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Stirring Formula for N

During the study of statistical mechanics, we usually consider system consisting of very large number of particles. If this number is represented by N, then as this number is very large, the calculation of In N ! (Which occurs quite often) becomes very laborious. Hence a simple approximation is used for the value of In N ! which is called Sterling approximation. This is obtained as follows:
         We know that
                      N ! = N X ( N-1) X (N-2)............3 X 2 X 1
  or                N ! = 1 X 2 X.......(N-2) X (N-1) X N
                N ! = In 1 + In 2 +.........+In (N-2) + In (N-1) + In N
In this summation, except for the first few terms whose values are small, as m increases and attains large values, the increase in the value of the In m for increase in the value of m can be approximately treated as continuous so the tit gives the area under the curve from m=1 to m=N obtained by plotting In m vs. m. This in turn is equal to the integration of In x dx between the limits x =1 to x = N . Hence eqn. can be approximated to
The integral on the R.H.S can be solved by integration by pats.* Thus
                                               =N In N-N
Hence for large values of N

This is called Sailing’s formula for large values of N.

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