Paul Trunfio

Complex dynamics in initially separated reaction-diffusion systems

We review recent developments in the study of the diffusion reaction systems of the type A + B → C in which the reactants are initially separated. We consider the case where the A and B particles are initially placed uniformly in Euclidean space at x > 0 and x < 0, respectively. We find that whereas for d ⩾ 2 the mean field exponent characterizes the width of the reaction zone, fluctuations are relevant in the one-dimensional system. We present analytical and numerical results for the reaction rate on fractals and percolation systems at criticality.We also study the case where the particles are Levy flights in d = 1. Finally, we consider experimentally, analytically, and numerically the reaction A + Bstatic → C, where species A diffuses from a localized source.

Diffusion-controlled reaction, A+B→C, with initially separated reactants

We review recent developments in the study of diffusion-reaction systems of the type A+B→C in which reactants are initially separated. We consider two initial boundary conditions: (i) the A and B particles are initially placed uniformly in Euclidean space at x>O and x<0 respectively, and (ii) the A particles are diffusing and inserted at a given site and the B particles are static and distributed uniformly in space. We present analytical and numerical results for both systems. We consider d = 1, 2, 3 dimensional systems as well as fractal lattices.

NetSci High: Bringing Network Science Research to High Schools

We present NetSci High, our NSF-funded educational outreach program that connects high school students who are underrepresented in STEM (Science Technology Engineering and Mathematics), and their teachers, with regional university research labs and provides them with the opportunity to work with researchers and graduate students on team-based, year-long network science research projects, culminating in a formal presentation at a network science conference. This short paper reports the content and materials that we have developed to date, including lesson plans and tools for introducing high school students and teachers to network science; empirical evaluation data on the effect of participation on students’ motivation and interest in pursuing STEM careers; the application of professional development materials for teachers that are intended to encourage them to use network science concepts in their lesson plans and curriculum; promoting district-level interest and engagement; best practices gained from our experiences; and the future goals for this project and its subsequent outgrowth.

Learning Fractals by “Doing Science”: Applying Cognitive Apprenticeship Strategies to Curriculum Design and Instruction

Science research professionals originated an education innovation project that adapts the mentoring model of graduate study in science to the high school, closely coupling experiment and computer visualization models devised originally for science research in natural or random fractals. Educational researchers who joined the project helped to interpret the effort in terms of “cognitive apprenticeship,” a teaching paradigm already known in the cognitive research literature. This article traces the evolution of the materials informed by this paradigm and the results of two sequential trials of a fractal dimension unit in a suburban high school. In the 2nd‐year trial, students began to act as independent investigators and the teacher gradually and spontaneously adopted the role of mentor.

Some recent variations on the expected number of distinct sites visited by an n-step random walk

Asymptotic forms for the expected number of distinct sites visited by an n-step random walk, being calculable for many random walks, have been used in a number of analyses of physical models. We describe three recent extensions of the problem, the first replacing the single random walker by N→∞ random walkers, the second to the study of a random walk in the presence of a trapping site, and the third to a random walk in the presence of a trapping hyperplane.

Anomalous dynamics in reaction-diffusion systems

SummaryWe review recent developments in the study of the diffusion reaction systems of the type A+B→C in which the reactants are initially separated. We consider the case where the A and B particles are initially placed uniformly in Euclidean space atx>0 andx<0, respectively. We find that whereas ford≥2 the mean-field exponent characterizes the width of the reaction zone, fluctuations are relevant in the one-dimensional system. We also presented analytical and numerical results for the reaction rate on fractals and percolation systems.

Learning science through guided discovery: liquid water and molecular networks

In every drop of water, down at the scale of atoms and molecules, there is a world that can fascinate anyone. The objective of “Learning science through guided discovery: liquid water and molecular networks” is to use advanced technology to provide a window into the submicroscopic, and thereby allow students to discover by themselves an entire new world. We are developing a coordinated two-fold approach to high school science teaching in which a cycle of hands-on activities, games, and experimentation is followed by a cycle of computer simulations employing the full power of computer animation to “ZOOM” into the depths of this newly discovered world. Pairing of laboratory experiments with corresponding simulations challenges students to understand multiple representations of concepts. We thereby provide students with the opportunity to work in a fashion analogous to that in which practicing scientists work - e.g., by “building up” to general principles from specific experiences. Moreover, the ability to visualize “real-time” dynamic motions allows for student-controlled graphic simulations on the molecular scale, and interactive guided lessons superior to those afforded by even the most artful of texts. While our general approach could be applied to a variety of topics, we have chosen to focus first on the most familiar of molecular networks, that of liquid water. Later we will test the generality of the approach by exploring macromolecules such as proteins and DNA.

Self-Organized Criticality: The Origin of Fractal Growth

We show that aggregation processes naturally evolve into self-organized critical states. The associated critical exponents provide a new characterization of fractal growth. We consider diffusion-limited aggregation (DLA) and compare our description with that based on the f(α) spectrum. We find that a critical value α c of a exists, above which the spectrum fails to characterize the growth, and below which only a spiky part of the aggregate is described. For DLA, α c = 1.

SPREADING OF N DIFFUSING SPECIES WITH DEATH AND BIRTH FEATURES

The number of distinct sites visited by a random walker after t steps is of great interest, as it provides a direct measure of the territory covered by a diffusing particle. We review the analytical solution to the problem of calculating SN(t), the mean number of distinct sites visited by N random walkers on a d-dimensional lattice, for d=1, 2, 3 in the limit of large N. There are three distinct time regimes for SN(t). A remarkable transition, for dimension ≥2, in the geometry of the set of visited sites is found. This set initially grows as a disk with a relatively smooth surface until it reaches a certain size, after which the surface becomes increasingly rough. We also review the results for a model for migration and spreading of populations and diseases. The model is based on N diffusing species, where each species has a probability α- of dying (or recovery from a disease) and a probability α+ to give birth (or to infect another species). It is found analytically that when α+ ≈ α- ≠ 0, after a crossover time t× ~ N/2α-, the territory covered by the population is localized around its center of mass while the center of mass diffuses regularly. When α+ > α-, the localization breaks down after a second crossover time and the species diffuse and spread around their center of mass. These results may explain the phenomena of migration and spreading of diseases and population appearing in nature.

Diffusion Reaction A + B → C, With A & B Initially-Separated

The dynamics of diffusion controlled reactions of the type A + B → C has been studied extensively since the pioneering work of Smoluchowski [1,2]. Most studies have focused on homogeneous systems, i.e., when both reactants are initially uniformly mixed in a d-dimensional space, and interesting theoretical results have been obtained. When the concentrations of the A and B reactants are initially equal, i.e., c A (0) = c B (0) = c(0), the concentration of both species is found to decay with time as, c(t) ~ t -d/4 for Euclidean d ≤ 4-dimensional systems [3–10] and as for fractals [5,6] with fracton dimension d s ≤ 2. Also, self-segregated regions of A and B in low dimensions (d ≤ 3) [4] and in fractals [9] have been found. Quantities such as the distributions of domain sizes of segregated regions and interparticle distances between species of the same type and different types have been calculated [11–13]. These systems were also studied theoretically and numerically under steady state conditions and interesting predictions have been obtained [14–17]. However, the above numerical and theoretical predictions have not been observed in experiments, in part because of difficulties to implement the initially uniformly-mixed distributions of reactants.