  We determine the log-Sobolev constant of the multi-urn Bernoulli-Laplace
diffusion model with arbitrary parameters, up to a small universal
multiplicative constant. Our result extends a classical estimate of Lee and Yau
(1998) and confirms a conjecture of Filmus, O'Donnell and Wu (2018). Among
other applications, we completely quantify the "small-set expansion" phenomenon
on the multislice, and obtain sharp mixing-time estimates for the colored
exclusion process on various graphs.
