  We discuss Monte Carlo dynamics based on <N(sigma, Delta E)>_E, the
(microcanonical) average number of potential moves which increase the energy by
Delta E in a single spin flip. The microcanonical average can be sampled using
Monte Carlo dynamics of a single spin flip with a transition rate min(1,
<N(sigma', E-E')>_E' / <N(sigma, E'-E) >_E) from energy E to E'. A cumulative
average (over Monte Carlo steps) can be used as a first approximation to the
exact microcanonical average in the flip rate. The associated histogram is a
constant independent of the energy. The canonical distribution of energy can be
obtained from the transition matrix Monte Carlo dynamics. This second dynamics
has fast relaxation time - at the critical temperature the relaxation time is
proportional to specific heat. The dynamics are useful in connection with
reweighting methods for computing thermodynamic quantities.
