For many chaotic systems there appears to be a single asymptotic distribution describing the frequency with which almost all trajectories visit different regions of state space.
Such a distribution is commonly called the ``physical'' invariant measure or ``natural'' invariant measure of the system.
Consider the evolution of a cluster of initial conditions under the action of the system.
This cluster will quickly spread out and begin to distribute itself according to the asymptotic distribution of the system.
When the cluster reaches the asymptotic distribution, we say that the cluster of initial conditions has attained the equilibrium state of the system.
In this paper we are concerned with the rate at which initial distributions approach equilibrium.
Often this rate is exponential, and we describe a new numerical method for rigorously bounding (and sometimes estimating) the rate.
Our method provides a considerable improvement over current theoretical bounds.