  We discuss the problem of counting the maximum number of distinct states that
an isolated physical system can pass through in a given period of time---its
maximum speed of dynamical evolution. Previous analyses have given bounds in
terms of the standard deviation of the energy of the system; here we give a
strict bound that depends only on E-E0, the system's average energy minus its
ground state energy. We also discuss bounds on information processing rates
implied by our bound on the speed of dynamical evolution. For example, adding
one Joule of energy to a given computer can never increase its processing rate
by more than about 3x10^33 operations per second.
