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Averages in Statistical Mechanics

Copyright ï³°an> 2011 by Robert Finkel

Averages

As an example of averaging with probabilities, consider averaging quiz grades for a hypothetical class of 10 students. The possible grades are 0, 50, and 100. One student scored 0, four got 50, and five got 100. Here is the class average:

Another way to look at the same average is:

.

Notice that the probability of a student getting 0 is 1/10 and the probability of getting 50 is 4/10, etc. We write

.

This is a special case of a general way to take an average over ï³°an>when the corresponding probabilities are known:

ã°¡n style='font:7.0pt "Times New Roman"'>       A molecule can exist in two isomeric states in equilibrium. Isomer A has energy kT and isomer B has energy 0.31kT with respective probabilities 0.33 and 0.67. Find the average energy.

[ans. 0.52kT]

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