  A reliable description of surfaces structures in a reactive environment is
crucial to understand materials functions. We present a first-principles theory
of replica-exchange grand-canonical-ensemble molecular dynamics (REGC-MD) and
apply it to evaluate phase equilibria of surfaces in reactive gas-phase
environment. We identify the different surface phases and locate phase
boundaries including triple and critical points. The approach is demonstrated
by addressing open questions for the Si(100) surface in contact with a hydrogen
atmosphere. In the range from 300 to 1 000 K, we find 25 distinct
thermodynamically stable surface phases, for which we also provide microscopic
descriptions. Most of the identified phases, including few order-disorder phase
transitions, have not yet been observed experimentally. The REGC-MD-derived
phase diagram shows significant, qualitative differences to the description by
the state-of-the-art "ab initio atomistic thermodynamics" approach.
