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Geometric criteria for the absence of effective many-body interactions in nonadditive hard particle mixtures

We consider a mixture of small and big classical particles in arbitrary spatial dimensions interacting via hard-body potentials with non-additive excluded-volume interactions. In particular, we focus on variants of the Asakura--Oosawa (AO) model where the interaction between the small particles is neglected but the big-small and big-big interactions are present and can be condensed into an effective depletion interaction among the big particles alone. The original AO model involves hard spherical particles in three spatial dimensions with interaction diameters $σ_\text{pp}=0$, $σ_\text{cc}>0$ and $σ_\text{pc}>σ_\text{cc}/2$ respectively, where $σ_{ij}$ with $\{i,j\}=\{\text{p},\text{c}\}$ (indicating the physical interpretation of the small and big particles as polymers (p) and colloids (c), respectively) is the minimum possible center-to-center distance between particle $i$ and particle $j$ allowed by the excluded-volume constraints. It is common knowledge that there are only pairwise effective depletion interactions between the big particles if the geometric condition $σ_\text{pc}/σ_\text{cc} < 1/\sqrt{3}$ is fulfilled. In this case, triplet and higher-order many body interactions are vanishing and the equilibrium statistics of the binary mixture can exactly be mapped onto that of an effective one-component system with the effective depletion pair-potential. Here we prove this geometric criterion rigorously and generalize it to polydisperse mixtures and to anisotropic particle shapes in any dimension, providing geometric criteria sufficient to guarantee the absence of triplet and higher-order many body interactions. For an external hard wall confining the full mixture, we also give criteria which guarantee that the system can be mapped onto one with effective external one-body interactions.