Exact Evolution versus Mean Field with Second-order correction for Bosons Interacting via Short-range Two-body Potential

We consider the evolution of N bosons, where N is large, with two-body interactions of the form $N^{3\beta}v(N^\beta \cdot)$, $0\leq\beta\leq 1$. The parameter $\beta$ measures the strength of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see Commun. Math. Phys. 294: 273-301, 2010 and Adv. Math. 228: 1788-1815, 2011. For $0\leq\beta < 1/2$, we derive an error bound of the form $p(t)/N^\alpha$, where $\alpha > 0$ and $p(t)$ is a polynomial, which implies a specific rate of convergence as $N\rightarrow\infty$.