  We present diffusion Monte Carlo (DMC) and path-integral Monte Carlo (PIMC)
calculations of a one-dimensional Bose system with realistic interparticle
interactions in a periodic external potential. Our main aim is to test the
predictions of the Luttinger liquid (LL) theory, in particular with respect to
the superfluid-Mott insulator transition at both zero and finite temperatures,
in the predicted robust and fragile superfluid regimes. For that purpose, we
present our results of the superfluid fraction $\rho_s/\rho_0$, the one-body
density matrix, the two-body correlation functions, and the static structure
factor. The DMC and PIMC results in the limit of very low temperature for
$\rho_s/\rho_0$ agree, but the LL model for scaling $\rho_s/\rho_0$ does not
fit the data well. The critical depth of the periodic potential is close to the
values obtained for ultracold gases with different models of interaction, but
with the same value of the bare LL parameter, demonstrating the universality of
LL description. Algebraic decay of correlation functions is observed in the
superfluid regime and exponential decay in the Mott-insulator one, as well as
in all regimes at finite temperature for distances larger than a characteristic
length.
