  It is frequently assumed that in the limit of vanishing cooling rate, the
glass transition phenomenon becomes a thermodynamic transition at a temperature
$T_{K}$. However, with any finite cooling rate, the system falls out of
equilibrium at temperatures near $T_g(>T_{K})$, implying that the very
existence of the putative thermodynamic phase transition at $T_{K}$ can be
questioned. Recent studies of systems with randomly pinned particles have
hinted that the thermodynamic glass transition may be observed in simulations
and experiments carried out for liquids with randomly pinned particles. This
expectation is based on the results of approximate calculations that suggest
that the temperature of the thermodynamic glass transition increases as the
concentration of pinned particles is increased and it may be possible to
equilibrate the system at temperatures near the increased transition
temperature. We test the validity of this prediction through extensive
molecular dynamics simulations of two model glass-forming liquids in the
presence of random pinning. We fit the temperature-dependence of the structural
relaxation time to the Vogel-Fulcher-Tammann form that predicts a divergence of
the relaxation time at a temperature $T_{VFT}$ and identify this temperature
with the thermodynamic transition temperature $T_K$. We find that $T_{VFT}$
does not show any sign of increasing with increasing concentration of pinned
particles. The main effect of pinning is found to be a rapid decrease in the
kinetic fragility of the system with increasing pin concentration. Implications
of these observations for current theories of the glass transition are
discussed.
