Page:Elementary Principles in Statistical Mechanics (1902).djvu/93

or, since

$\psi _{q}={\overline {\epsilon }}_{q}+\Theta {\overline {\eta }}_{q},$

(183)

and

$d\psi _{q}=d{\overline {\eta }}_{q}+{\overline {\eta }}_{q}\,d\Theta +\Theta \,d{\overline {\eta }}_{q},$

(183)

$d{\overline {\epsilon }}_{q}=-\Theta d{\overline {\eta }}_{q}-{\overline {A}}_{1}\,da_{1}-{\overline {A}}_{2}\,da_{2}-\mathrm {etc.}$

(184)

It appears from this equation that the differential relations subsisting between the average potential energy in an ensemble of systems canonically distributed, the modulus of distribution, the average index of probability of configuration, taken negatively, and the average forces exerted on external bodies, are equivalent to those enunciated by Clausius for the potential energy of a body, its temperature, a quantity which he called the disgregation, and the forces exerted on external bodies.^[1]

For the index of probability of velocity, in the case of canonical distribution, we have by comparison of (144) and (163), or of (145) and (164),

$\eta _{p}={\frac {\psi _{p}-\epsilon _{p}}{\Theta }}$

(185)

which gives

${\overline {\eta }}_{p}={\frac {\psi _{p}-{\overline {\epsilon }}_{p}}{\Theta }};$

(186)

we have also

${\overline {\epsilon }}_{p}={\tfrac {1}{2}}n\Theta ,$

(187)

and by (140),

$\psi _{p}=-{\tfrac {1}{2}}n\Theta \log(2\pi \Theta ).$

(188)

From these equations we get by differentiation

$d\psi _{p}={\overline {\eta }}_{p}\,d\Theta ,$

(189)

and

$d{\overline {\epsilon }}_{p}=-\Theta \,d{\overline {\eta }}_{p}.$

(190)

The differential relation expressed in this equation between the average kinetic energy, the modulus, and the average index of probability of velocity, taken negatively, is identical with that given by Clausius locis citatis for the kinetic energy of a body, the temperature, and a quantity which he called the transformation-value of the kinetic energy.^[2] The relations

${\overline {\epsilon }}={\overline {\epsilon }}_{q}+{\overline {\epsilon }}_{p},\quad {\overline {\eta }}={\overline {\eta }}_{q}+{\overline {\eta }}_{p}$

1. Pogg. Ann., Bd. CXVI, S. 73, (1862); ibid., Bd. CXXV, S. 353, (1865). See also Boltzmann, Sitzb. der Wiener Akad., Bd. LXIII, S. 728, (1871).
2. Verwandlungswerth des Wärmeinhaltes.