  In a previous paper [Gholami et al. Adv.Th.Sim.4, 2000303 (2021)], we have
identified a precise relation between the chemical potential of a fully
atomistic simulation and the simulation of an open system in the adaptive
resolution method (AdResS). The starting point was the equivalence derived from
the statistical partition functions between the grand potentials, $\Omega$, of
the open system and of the equivalent subregion in the fully atomistic
simulation of reference. In this work, instead, we treat the identity for the
grand potential based on the thermodynamic relation $\Omega=-pV$ and
investigate the behaviour of the pressure in the coupling region of the
adaptive resolution method (AdResS). We confirm the physical consistency of the
method for determining the chemical potential described by the previous work
and strengthen it further by identifying a clear numerical relation between the
potential that couples the open system to the reservoir on the one hand and the
local pressure of the reference fully atomistic system on the other hand. Such
a relation is of crucial importance in the perspective of coupling the AdResS
method for open system to the continuum hydrodynamic regime.
