  The Time Dependent Boltzmann equation (TDBE) is a viable option to study
strongly out-of-equilibrium thermalization dynamics which are becoming
increasingly critical for many novel physical applications like Ultrafast
thermalization, Terahertz radiation etc. However its applicability is greatly
limited by the impractical scaling of the solution to its scattering integral
term. In our previous work\cite{Michael} we had proposed a numerical solver to
calculate the scattering integral term in the TDBE and then improved on
it\cite{1DPaper} to include second degree momentum discretisation and adaptive
time stepping. Our solver requires no close-to-equilibrium assumptions and can
work with realistic band structures and scattering amplitudes. Moreover, it is
numerically efficient and extremely robust against inherent numerical
instabilities. While in our previous work \cite{1DPaper} we showcased the
application of our solver to 1D materials, here we showcase its applications to
a simple 2D system and analyse thermalisations of the introduced
out-of-equilibrium excitations. The excitations added at higher energies were
found to thermalise faster than those introduced at relatively lower energies.
Also, we conclude that the thermalisation of the out-of-equilibrium population
to equilibrium values is not a simple exponential decay but rather a
non-trivial function of time. Nonetheless, by fitting a double exponential
function to the decay of the out-of-equilibrium population with time we were
able to generate quantitative insights into the time scales involved in the
thermalisations.
