  We investigate the behavior of the Gibbs-Shannon entropy of the stationary
nonequilibrium measure describing a one-dimensional lattice gas, of L sites,
with symmetric exclusion dynamics and in contact with particle reservoirs at
different densities. In the hydrodynamic scaling limit, L to infinity, the
leading order (O(L)) behavior of this entropy has been shown by Bahadoran to be
that of a product measure corresponding to strict local equilibrium; we compute
the first correction, which is O(1). The computation uses a formal expansion of
the entropy in terms of truncated correlation functions; for this system the
k-th such correlation is shown to be O(L^{-k+1}). This entropy correction
depends only on the scaled truncated pair correlation, which describes the
covariance of the density field. It coincides, in the large L limit, with the
corresponding correction obtained from a Gaussian measure with the same
covariance.
