| Abstract: Many slowly evolving, deterministic systems with negligible noise—from soap foams to the cytoskeleton to the stock market—nevertheless display rapid random motions with non-Gaussian displacements, intermittent dynamics and power-law time correlations. Recent simulations have suggested that in a soap foam such motion is due to the system's configuration moving on a fractal energy landscape embedded in a high-dimensional configuration space via energy minimization. Here we show that such energy landscapes are not merely abstractions that can only be explored in a computer, but rather empirically accessible objects that can be explored and quantified using particle tracking data. In a foam-like dense emulsion, the motion of oil droplets measured using confocal particle tracking reveals the emergent fractal geometry of the energy landscape, confirming that this fractal geometry does explain the emulsion’s microscopic dynamics and mechanical properties. This new empirical approach allows a similar hypothesis to be tested in other systems; we apply it to the price motions of the component stocks of the SP500 index. Notably, analysis suggests that stock prices move between transient equilibria on a high-dimensional surface with emergent fractal characteristics similar to that of potential energy landscapes. In the second part of the talk, we describe using a novel computational approach to effectively explore high-dimensional fractal surfaces. Exploring the energy landscape for foams using a biased relaxation method termed metadynamics, we discover and descend deep and narrow meandering ‘canyons’ that contain dense clusters of minima along their floors. Similar canyon structures in the energy landscapes of two model glass formers—hard sphere fluids and the Kob-Andersen glass—allow us to reach high packing densities and low energies, respectively. In the hard sphere system, fluid configurations are found to form continuous thread-like domains that cover the canyon floors up to densities well above the jamming transition. For the Kob-Andersen glass former, our technique samples low energy states with modest computational effort, with the lowest energies found approaching the predicted Kauzmann limit. Remarkably, the fractal structure of the canyons in all three models are almost identical, perhaps providing an explanation for the similarity of glassy phenomena in systems with very different microscopic structures. |