{{page>wiki:headers:hheader}} Navigate [[..:..:activities:link|back to the activity]] ===== Demonstrating Extensivity in the Fairness Function: Instructor's Guide ===== ==== Main Ideas ==== * Statistical mechanics * Extensive vs. intensive properties * Probabilities in the fairness function ==== Students' Task ==== //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 ==== Props/Equipment ==== * [[:Props:start#whiteboards|Tabletop Whiteboard]] with markers ==== 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. $$ F_{A}=-k\sum_{i} P_{i} \ln P_{i} \; \; \; . $$ $$ F_{B}=-k\sum_{j} P_{j} \ln P_{j} \; \; \; . $$ $$ F_{AB}=-k\sum_{i,j} P_{i}P_{j} \ln P_{i}P_{j} \; \; \; . $$ $$ F_{AB}=-k\sum_{i,j} P_{i}P_{j} (\ln P_{i} + \ln P_{j}) \; \; \; . $$ $$ F_{AB}=-k\left(\sum_{i,j} P_{i}P_{j} \ln P_{i}\right) - k\left(\sum_{i,j} P_{i}P_{j} \ln P_{j}\right) \; \; \; . $$ $$ F_{AB}=-k\left(\sum_{i} P_{i}\ln P_{i}\right)\left(\sum_{j}P_{j}\right) - k\left(\sum_{j} P_{j} \ln P_{j}\right)\left(\sum_{i} P_{i} \right) \; \; \; . $$ $$ F_{AB}=-k\left(\sum_{i} P_{i}\ln P_{i}\right) - k\left(\sum_{j} P_{j} \ln P_{j}\right) \; \; \; . $$ $$F_{AB}=F_{A}+F_{B} \; \; \; .$$ ==== Extensions ====