Darwin–Fowler method

Darwin–Fowler method

In most texts on statistical mechanics the statistical distribution functions

f

{\displaystyle f}

in Maxwell–Boltzmann statistics, Bose–Einstein statistics, Fermi–Dirac statistics) are derived by determining those for which the system is in its state of maximum probability. But one really requires those with average or mean probability, although – of course – the results are usually the same for systems with a huge number of elements, as is the case in statistical mechanics. The method for deriving the distribution functions with mean probability has been developed by C. G. Darwin and Fowler and is therefore known as the Darwin–Fowler method. This method is the most reliable general procedure for deriving statistical distribution functions. Since the method employs a selector variable (a factor introduced for each element to permit a counting procedure) the method is also known as the Darwin–Fowler method of selector variables. Note that a distribution function is not the same as the probability – cf. Maxwell–Boltzmann distribution, Bose–Einstein distribution, Fermi–Dirac distribution. Also note that the distribution function

f

i

{\displaystyle f_{i}}

which is a measure of the fraction of those states which are actually occupied by elements, is given by

f

i

=

n

i

/

g

i

{\displaystyle f_{i}=n_{i}/g_{i}}

or

n

i

=

f

i

g

i

{\displaystyle n_{i}=f_{i}g_{i}}

, where

g

i

{\displaystyle g_{i}}

is the degeneracy of energy level

i

{\displaystyle i}

of energy

ε

i

{\displaystyle \varepsilon _{i}}

and

n

i

{\displaystyle n_{i}}

is the number of elements occupying this level (e.g. in Fermi–Dirac statistics 0 or 1). Total energy

E

{\displaystyle E}

and total number of elements

N

{\displaystyle N}

are then given by

E

=

∑

i

n

i

ε

i

{\displaystyle E=\sum _{i}n_{i}\varepsilon _{i}}

and

N

=

∑

n

i

{\displaystyle N=\sum n_{i}}

.

The Darwin–Fowler method has been treated in the texts of E. Schrödinger, Fowler and Fowler and E. A. Guggenheim, of K. Huang, and of H. J. W. Müller–Kirsten. The method is also discussed and used for the derivation of Bose–Einstein condensation in the book of R. B. Dingle.

Classical statistics

For

N

=

∑

i

n

i

{\displaystyle N=\sum _{i}n_{i}}

independent elements with

n

i

{\displaystyle n_{i}}

on level with energy

ε

i

{\displaystyle \varepsilon _{i}}

and

E

=

∑

i

n

i

ε

i

{\displaystyle E=\sum _{i}n_{i}\varepsilon _{i}}

for a canonical system in a heat bath with temperature

T

{\displaystyle T}

we set

Z

=

Quantum statistics

We have as above

Z

ω

=

∑

∏

(

ω

z

i

)

n

i

,

z

i

=

e

−

ε

i

/

k

T

,

{\displaystyle Z_{\omega }=\sum \prod (\omega z_{i})^{n_{i}},\;\;z_{i}=e^{-\varepsilon _{i}/kT},}

where

n

i

{\displaystyle n_{i}}

is the number of elements in energy level

ε

i

{\displaystyle \varepsilon _{i}}