An interacting particle system (Glauber dynamics) which evolves on a finite subset in the d-dimensional integer lattice is considered. It is known that a mixing property of the Gibbs state in the sense of Dobrushin and Shlosman is equiwdent to several very strong estimates in terms of the Glauber dynamics. We show that similar, but seemingly much milder estimates are again equivalent to the Dobrushin-Shlosman mixing condition, hence to the origimd ones found by Stroock and Zegarlinski. This may be understood as the absence of intermediate speed of convergence to equilibrium.