88
CERTAIN IMPORTANT FUNCTIONS
function of , which becomes infinite with , and vanishes for the smallest possible value of , or for , if the energy may be diminished without limit.
Let us also set
|
(266)
|
The extension in phase between any two limits of energy,
and
, will be represented by the integral
|
(267)
|
And in general, we may substitute
for
in a
-fold integral, reducing it to a simple integral, whenever the limits can be expressed by the energy alone, and the other factor under the integral sign is a function of the energy alone, or with quantities which are constant in the integration.
In particular we observe that the probability that the energy of an unspecified system of a canonical ensemble lies between the limits and will be represented by the integral[1]
|
(268)
|
and that the average value in the ensemble of any quantity which only varies with the energy is given by the equation
[2]
|
(269)
|
where we may regard the constant
as determined by the equation
[3]
|
(270)
|
In regard to the lower limit in these integrals, it will be observed that
is equivalent to the condition that the value of
is the least possible.
- ↑ Compare equation (93).
- ↑ Compare equation (108).
- ↑ Compare equation (92).