Setting Boundaries for Statistical Mechanics

Abstract

Statistical mechanics has grown without bounds in space. Statistical mechanics of noninteracting point particles in an unbounded perfect gas is widely used to describe liquids like concentrated salt solutions of life and electrochemical technology, including batteries. Liquids are filled with interacting molecules. A perfect gas is a poor model of a liquid. Statistical mechanics without spatial bounds is impossible as well as imperfect, if 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 Maxwell’s partial differential equations. Partial differential equations require boundary structures and conditions. Boundary conditions cannot be defined uniquely ‘at infinity’ because the limiting process that defines ‘infinity’ includes such a wide variety of structures and behaviors, from elongated ellipses to circles, from light waves that never decay, to dipolar fields that decay steeply, to Coulomb fields that hardly decay at all. Boundaries and boundary conditions needed to describe matter are not prominent in classical statistical mechanics. Statistical mechanics of bounded systems is described in the EnVarA system of variational mechanics developed by Chun Liu, more than anyone else. EnVarA treatment does not yet include Maxwell equations.

H-Bonds in Crambin: Coherence in an α-Helix

“…A single input–output description is natural for a chemical bond, which is classically defined as a structure between a pair of atoms . We do not deal with the mathematical and chemical difficulties in defining a two-body property like an ionic bond for a large three-dimensional system of charges, periodic or not, in which the system extends “to infinity,” with unavoidable consequences. , We hope that a two-body analysis will be as successful in our chemical context as two-body (input–output) analysis has been of engineering devices. Chemical bonds are not isolated structures, particularly in biological contexts, but neither are engineering devices.…”

“…A system can be determined to be linear based on the coherence function. − The coherence function is defined in terms of power and cross-power spectral densities, G xx ( f ) and G xy ( f ), respectively, as C italicxy ( f ) = false| G italicxy ( f ) false| 2 G italicxx ( f ) G italicyy ( f ) Suppose that the input function is truncated such that x T ( t ) = x ( t ) if | t | < T /2; and 0 otherwise. Its Fourier transform is X T ( f ).…”

“… 54 Thus, the topology of the network and the values of its parameters cannot be determined without additional information. As has been reviewed, 14 , 15 , 18 , 44 structural (i.e., anatomical and histological) information has been used to determine circuit structures for many biological systems (skeletal muscle, cardiac muscle, epithelia, lens of the eye, syncytia in general, etc. 14 , 19 , 44 , 55 ).…”

“…Use of the coherence function is well established in engineering. Estimators of coherence are described in classic literature , and are used with little change in modern literature. − One of us (Eisenberg) has used these methods in physiology to analyze the electrical structure of cells and tissues, , as has been reviewed. − …”

Maxwell’s True Current

“…Maxwell's definition helps [2,4] in dealing with the challenges of calculating and simulating currents (carried by significantly sized ions) over the wide range of scales found in biology in complex structures. These challenges must be met if theory and simulation are to deal with the reality of how biology and evolution have combined structure and physics to create physiological systems of great importance in health and disease [9][10][11].…”