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We have constructed $N$-body particle-mesh simulations of disk galaxies in which the relaxation times of the simulated disks (as measured by thermalization of the disk, i.e. increase in Toomre's $Q$ parameter) is comparable to the actual relaxation time scale in actual disk galaxies (several tens of rotation periods). These simulations require 1M to 4M particles (1M $= 2^{20}$), consistent with the work of White and of Comins and Schroeder on the dependence of relaxation time on $N$. We observe that during the interval when $Q$ is increasing, that the Fourier power associated with spiral modes is large. When $Q$ has risen to its asymptotic value in the simulation, the Fourier power diminishes to a low level. This suggests a scenario in which stars (simulation particles) scatter off the time-varying spiral potential, as suggested by Carlberg and Sellwood. Eventually random velocities of stars increase to a value which quenches the spiral instability. We compare the heating rates in our simulations at observed spiral wave amplitudes to the expected growth rates as given by Carlberg and Sellwood.
