  A Multi-scale Boltzmann Equation (MBE) is found from the gas-kinetic theory
and the direct modeling philosophy as a master equation for complex physical
systems of neutral gases across all flow regimes, which locates between the
continuum limit and the free-molecular limit, covering a vast range of
applications such as hypersonic flows over aerospace crafts and delicate flows
around MEMS. The most explicit characteristic of MBE is evolving the variable
observation time in the expression, which distinguishes the MBE from the
single-scale master or governing equation where a fixed scale is implied in the
assumptions. The fundamental properties of MBE, such as the asymptotic
property, are proved theoretically, while a concise numerical scheme is
developed for MBE to demonstrate its validity by benchmark multi-scale
problems.
