  Observations of the Lyman-$\alpha$ forest in distant quasar spectra with
upcoming surveys are expected to provide significantly larger and
higher-quality datasets. To interpret these datasets, it is imperative to
develop efficient simulations. One such approach is based on the assumption
that baryonic densities in the intergalactic medium (IGM) follow a lognormal
distribution. We extend our earlier work to assess the robustness of the
lognormal model of the Lyman-$\alpha$ forest in recovering the parameters
characterizing IGM state, namely, the mean-density IGM temperature ($T_0$), the
slope of the temperature-density relation ($\gamma$), and the hydrogen
photoionization rate ($\Gamma_{12}$), by comparing with high-resolution
Sherwood SPH simulations across the redshift range $2 \leq z \leq 2.7$. These
parameters are estimated through a Markov Chain Monte Carlo technique, using
the mean and power spectrum of the transmitted flux. We find that the usual
lognormal distribution of IGM densities cannot recover the parameters of the
SPH simulations. This limitation arises from the fact that the SPH baryonic
density distribution cannot be described by a simple lognormal form. To address
this, we extend the model by scaling the linear density contrast by a parameter
$\nu$. While the resulting baryonic density is still lognormal, the additional
parameter gives us extra freedom in setting the variance of density
fluctuations. With this extension, values of $T_0$ and $\gamma$ implied in the
SPH simulations are recovered at $\sim 1-\sigma$ ($\lesssim$ 10%) of the median
(best-fit) values for most redshifts bins. However, this extended lognormal
model cannot recover $\Gamma_{12}$ reliably, with the best-fit value discrepant
by $\gtrsim 3-\sigma$ for $z > 2.2$. Despite this limitation in the recovery of
$\Gamma_{12}$, we argue that the model remains useful for constraining
cosmological parameters.
