An ensemble is specified by fixing the list of allowed states, and the statistical weight for each state. This book relies primarily on the canonical ensemble, in which the number of particles N, system volume V, and temperature T are constraints. In an allowed state, all N particles of the system lie within V. There are no constraints on the particle momenta. The shape of V is usually not given. Excepting surface energies, thermodynamic properties do not depend on the shape of the substance being studied, but only on the amount of substance that is present. So long as V is of macroscopic dimensions in all directions, the shape of V ought not to be important for computing thermodynamic properties. (Statistical mechanics can treat surface issues [4], but surfaces are not emphasized here.) T does not affect the list of allowed states. T only enters the calculation through the statistical weight.
For the canonical ensemble, the correct statistical weight of a state j was given by Gibbs as
Here 
is the total energy of the state j, while 
. T is the absolute temperature; 
is Boltzmann's
constant, which converts T from temperature units to energy units. This
statistical weight is not normalized. Some European authors use peculiar
temperature units in which 
, so that the rhs of 3.1 becomes
. C is a constant, varying from system to system, treated
in later Lectures. Experimental tests of rigid-bar gravitational wave
detectors (which appear not to have detected gravity waves) make eq.
3.1 among the most accurately tested of all physical laws.
[5]
The normalizing factor in the canonical ensemble average is
Q is known as the canonical partition function. Q serves to connect statistical mechanics to thermodynamics. A later Lecture argues
where A is the Helmholtz free energy of the system.
Gibbs advances strong plausibility arguments for believing eq. 3.3,
in that a system in which 
has the properties one would
expect of a thermodynamic system. A plausibility argument is not a proof.
Eqs. 3.1 and 3.3, and the theoretical construct which
defines their applications, are a new law of nature, like Maxwell's equations
or the gravitational field equations. The validity of these equations are
attested by their strong experimental support under a wide range of conditions.
Eq. 3.1 and the ensemble averaging concept cannot be derived (at least as of this writing) from the mechanical laws of motion, either Hamilton's or Schrodinger's. At our present state of understanding, the status of eq. 3.1 in mechanics is much like the status of the parallel postulate in ancient Greek plane geometry. The parallel postulate is consistent with the remainder of geometry, but ancient and medieval efforts to derive the parallel postulate from the other postulates were necessarily failures. Finally Riemann and Bolyai demonstrated that one can construct consistent non-Euclidean geometries, in which the other postulates of geometry are true, but the parallel postulate is false. Whether some future Reimann or Bolyai can close this circle by introducing non-thermodynamic statistical weights, thereby clarifying the interpretation of the usual statistical weights, remains to be seen. An obvious area to search for such statistical weights is in the treatment of many-particle gravitating systems, whose behavior can not be described by normal statistical mechanics.[6]
