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Considering finite particle systems, we elaborate on various ways to pass tothe limit as thenumber of agents tends to infinity, either by mean field limit,deriving the Vlasov equation,or by hydrodynamic or graph limit, obtaining theEuler equation. We provide convergenceestimates. We also show how to pass fromLiouville to Vlasov or to Euler by taking adequatemoments. Our resultsencompass and generalize a number of known results of the literature.As asurprising consequence of our analysis, we show that sufficiently regularsolutions of anylinear PDE can be approximated by solutions of systems of Nparticles, to within 1/ log log(N ).

From microscopic to macroscopic scale equations: mean field, hydrodynamic and graph limits