The maximum-term method is a consequence of the large numbers encountered in statistical 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.
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These conditions are (see also proof below) that (1) the number of terms in the sum is large and (2) the terms themselves scale exponentially with this number. A typical application is the calculation of a thermodynamic potential from a partition function. These functions often contain terms with factorials
Example
Proof
Consider the sum
where
Taking logarithm gives
In statistical mechanics often
Here we have
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