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The study of molecular polaritons beyond simple quantum emitter ensemblemodels (e.g., Tavis-Cummings) is challenging due to the large dimensionality ofthese systems (the number of molecular emitters is $N\approx 10^{6}-10^{10}$)and the complex interplay of molecular electronic and nuclear degrees offreedom. This complexity constraints existing models to either coarse-grain therich physics and chemistry of the molecular degrees of freedom or artificiallylimit the description to a small number of molecules. In this work, we exploitpermutational symmetries to drastically reduce the computational cost of\textit{ab-initio} quantum dynamics simulations for large $N$. Furthermore, wediscover an emergent hierarchy of timescales present in these systems, thatjustifies the use of an \textit{effective} single molecule to approximatelycapture the dynamics of the entire ensemble, an approximation that becomesexact as $N\rightarrow \infty$. We also systematically derive finite $N$corrections to the dynamics, and show that addition of $k$ extra effectivemolecules is enough to account for phenomena whose rates scale as$\mathcal{O}(N^{-k})$. Based on this result, we discuss how to seamlesslymodify existing single-molecule strong coupling models to describe the dynamicsof the corresponding ensemble, as well as the crucial differences in phenomenapredicted by each model. We call this approach Collective dynamics UsingTruncated Equations (CUT-E), benchmark it against well-known results ofpolariton relaxation rates, and apply it to describe a universalcavity-assisted energy funneling mechanism between different molecular species.Beyond being a computationally efficient tool, this formalism provides anintuitive picture for understanding the role of bright and dark states inchemical reactivity, necessary to generate robust strategies for polaritonchemistry.

Collective dynamics Using Truncated Equations (CUT-E): simulating the collective strong coupling regime with few-molecule models