Maximum term method

Maximum term method is a consequence of the large numbers encountered in stastistical mechanics. It states that under appropriate conditions the logarithm of a summation is essentially equal to the logarithm of the maximum term in the summation.

Proof

Consider the sum

$S=\sum _{N=1}^{M}T_{N}$

where $T_{N}$>0 for all N. Since all the terms are positive, the value of S must be greater than the value of the largest term, $T_{max}$, and less than the product of the number of terms and the value of the largest term. So we have

$T_{max}<=S<=MT_{max}$

Taking logarithm gives

$\ln T_{max}<=\ln S<=\ln T_{max}+\ln M$

This is almost always the case in statistical mechanics that $T_{max}$ will be $O(e^{M})$ .

Here we have

$O(M)<=\ln S<=O(M)+\ln M$

For large M, ln M is negligible with respect to M itself, and so we can see that ln S is bounded from above and below by $\ln T_{max}$, and so

$\ln S=\ln T_{max}$