  The topological charge distribution P(Q) is calculated for lattice ${\rm
CP}^{N-1}$ models. In order to suppress lattice cut-off effects we employ a
fixed point (FP) action. Through transformation of P(Q) we calculate the free
energy $F(\theta)$ as a function of the $\theta$ parameter. For N=4, scaling
behavior is observed for P(Q), $F(\theta)$ as well as the correlation lengths
$\xi(Q)$. For N=2, however, scaling behavior is not observed as expected. For
comparison, we also make a calculation for the ${\rm CP}^{3}$ model with
standard action. We furthermore pay special attention to the behavior of P(Q)
in order to investigate the dynamics of instantons. For that purpose, we
carefully look at behavior of $\gamma_{\it eff}$, which is an effective power
of P(Q)($\sim \exp(-CQ^{\gamma_{\it eff}})$), and reflects the local behavior
of P(Q) as a function of Q. We study $\gamma_{\it eff}$ for two cases, one of
which is the dilute gas approximation based on the Poisson distribution of
instantons and the other is the Debye-H\"uckel approximation of instanton
quarks. In both cases we find similar behavior to the one observed in numerical
simulations.
