  We investigate Sine$_\beta$, the universal point process arising as the
thermodynamic limit of the microscopic scale behavior in the bulk of
one-dimensional log-gases, or $\beta$-ensembles, at inverse temperature
$\beta>0$. We adopt a statistical physics perspective, and give a description
of Sine$_\beta$ using the Dobrushin-Lanford-Ruelle (DLR) formalism by proving
that it satisfies the DLR equations: the restriction of Sine$_\beta$ to a
compact set, conditionally to the exterior configuration, reads as a Gibbs
measure given by a finite log-gas in a potential generated by the exterior
configuration. Moreover, we show that Sine$_\beta$ is number-rigid and tolerant
in the sense of Ghosh-Peres, i.e. the number, but not the position, of
particles lying inside a compact set is a deterministic function of the
exterior configuration. Our proof of the rigidity differs from the usual
strategy and is robust enough to include more general long range interactions
in arbitrary dimension.
