132
3 Ensembles: Systems of Particles
i =
N!
N 0 !N 1 ! · · · N q !
,
i = I, II, . . . .
Thus, for example, for the sharing of three quanta of energy amongst three
oscillators, we obtain
I =
3!
2!0!0!1!
= 3, , II =
3!
1!1!1!0!
= 6, , III =
3!
0!3!0!0!
= 1 ,
to give a total of 10 microstates for the three macrostates. The corresponding
probabilities (relative occurrences) for this three-oscillator system to be found in
one of the three macrostates are thus p I = 0.30, p II = 0.60, and p III = 0.10.
(ii) Five oscillators sharing five quanta of energy:
For five quanta of energy shared amongst five equivalent yet distinguishable
oscillators the same combinatoric procedure leads to seven generic macrostates,
corresponding systematically to one oscillator receiving all five energy quanta
(macrostate I), one oscillator receiving four energy quanta plus one oscillator
receiving one energy quantum (macrostate II), one oscillator receiving three quanta
of energy plus one oscillator receiving two quanta of energy (macrostate III), one
oscillator receiving three quanta of energy plus two oscillators each receiving one
quantum of energy (macrostate IV), two oscillators each receiving two quanta of
energy plus one oscillator receiving one quantum of energy (macrostate V), one
oscillator receiving two quanta of energy plus three oscillators each receiving one
quantum of energy (macrostate VI), and all five oscillators each receiving one
quantum of energy (macrostate VII). The corresponding degeneracies, obtained
from our combinatoric formula, are
I = 5, , II = 20, , III = 20, , IV = 30, , V = 30, , VI = 20, , VII = 1,
to give a total of 126 microstates and relative probabilities
p I = 0.0397, p II = p III = p VI = 0.1587, p IV = p V = 0.2381, p VII = 0.0077,
for the seven macrostates.
(iii) Ten oscillators sharing five energy quanta:
If we carry out the same analysis for the sharing of five quanta of energy amongst a
set of ten equivalent yet distinguishable oscillators instead of five, we find that there
are still seven macrostates, determined essentially by the manner in which the five
quanta of energy are shared amongst the set of ten oscillators. The same systematic
determination of the generic macrostates differs only in the numbers of oscillators
that receive zero quanta of energy, with N 0 (i) = N
0 (i) + 5, i = I, II, . . . , VII, and
N
0 (i) the number of oscillators receiving zero quanta of energy in the sharing of
five quanta of energy amongst five equivalent yet distinguishable oscillators. There
3 Ensembles: Systems of Particles
i =
N!
N 0 !N 1 ! · · · N q !
,
i = I, II, . . . .
Thus, for example, for the sharing of three quanta of energy amongst three
oscillators, we obtain
I =
3!
2!0!0!1!
= 3, , II =
3!
1!1!1!0!
= 6, , III =
3!
0!3!0!0!
= 1 ,
to give a total of 10 microstates for the three macrostates. The corresponding
probabilities (relative occurrences) for this three-oscillator system to be found in
one of the three macrostates are thus p I = 0.30, p II = 0.60, and p III = 0.10.
(ii) Five oscillators sharing five quanta of energy:
For five quanta of energy shared amongst five equivalent yet distinguishable
oscillators the same combinatoric procedure leads to seven generic macrostates,
corresponding systematically to one oscillator receiving all five energy quanta
(macrostate I), one oscillator receiving four energy quanta plus one oscillator
receiving one energy quantum (macrostate II), one oscillator receiving three quanta
of energy plus one oscillator receiving two quanta of energy (macrostate III), one
oscillator receiving three quanta of energy plus two oscillators each receiving one
quantum of energy (macrostate IV), two oscillators each receiving two quanta of
energy plus one oscillator receiving one quantum of energy (macrostate V), one
oscillator receiving two quanta of energy plus three oscillators each receiving one
quantum of energy (macrostate VI), and all five oscillators each receiving one
quantum of energy (macrostate VII). The corresponding degeneracies, obtained
from our combinatoric formula, are
I = 5, , II = 20, , III = 20, , IV = 30, , V = 30, , VI = 20, , VII = 1,
to give a total of 126 microstates and relative probabilities
p I = 0.0397, p II = p III = p VI = 0.1587, p IV = p V = 0.2381, p VII = 0.0077,
for the seven macrostates.
(iii) Ten oscillators sharing five energy quanta:
If we carry out the same analysis for the sharing of five quanta of energy amongst a
set of ten equivalent yet distinguishable oscillators instead of five, we find that there
are still seven macrostates, determined essentially by the manner in which the five
quanta of energy are shared amongst the set of ten oscillators. The same systematic
determination of the generic macrostates differs only in the numbers of oscillators
that receive zero quanta of energy, with N 0 (i) = N
0 (i) + 5, i = I, II, . . . , VII, and
N
0 (i) the number of oscillators receiving zero quanta of energy in the sharing of
five quanta of energy amongst five equivalent yet distinguishable oscillators. There