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Abstract We propose a new approach to deriving quantitative mean field approximations for any probability measure $$P$$ on $$\mathbb {R}^{n}$$ with density proportional to $$e^{f(x)}$$, for $$f$$ strongly concave. We bound the mean field approximation for the log partition function $$\log \int e^{f(x)}dx$$ in terms of $$\sum _{i \neq j}\mathbb {E}_{Q^{*}}|\partial _{ij}f|^{2}$$, for a semi-explicit probability measure $$Q^{*}$$ characterized as the unique mean field optimizer, or equivalently as the minimizer of the relative entropy $$H(\cdot \,|\,P)$$ over product measures. This notably does not involve metric-entropy or gradient-complexity concepts which are common in prior work on nonlinear large deviations. Three implications are discussed, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems. Our arguments are based primarily on functional inequalities and the notion of displacement convexity from optimal transport.