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We consider a noninteracting unbounded spin system with conservation of themean spin. We derive a uniform logarithmic Sobolev inequality (LSI) providedthe single-site potential is a bounded perturbation of a strictly convexfunction. The scaling of the LSI constant is optimal in the system size. Theargument adapts the two-scale approach of Grunewald, Villani, Westdickenbergand the second author from the quadratic to the general case. Using anasymmetric Brascamp-Lieb-type inequality for covariances, we reduce the task ofderiving a uniform LSI to the convexification of the coarse-grainedHamiltonian, which follows from a general local Cram\'{e}r theorem.

Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential