A Three-Fold Approach to the Heat Equation: Data, Modeling, Numerics

This article describes our modeling approach to teaching the one-dimensional heat (diffusion) equation in a one-semester undergraduate partial differential equations course. We constructed the apparatus for a demonstration of heat diffusion through a long, thin metal rod with prescribed temperatures at each end. The students observed the physical phenomenon, collected temperature data along the rod, then referenced the demonstration for purposes in and out of the classroom. Here, we discuss the experimental setup, how the demonstration informed practices in the classroom and a project based on the collected data, including analytical and computational components

Local origins of volume fraction fluctuations in dense granular materials

Fluctuations of the local volume fraction within granular materials have previously been observed to decrease as the system approaches jamming. We experimentally examine the role of boundary conditions and inter-particle friction $\mu$ on this relationship for a dense granular material of bidisperse particles driven under either constant volume or constant pressure. Using a radical Vorono\"i tessellation, we find the variance of the local volume fraction $\phi$ monotonically decreases as the system becomes more dense, independent of boundary condition and $\mu$. We examine the universality and origins of this trend using experiments and the recent granocentric model \cite{Clusel-2009-GMR,Corwin-2010-MRP}, modified to draw particle locations from an arbitrary distribution ${\cal P}(s)$ of neighbor distances $s$. The mean and variance of the observed ${\cal P}(s)$ are described by a single length scale controlled by $\bar \phi$. Through the granocentric model, we observe that diverse functional forms of ${\cal P}(s)$ all produce the trend of decreasing fluctuations, but only the experimentally-observed ${\cal P}(s)$ provides quantitative agreement with the measured $\phi$ fluctuations. Thus, we find that both ${\cal P}(s)$ and ${\cal P}(\phi)$ encode similar information about the ensemble of observed packings, and are connected to each other by the local granocentric model

Time-Frequency Analysis Reveals Pairwise Interactions in Insect Swarms

The macroscopic emergent behavior of social animal groups is a classic example of dynamical self-organization, and is thought to arise from the local interactions between individuals. Determining these interactions from empirical data sets of real animal groups, however, is challenging. Using multicamera imaging and tracking, we studied the motion of individual flying midges in laboratory mating swarms. By performing a time-frequency analysis of the midge trajectories, we show that the midge behavior can be segmented into two distinct modes: one that is independent and composed of low-frequency maneuvers, and one that consists of higher-frequency nearly harmonic oscillations conducted in synchrony with another midge. We characterize these pairwise interactions, and make a hypothesis as to their biological function

Statistical theory of correlations in random packings of hard particles

A random packing of hard particles represents a fundamental model for granular matter. Despite its importance, analytical modeling of random packings remains difficult due to the existence of strong correlations which preclude the development of a simple theory. Here, we take inspiration from liquid theories for the $n$-particle angular correlation function to develop a formalism of random packings of hard particles from the bottom-up. A progressive expansion into a shell of particles converges in the large layer limit under a Kirkwood-like approximation of higher-order correlations. We apply the formalism to hard disks and predict the density of two-dimensional random close packing (RCP), $\phi_{\rm rcp} = 0.85\pm0.01$, and random loose packing (RLP), $\phi_{\rm rlp} = 0.67\pm0.01$. Our theory also predicts a phase diagram and angular correlation functions that are in good agreement with experimental and numerical data.Comment: 9 pages, 6 figures, to appear in PR

Collective Gradient Sensing in Fish Schools

Throughout the animal kingdom, animals frequently benefit from living in groups. Models of collective behaviour show that simple local interactions are sufficient to generate group morphologies found in nature (swarms, flocks and mills). However, individuals also interact with the complex noisy environment in which they live. In this work, we experimentally investigate the group performance in navigating a noisy light gradient of two unrelated freshwater species: golden shiners (Notemigonuscrysoleucas) and rummy nose tetra (Hemigrammus bleheri). We find that tetras outperform shiners due to their innate individual ability to sense the environmental gradient. Using numerical simulations, we examine how group performance depends on the relative weight of social and environmental information. Our results highlight the importance of balancing of social and environmental information to promote optimal group morphologies and performance

