  Statistical mechanics has grown without bounds in space. Statistical
mechanics of point particles in an unbounded perfect gas is commonly accepted
as a foundation for understanding many systems, including liquids like the
concentrated salt solutions of life and electrochemical technology, from
batteries to nanodevices. Liquids, however, are not gases. Liquids are filled
with interacting molecules and so the model of a perfect gas is imperfect. Here
we show that statistical mechanics without bounds (in space) is impossible as
well as imperfect, if the molecules interact as charged particles, as nearly
all atoms do. The behavior of charged particles is not defined until boundary
structures and values are defined because charges are governed by the Maxwell
partial differential equations. Partial differential equations require boundary
conditions to be computable or well defined. The Maxwell equations require
boundary conditions on finite sized spatial boundaries (i.e., structures).
Boundary conditions cannot be defined 'at infinity' in a general (i.e., unique)
way because the limiting process that defines infinity includes such a wide
variety of behavior, from light waves that never decay, to fields from dipole
and multipolar charges that decay steeply, to Coulomb fields that decay but not
so steeply. Statistical mechanics involving charges thus involves spatial
boundaries and boundary conditions of finite size. Nearly all matter involves
charges, thus nearly all statistical mechanics requires structures and boundary
conditions on those structures. Boundaries and boundary conditions are not
prominent in classical statistical mechanics. Including boundaries is a
challenge to mathematicians. Statistical mechanics must describe bounded
systems if it is to provide a proper foundation for studying matter.
