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Setting Boundaries for Statistical Mechanics

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.

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