  We analytically study a one dimensional compaction model in the glassy
regime. Both correlation and response functions are calculated exactly in the
evolving dense and low tapping strength limit, where the density relaxes in a
$1/\ln t$ fashion. The response and correlation functions turn out to be
connected through a non-equilibrium generalisation of the
fluctuation-dissipation theorem. The initial response in the average density to
an increase in the tapping strength is shown to be negative, while on longer
timescales it is shown to be positive. On short time scales the
fluctuation-dissipation theorem governs the relation between correlation and
response, and we show that such a relationship also exists for the slow degrees
of freedom, albeit with a different temperature. The model is further studied
within the statistical theory proposed by Edwards and co-workers, and the
Edwards entropy is calculated in the large system limit. The fluctuations
described by this approach turn out to match the fluctuations as calculated
through the dynamical consideration. We believe this to be the first time these
ideas have been analytically confirmed in a non-mean-field model.
