow and become known and used in the school system and community. Although the move from school to work for special education students has no clear progression, this unfortunate finding can result in a positive outcome since it highlighted the need for public school systems to establish and use special education parent advisory committees as vehicles for home-school-community interaction. This research closes with a recommendation for a follow-up or longitudinal study of Rush County’s Special Education PAC as well as for research that would include teachers, school administrators, and the parents of other than school-aged people with disabilities. A related study that specifically correlates parental involvement with outcomes for families could also complement this research

Quasi-deterministic transport of Brownian particles in an oscillating periodic potential

We consider overdamped Brownian dynamics in a periodic potential with temporally oscillating amplitude. We analyze the transport which shows effective diffusion enhanced by the oscillations and derive approximate expressions for the diffusion coefficient. Furthermore we analyze the effect of the oscillating potential on the transport if additionally a constant force is applied. We show the existence of synchronization regimes at which the deterministic dynamics is in resonance with the potential oscillations giving rise to transport with extremely low dispersion. We distinguish slow and fast oscillatory driving and give analytical expressions for the mean velocity and effective diffusion.Comment: submitted: Feb 12th, 201

On the family of cubical multivariate cryptosystems based on the algebraic graph over finite commutative rings of characteristic 2

The family of algebraic graphs A(n;K) defined over the finite commutative ring K were used for the design of different multivariate cryptographical algorithms (private and public keys, key exchange protocols). The encryption map corresponds to a special walk on this graph. We expand the class of encryption maps via the use of an automorphism group of A(n;K). In the case of characteristic 2 the encryption transformation is a Boolean map. We change finite field for the commutative ring of characteristic 2 and consider some modifications of algorithm which allow to hide a ground commutative ring

Phase Transitions and Criticality in the Collective Behavior of Animals -- Self-organization and biological function

Collective behaviors exhibited by animal groups, such as fish schools, bird flocks, or insect swarms are fascinating examples of self-organization in biology. Concepts and methods from statistical physics have been used to argue theoretically about the potential consequences of collective effects in such living systems. In particular, it has been proposed that such collective systems should operate close to a phase transition, specifically a (pseudo-)critical point, in order to optimize their capability for collective computation. In this chapter, we will first review relevant phase transitions exhibited by animal collectives, pointing out the difficulties of applying concepts from statistical physics to biological systems. Then we will discuss the current state of research on the "criticality hypothesis", including methods for how to measure distance from criticality and specific functional consequences for animal groups operating near a phase transition. We will highlight the emerging view that de-emphasizes the optimality of being exactly at a critical point and instead explores the potential benefits of living systems being able to tune to an optimal distance from criticality. We will close by laying out future challenges for studying collective behavior at the interface of physics and biology.Comment: to appear in "Order, disorder, and criticality", vol. VII, World Scientific Publishin