We develop a translational-rotational cage model that describes the behavior of dense two dimensional (2D) Brownian systems of hard annular sector particles (ASPs), resembling "C"-shapes. At high particle densities, pairs of ASPs can form mutually interdigitating lock-and-key dimers. This cage model considers either one or two mobile central ASPs which can translate and rotate within a static cage of surrounding ASPs that mimics the system's average local structure and density. By comparing with recent measurements made on dispersions of microscale lithographic ASPs (P.-Y. Wang and T.G. Mason, J. Am. Chem. Soc. 137 15308 (2015)), we show that mobile two-particle predictions of the probability of dimerization P dimer , equilibrium constant K, and 2D osmotic pressure Π2D, as a function of the particle area fraction φA, correspond closely to these experiments. By contrast, predictions based on only a single mobile particle do not agree well with either the two-particle predictions or the experimental data. Thus, we show that collective entropy can play an essential role in the behavior of dense Brownian systems composed of non-trivial hard shapes, such as ASPs.
We study the energy dynamics of a particle in a billiard subject to a rapid periodic drive. In the regime of large driving frequencies ω, we find that the particle's energy evolves diffusively, which suggests that the particle's energy distribution η(E, t) satisfies a Fokker-Planck equation. We calculate the rates of energy absorption and diffusion associated with this equation, finding that these rates are proportional to ω −2 for large ω. Our analysis suggests three phases of energy evolution: Prethermalization on short timescales, then slow energy absorption in accordance with the Fokker-Planck equation, and finally a breakdown of the rapid driving assumption for large energies and high particle speeds. We also present numerical simulations of the evolution of a rapidly driven billiard particle, which corroborate our theoretical results.
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