  Recently, a new non-thermal mechanism for dark matter production has been
proposed which results in its exponential growth with the expansion of the
universe. This mechanism works provided a small but non-zero initial dark
matter ($\chi$) number density exists in the early universe which scatters of
the bath particles ($\phi$) to generate more dark matter particles ($\chi \phi
\to \chi \chi$). The process ends when the scattering rate becomes Boltzmann
suppressed. The analysis, in literature, is performed on the simplifying
assumption of the dark matter phase space tracing the equilibrium distribution
of either standard model or a hidden sector bath. Owing to the non-thermal
nature of the production mechanism, this assumption may not hold. In this
paper, we compute the distribution function of dark matter by solving the
Boltzmann-equation at the operator level analytically and/or numerically. We
find that the obtained distribution exhibits different behavior from the
equilibrium pattern and is sensitive to the mechanism populating the initial
momentum modes of the dark matter distribution function. It becomes more useful
to characterize the growth of the momentum modes of the dark matter
distribution function with the expansion of the universe. The growth of these
modes can be parameterised by an exponential factor i.e. $\exp{[A(x)]}$ where
$A(x)$ depends on the details of the model.
