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## Entropy of a Microcanonical Ensemble: Instructor's Guide

### Main Ideas

• Maximum fairness function
• Eigenstates
• Probabilities
• Expression on Boltzmann's tombstone
• Microcanonical ensemble
• Canonical ensemble
• Grand canonical ensemble

Estimated Time: 10 minutes

Students, in small groups, will answer the following question:

Consider a system with W eigenstates, all of which have the same energy. What is the system's entropy?

### Prerequisite Knowledge

• Knowledge that the maximum fairness function is equivalent to entropy.
• Experience manipulating variables in summation notation.
• $S=F_{max}=-k_{B} \sum_{i} P_{i} \ln P_{i}$

### Activity: Introduction

Little introduction is needed for this activity; students should have seen at some point before this activity that the maximized fairness function is equal to the entropy. Place the students into small groups and prompt the groups with:

Consider a system with W eigenstates, all of which have the same energy. What is the system's entropy?

### Activity: Wrap-up

Emphasize to the students that since the eigenstates have the same energy, each eigenstate must be equally probable. So, for this system,

$$P_{i}=\frac{1}{W} \; \; \; .$$

Show students that inserting this expression into the equation for entropy and simplifying yields:

$$S=k_{B} \ln W \; \; \; .$$

As a fun fact, tell the class that this result is the same one written on Boltzmann's tombstone, and is sometimes taken as the definition for entropy. Note that this expression is more specific than the one students have used thus far because it describes only a microcanonical ensemble.

This is also a good time to touch on the differences between a microcanonical ensemble, a canonical ensemble, and a grand canonical ensemble. Although these terms are not likely to show up again in this course, students will likely hear them in future thermodynamic courses.

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