The University of St. Thomas


Complex Systems

The interdisciplinary field of Complex Systems studies phenomena which involve many independently interacting individuals. Some examples include the economy, traffic, and flocks of animals. If we treat each individual in a complex system as a molecule, the group may be treated as a peculiar state of matter - indeed, complex systems often demonstrate phenomena such as phase transitions. In this way we may use techniques developed in statistical and condensed matter physics to address systems which are not physical in the usual sense.

Bird Flocking

In particular, we have studied theoretical models of bird flocking. Starling flocks - known as murmurations - have led to large scale empirical studies and even inspired viral videos online. Even though there may be many complicated factors determining a bird's real flight path in a flock, it is often instructive to study a simplified model which only includes a few salient aspects. Such a model may be feasible to study through computer simulation, and it clarifies which features of the individual behavior affect the group behavior. As such, we have focused on simulating variations of an abstract flocking model known as the Vicsek model.

In the Vicsek model, birds update their velocities after discrete steps in time. At each time step, the birds match their direction to the average velocity of their neighbors. This matching interaction is not exact - it is perturbed by a random noise term which may represent more complicated interactions such as air turbulence or a bird's independent whims. As this noise level is decreased, the system undergoes a phase transition between disordered individual flight and an ordered flocking phase. Previous research has led to the measurement of quantities known as critical exponents which describe the large scale features of the phase transition.

Importantly for our research, a bird's neighbors in the Vicsek model are defined to be all birds within a fixed perceptual distance. As the flock becomes more dense, a bird has more neighbors to interact with and so the flock becomes more ordered. However, in the empirical research on starlings, flocks with different densities behaved similarly and starlings appeared to interact with only their closest neighbors regardless of the distance between them. Inspired by this research, we developed a new model in which birds have flexible interaction ranges varying with the local density.

In this model, a bird interacts with those neighbors defined through a natural geometric concept known as Delaunay triangulation. To study this system feasibly, new algorithms were developed to update a bird's nearest neighbors over time. To get statistically significant results, a large number of simulations were carried out on the Physics department's 16-processor computer cluster.

As in the Vicsek model, the new model exhibits a phase transition from disordered motion to ordered flocking behavior. However, measurements of the critical exponents indicate that the physics of the phase transition is distinct from earlier models. We have also modified the model to include more realistic interactions such as a repulsion term that prevents birds from colliding. However the large scale flocking behavior was found to be quantitavely unchanged: a surprising phenomenon often seen in complex systems known as universality.

The starling research had also suggested that the distance over which the birds' velocities are correlated grows with the size of the flock. Certain features of the group behavior appeared to be independent of the metric size or number of birds in a flock. In a certain sense, the behavior of a flock of 100,000 birds may be similar to that of a flock of 1000 birds. This phenomena of scale invariance has also been observed in simulations of the theoretical model, and has been depicted graphically for flock sizes which are impossible to observe in the real world.

A more technical discussion of these results may be found here. For more information please contact Source code available upon request.


We have involved students in several meteorological projects at St. Thomas. These include investigations into the Lorenz and the Rayleigh-Bénard models of the atmosphere. These models are broad simplifications of the full Navier-Stokes equations that describe the atmosphere, yet hopefully retain some of its characteristic behaviors. Hence by studying these models we hope to glean, in a manageable way, an intuition about how the atmosphere works. As a complement, we also have investigated a more complete version of the atmospheric equations using the Advanced Regional Prediction System (ARPS), a very involved computer program developed at the University of Oklahoma. Using this simulation, we are able to take real-time atmospheric data (called a "sounding") and predict whether or not a storm will develop. Comparing our results to ground truth (i.e. does a storm actually occur?) allows us to better understand the degree to which the atmosphere is well characterized by the simulation as well as the sensitivity to its input values (i.e. the sounding).

For variety (and fun), we also constructed a tornado vortex chamber (TVC) - a non-computational project - in order to visually represent tornadoes.

Our work in this area led to the following journal article:
Knox, J.A., and Ohmann, P.R. (2006). Iterative solutions of the gradient wind equation. Computers and Geosciences 32, 656-662.

Soil Science

Acid rain is an environmental problem that has garnered international attention for the past 30 years.  One problem it causes is depletion of calcium in forest systems, leading to a decrease in soil pH levels. This has resulted in great losses of vegetation in some areas. However, not all regions have been strongly affected by calcium depletion; some have appeared to find alternative sources of calcium to counter these losses. One hypothesis is that calcium may be replenished through diffusion from underground sources. Our research shows a consistency between the concentration of calcium in the soil at Walker Branch Watershed (WBW) in Tennessee and the replenishment of lost calcium via diffusion from the underlying bedrock. We constructed a computational model of calcium diffusion to model the WBW system; results include an estimate of the upward calcium transport of 6.5 kg/ha/yr at the soil surface through diffusion from bedrock 20 meters underneath. This may be sufficient to make up for the calcium leached away by acid rain.

Our research work was published in the following article:
Grigal, D.F., and Ohmann, P.R. (2005). Calcium and Forest Systems: Diffusion from Deep Sources. Soil Science 170 (2), 129-136.


In the spirit of scientific inquiry, we explore other topics of interest as opportunities arise. For example, this has included modeling equilibrium charge distributions on conductors, which we have turned into a computational project in Phys 341: Electricity and Magnetism. We also play a supportive role in other departmental investigations, turning our collaboration into a particularly rich experience for our students.

Applets and Videos

Video: Starling Murmurations

Applet: Vicsek Model

Applet: Density-Independent Model

Video: Neighborhood Construction

Scale Invariance

Computation Posters

Chaotic Flow in the Lorenz Model; Luke EdholmChaotic Flow in the Lorenz Model; Luke Edholm

Charge Distribution on 1- and 2- Dimensional Surfaces; Lauren Edge

Charge Distribution on 1- and 2- Dimensional Surfaces; Lauren Edge

Calcium Diffusion in Ecological Systems, Molly Andreason

Calcium Diffusion in Ecological Systems, Molly Andreason

Mathematical Models of Nerves: Translating Biochemistyr to Physics and Mathematics, Rebecca Lucast

Mathematical Models of Nerves: Translating Biochemistyr to Physics and Mathematics, Rebecca Lucast

Stormy Skies...Or Not; Katy Micek

Stormy Skies...Or Not; Katy Micek