On the spectral gap of Kawasaki dynamics under a mixing condition revisited

We consider a conservative stochastic spin exchange dynamics which is reversible with respect to the canonical Gibbs measure of a lattice gas model. We assume that the corresponding grand canonical measure satisfies a suitable strong mixing condition. We give an alternative and quite natural, from the physical point of view, proof of the famous Lu-Yau result which states that the relaxation time in a box of side L scales like L-2. We then show how to use such an estimate to prove a decay to equilibrium for local functions of the form 1/t(alpha-epsilon), where epsilon is positive and arbitrarily small and alpha=1/2 for d=1, alpha=1 for d greater than or equal to 2.