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Maximizing the Lagrangian: Instructor's Guide

Main Ideas

  • This activity may be the first time students are seeing a Lagrangian.
  • Demonstrate the usefulness of the method of Lagrange Multipliers to acquire additional information about a system
  • Provide students the opportunity to manipulate the fairness function
  • Help define the Partition function

Students' Task

Estimated Time: 30 minutes

Students are given a lagrangian with a probability and energy constraint, written as

$$ L = -k_{B}\sum_{i}P_{i}\ln P_{i} + \alpha k_{B}\left(1-\sum_{i}P_{i}\right)+\beta k_{B}\left(U-\sum_{i}P_{i}E_{i}\right) \; \; \; , $$

and derive as much information as possible about the probability $P_{i}$ by using the problem's constraints. At the end of the activity, the partition function can also be defined.

Prerequisite Knowledge

  • Experience with calculus, particularly maximizing a function by taking derivatives.
  • An ability to interpret and manipulate summation notation.
  • Any prior experience with Lagrangians is useful.
  • Prior exposure to the fairness function is also recommended.


Activity: Introduction

It is recommended that the lecture on the method of Lagrange multipliers be presented before having students do this activity. After the class has been introduced to the method of Lagrange multipliers, place students into small groups and write on the board the Lagrangian with the energy constraint included (see Student's Task section for equation). Ask the groups to maximize the Lagrangian and see what kind of relations they can find using the probability and energy constraints.

Activity: Student Conversations

Activity: Wrap-up

Typically, students will have less trouble using the probability constraint than the energy constraint. After deriving the Lagrangian, most groups will have

$$P_{i}=e^{-1- \alpha - \beta E_{i}} \; \; \; . $$

Now, add a sum over i to both sides and move the exponential to the other side to show the class that

$$e^{1+\alpha}=\sum_{i}e^{\beta E_{i}} \; \; \; . $$

This is the perfect time to define the partition function to the class as

$$Z \equiv \sum_{i}^{All \: states}e^{\beta E_{i}} \; \; \; , $$

and show that this can be alternately written through manipulation as

$$P_{i}=\frac{e^{-\beta E_{i}}}{Z} \; \; \; , $$

which can be put into words as

$$P_{i}=\frac{Boltzmann \: factor}{Partition \: function} \; \; \; . $$

Now,the energy constraint equation, written as

$$U=\sum_{i} P_{i}E_{i} \; \; \; , $$

can be rewritten with the new expression for the probability as

$$U=\sum_{i} \frac{E_{i} e^{-\beta E_{i}}}{Z} \; \; \; . $$

At this point, the students now have expressions for both the probability and the internal energy of an arbitrary system in terms of the Lagrangian. However, point out to the students that we have not yet found what $\beta$ is and must still complete this task.


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