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Demonstrating Extensivity in the Fairness Function: Instructor's Guide

Main Ideas

• Statistical mechanics
• Extensive vs. intensive properties
• Probabilities in the fairness function

Estimated Time: 15 minutes

Prerequisite Knowledge

• Familiarity with how probabilities of two uncorrelated systems interact
• Basic understanding of the fairness function is helpful
• Familiarity with summation notation, particularly manipulation of terms in sums

Activity: Introduction

Before performing this activity, a small lecture or an activity (i.e. [[..:..:courses:activities:eeact:combineprob|Combining Probabilities) about the probabilities of two uncorrelated systems is recommended. If the expression is not on the board yet, write that the fairness function is defined as

$$F=-k\sum_{i}^{All \: states} P_{i} \ln P_{i} \; \; \; .$$

Then, prompt the students to show the fairness function is extensive. That is, show that

$$F_{AB}=F_{A}+F_{B} \; \; \; .$$

Activity: Student Conversations

I am not convinced that an information-theory first approach is ideal, so I do not intend to use this activity in Spring 2012. – David Roundy

The discovery that Fairness is extensive is easily discovered or mentioned during other activities, such as the Students as molecules dice activity .

Activity: Wrap-up

To conclude this activity, work through the solution to the problem on the board.

$$F_{A}=-k\sum_{i} P_{i} \ln P_{i} \; \; \; .$$

$$F_{B}=-k\sum_{j} P_{j} \ln P_{j} \; \; \; .$$

$$F_{AB}=-k\sum_{i,j} P_{i}P_{j} \ln P_{i}P_{j} \; \; \; .$$

$$F_{AB}=-k\sum_{i,j} P_{i}P_{j} (\ln P_{i} + \ln P_{j}) \; \; \; .$$

$$F_{AB}=-k\left(\sum_{i,j} P_{i}P_{j} \ln P_{i}\right) - k\left(\sum_{i,j} P_{i}P_{j} \ln P_{j}\right) \; \; \; .$$

$$F_{AB}=-k\left(\sum_{i} P_{i}\ln P_{i}\right)\left(\sum_{j}P_{j}\right) - k\left(\sum_{j} P_{j} \ln P_{j}\right)\left(\sum_{i} P_{i} \right) \; \; \; .$$

$$F_{AB}=-k\left(\sum_{i} P_{i}\ln P_{i}\right) - k\left(\sum_{j} P_{j} \ln P_{j}\right) \; \; \; .$$

$$F_{AB}=F_{A}+F_{B} \; \; \; .$$

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