List of examples

Here are several entries from an ever-growing list of examples, which we will add to the website over time:

• Smarticles Robot Swarm A swarm of simple robots that we could control and predict using the Rattling theory. This was the original example that we used to develop the Rattling framework, and was implemented in labs in GA Tech and Northwestern – see our Science paper
• Models of animal collective behavior. Rattling predicts the steady-state distributions of many toy models of animal collective behavior for relevant ranges of model parameters. Examples include the Föllmer–Kirman model of binary decision-making by ant colonies and the Beekman–Ratnieks–Sumpter model of trail formation. Additional examples include the Jhawar et al. model of noise-induced schooling in fish and the Amé et al. model of shelter selection by cockroaches. There are many more!
  • Vicsek model we name separately because while it was originally developed as as model of bird flocking, it has since become a paradigmatic model of nonequilibrium self-organization and active spins. Here we showed that Rattling theory can analytically predict the flocking phase transition from a first-principles calculation (up to 2 fitting parameters).
• Models of ecological succession. Ecological succession entails changes in the composition of an ecosystem over time. Rattling can also predict the steady-state distribution of these compositions. Examples include succession in communities of microbes, mites, and subtidal and forest ecologies.
• Various random walk models. Rattling can be studied in large classes of random Markov chains by putting various distributions (e.g., power law) on the transition rates, with various underlying connectivities (e.g., densely connected or sparsely connected).
• Diffusion models: particle diffusing in a landscape of a random force-field or random spatially-varying diffusivity. Depending on the variability of the random fields, their correlation structure, dimensionality, we can get different behaviors and study where and how exactly Rattling hypothesis holds or breaks down.
• Equilibrium systems. While Rattling it a theory of nonequilibrium self-organization, we verified that it maintains its predictive power in some classic equilibrium models as well – and may sometimes be easier to use than the Boltzmann distribution. For example, we considered the random energy model (REM) (see paper). More concretely, this works for the Sherrington–Kirkpatrick model of spin-glasses – where many “spins” that take values of $\pm 1$ are randomly connected with aligning or anti-aligning forces. One way to sample equilibrium configurations of such systems is to run a Markov chain whose stationary distribution is the appropriate Boltzmann distribution. Rattling predicts the stationary distribution of this Markov chain well, without needing to know the energies of the configurations. The same is true of dynamics of the Sherrington–Kirkpatrick model in various parameter regimes.
• Random chemical reaction system A model of a chemical reaction system in which a fixed number of species react with randomly sampled rate constants, with bounded maximum molecularity of the reactions. We saw that depending on the parameters, Rattling works better or worse to predict the steady state – in agreement with the intuition that the more randomness we allow, the better Rattling hypothesis should apply.
• Random Rattling Model: In parallel to the conception of the Random Energy Model (REM) in equilibrium, we argue that it might be plausible for a broad class of complex systems to approximate the Rattling values of its configurations as Gaussian random variable. This would then imply the possibility of nonequilibrium phase transition in all such systems, leading to self-organization. Besides being an interesting example, this also paves the way of importing a plethora of results from equilibrium statistical mechanics and thermodynamics into non-equilibrium world via Ratting.
• Dynamical systems with mixed chaos: ones that exhibit both regular and chaotic behavior in different corners of their configuration space at the same time. When coupled to a thermal bath (Gaussian noise and damping), their stationary distribution can be predicted by Rattling theory. Systems we studied here are Kicked rotatorr, Zaslavsky web map, Bouncing ball on oscillating plate, and a few systems we have not seen in the literature before, such as “Kicked Harmonic Net” or “Wiggle string” or “Kicked lattice.”
• Driven field theories: Take some network of simple harmonic oscillators, add a small non-linearity, and apply a periodic driving force to the whole thing. This is basically the equivalent of any weakly-interacting field theory, but now driven out of equilibrium by the applied force. This is a rich and powerful class of examples where we can study how Rattling may be calculated perturbatively for a weak non-linearity, perhaps leveraging ideas from Renormalization Group or other techniques from field theory. While not fully developed yet, some interesting mathematical results appear readily in this framework.