This lecture has presented three distinct treatment of gas pressure, based respectively on ensemble averaging, thermodynamic manipulation of the partition function, and kinetic theory. All three approaches get the same answer. The third approach, which shows molecules as mobile points that transfer momenta via collisions, is a conventional treatment found in many texts. The second approach illustrates the fundamental links between statistical mechanics and thermodynamics, a theme that will be developed systematically in later chapters. The thermodynamic approach is very powerful, but requires that the partition function is known. The thermodynamic method only lets one determine those variables -- the thermodynamic variables -- that can be obtained from the free energy.
The first approach, not found in most other sources, illustrates the Gibbsian
ensemble method in pure form. Unlike the kinetic theory, no reference was made
to an elapsed time 
. For an equilibrium quantity, no reference to
the passage of time should required, because no time-dependent process is
involved in an equilibrium quantity. To repeat an analogy proposed above, an
ensemble may be viewed as a set of photographs, one of each possible state of
the system. To calculate the ensemble average of an equilibrium quantity, it
is adequate to examine individual snapshots, and make a calculation on each
snapshot. An average over all possible photographs is then taken. In
equilibrium calculations one need not inquire as to the dynamics -- the order
in which one snapshot follows the next in a ``motion picture'' of the system.
All three approaches to P represent the ``kinetic model of the gas'', the
kinetic model having as its primary assumption that the gas Hamiltonian depends
only on kinetic energies and on the gas-wall potential. Each approach uses the
same physical model for the system. The approaches differ in their lines of
attack. Since Section 6.1 obtained P from a series of static pictures (Fig.
6.1) of atoms near walls, no atomic motion being exhibited during the
calculation, why does the description of Section 6.1 merit the apellation
``kinetic model''? The answer is that the
kinetic model of the ideal gas should be understood as
the antithesis of the ``static model of the ideal gas'', in which nearly
immobile gas atoms interact via a short-range springlike repulsive interaction.
In a statistical-mechanical treatment of the errorneous static theory of
gasses, the energy of the system would be dominated by harmonic potentials
(with extremely large equilibrium lengths) between nearby gas molecules
The static model of the gas had peak acceptance in the early 
century,
but is now nearly forgotten.