  We consider a unified model of interacting dark matter and dark energy to
account for coincidence of present day dark energy and dark matter densities.
We assume dark energy to be represented by a homogeneous scalar field $\phi$
whose dynamics is driven by a (non-canonical) $k$-essence Lagrangian with
constant potential and the particles of dark matter fluid undergoing velocity
diffusion in background medium of the $k-$essence scalar field $\phi$. This
results in a transfer of energy from the fluid of dark matter to that of dark
energy. This effect shows up as a source term in the continuity equation for
dark matter and dark energy fluids. The source term involves a diffusion
coefficient which is a measure of average energy transferred per unit time due
to diffusion. We use time evolutions of the scale factor of background FRW
spacetime, energy density and pressure of the dark fluid obtained from analysis
of Supernova Ia data to obtain bounds on the diffusion parameter. For a
constant potential in the $k$-essence Lagrangian, the temporal behaviour of a
homogeneous $k$-essence field $\phi$ is obtained for different values of the
diffusion parameter. The obtained temporal behaviour may be expressed as
$\phi(t/t_0) = \phi_0 + \varepsilon_1 (t/t_0 - 1) + \varepsilon_2 (t/t_0
-1)^2$, where $t_0$ is the time corresponding to present epoch. The
coefficients $\varepsilon_1$ and $\varepsilon_2$ have been found and obtained
as linear functions of diffusion parameter.