Searching for Effective Forces in Laboratory Insect Swarms

Collective animal behaviour is often modeled by systems of agents that interact via effective social forces, including short-range repulsion and long-range attraction. We search for evidence of such effective forces by studying laboratory swarms of the flying midge Chironomus riparius. Using multi-camera stereoimaging and particle-tracking techniques, we record three-dimensional trajectories for all the individuals in the swarm. Acceleration measurements show a clear short-range repulsion, which we confirm by considering the spatial statistics of the midges, but no conclusive long-range interactions. Measurements of the mean free path of the insects also suggest that individuals are on average very weakly coupled, but that they are also tightly bound to the swarm itself. Our results therefore suggest that some attractive interaction maintains cohesion of the swarms, but that this interaction is not as simple as an attraction to nearest neighbours

Evolution of Network Architecture in a Granular Material Under Compression

As a granular material is compressed, the particles and forces within the system arrange to form complex and heterogeneous collective structures. Force chains are a prime example of such structures, and are thought to constrain bulk properties such as mechanical stability and acoustic transmission. However, capturing and characterizing the evolving nature of the intrinsic inhomogeneity and mesoscale architecture of granular systems can be challenging. A growing body of work has shown that graph theoretic approaches may provide a useful foundation for tackling these problems. Here, we extend the current approaches by utilizing multilayer networks as a framework for directly quantifying the progression of mesoscale architecture in a compressed granular system. We examine a quasi-two-dimensional aggregate of photoelastic disks, subject to biaxial compressions through a series of small, quasistatic steps. Treating particles as network nodes and interparticle forces as network edges, we construct a multilayer network for the system by linking together the series of static force networks that exist at each strain step. We then extract the inherent mesoscale structure from the system by using a generalization of community detection methods to multilayer networks, and we define quantitative measures to characterize the changes in this structure throughout the compression process. We separately consider the network of normal and tangential forces, and find that they display a different progression throughout compression. To test the sensitivity of the network model to particle properties, we examine whether the method can distinguish a subsystem of low-friction particles within a bath of higher-friction particles. We find that this can be achieved by considering the network of tangential forces, and that the community structure is better able to separate the subsystem than a purely local measure of interparticle forces alone. The results discussed throughout this study suggest that these network science techniques may provide a direct way to compare and classify data from systems under different external conditions or with different physical makeup

Intrinsic Fluctuations and Driven Response of Insect Swarms

Animals of all sizes form groups, as acting together can convey advantages over acting alone; thus, collective animal behavior has been identified as a promising template for designing engineered systems. However, models and observations have focused predominantly on characterizing the overall group morphology, and often focus on highly ordered groups such as bird flocks. We instead study a disorganized aggregation (an insect mating swarm), and compare its natural fluctuations with the group-level response to an external stimulus. We quantify the swarm’s frequency-dependent linear response and its spectrum of intrinsic fluctuations, and show that the ratio of these two quantities has a simple scaling with frequency. Our results provide a new way of comparing models of collective behavior with experimental data

Long-Range Acoustic Interactions in Insect Swarms: An Adaptive Gravity Model

The collective motion of groups of animals emerges from the net effect of the interactions between individual members of the group. In many cases, such as birds, fish, or ungulates, these interactions are mediated by sensory stimuli that predominantly arise from nearby neighbors. But not all stimuli in animal groups are short range. Here, we consider mating swarms of midges, which are thought to interact primarily via long-range acoustic stimuli. We exploit the similarity in form between the decay of acoustic and gravitational sources to build a model for swarm behavior. By accounting for the adaptive nature of the midges\u27 acoustic sensing, we show that our \u27adaptive gravity\u27 model makes mean-field predictions that agree well with experimental observations of laboratory swarms. Our results highlight the role of sensory mechanisms and interaction range in collective animal behavior. Additionally, the adaptive interactions that we present here open a new class of equations of motion, which may appear in other biological contexts

Long-range Acoustic Interactions in Insect Swarms: An Adaptive Gravity Model

The collective motion of groups of animals emerges from the net effect of the interactions between individual members of the group. In many cases, such as birds, fish, or ungulates, these interactions are mediated by sensory stimuli that predominantly arise from nearby neighbors. But not all stimuli in animal groups are short range. Here, we consider mating swarms of midges, which interact primarily via long-range acoustic stimuli. We exploit the similarity in form between the decay of acoustic and gravitational sources to build a model for swarm behavior. By accounting for the adaptive nature of the midges' acoustic sensing, we show that our "adaptive gravity" model makes mean-field predictions that agree well with experimental observations of laboratory swarms. Our results highlight the role of sensory mechanisms and interaction range in collective animal behavior. The adaptive interactions that we present here open a new class of equations of motion, which may appear in other biological contexts.Comment: 25 pages, 15 figure