Thermal Physics · Problem C.2

Propagation — Charles Kittel & Herbert Kroemer

Physics — Statistical Mechanics · poisson-approach-to-gaussian

Problem

Approach to Gaussian distribution. Show that the Poisson function

P(N) = \langle N \rangle^N \exp(-\langle N \rangle)/N!

closely approaches a Gaussian function in form, for large

\langle N \rangle

. That is, show when

N

is close to

\langle N \rangle

that

P(N) \simeq A\exp[-B(N - \langle N \rangle)^2]\,,

where

A

B

are quantities to be determined by you. Hint: Work with

\log P(N)

; use the Stirling approximation. In the Gaussian form both

A

and

B

are functions of

\langle N \rangle

; in the development of the Poisson function you may find

A

B

are functions of

N

, but the two forms of

A

B

are closely equivalent over the range in which the exponential factor has significant values.