Example 4.10: Decrease of energy fluctuation with increasing N

(a) It is a typical characteristic of statistical distributions that, for instance, the percentage deviation of the total energy E of an N-particle system from its mean value E becomes increasingly negligible with rapidly increasing N. Verify this statement by verification of the relation D(E) =

1

√N D( ), where D2(E) =

(E − E)2

(E)2

=

E2 − E 2

(E)2

, (4.66)

i.e. show that D2(E) =

1 N

2 − 2

( )2

=

1 N

D2( ), (4.67)

where ≡ i and i is the energy of the i-th particle (classical statistics, take ni = 1, E =



i).