title: Maximum term method
text: 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.
  
  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  which scale as  (Stirling's approximation).
  
  Example
  
  Proof
  Consider the sum
  
  where >0 for all N. Since all the terms are positive, the value of S must be greater than the value of the largest term, , and less than the product of the number of terms and the value of the largest term. So we have
  
  Taking logarithm gives
  
  In statistical mechanics often  will be : see Big O notation.
  
  Here we have
  
  For large M,  is negligible with respect to M itself, and so we can see that ln S is bounded from above and below by , and so
  
  References
  D.A. McQuarrie, Statistical Mechanics. New York: Harper & Row, 1976.
  T.L. Hill, An Introduction to Statistical Thermodynamics. New York: Dover Publications, 1987
  
  Physical chemistry
  Statistical mechanics