Lab: Entropy

Goal: Measure the Entropy of a Set of Coins.

A microstate describes the location, orientation, direction and speed of motion of each member of the system. A macrostate describes the overall system or average properties of the system taken as a whole. A microstate description could be

1. penny #1: heads
2. penny #2: tails
3. penny #3: tails
4. penny #4: heads
5. penny #5: heads
6. penny #6: heads

A macrostate description would be "2/3 of the pennies in the system are heads up".

Some macrostate and microstate descriptions are equivalent: "All pennies are heads up". When a macrostate describes few micro-states, we say that the system has low entropy; if the macrosystem describes many micro-states, the macrosystem has high entropy.

If our purse can contain six pennies, then the most possible configurations occur if all pennies are present (high entropy), and the least number of configurations occur if no pennies are present (lowest entropy).

Materials

• Paper
• Twenty pennies
• Twenty-sided die.

Procedure

1. For two pennies, there are four possible micro-states: HH HT TH TT.
2. There are three macro-states (2H, 0T), (1H, 1T), (0H, 2T). There is only one microstate for macro-states (2H, 0T) and (0H, 2T). There are two micro-states for macrostate (1H, 1T) [HT, TH]. The multiplicity Ω of the macrostate is the number of micro-states, so (1H, 1T) has a multiplicity Ω of two. [Note that the multiplicity of a microstate is always 1, and therefore we only talk about the entropy of macro-states.]
3. The entropy S of a given macrostate is the natural log of the number of micro-states Ω it has: S = ln Ω.
4. The probability that a randomly selected microstate will occur depends on the total multiplicity of the system (Σ (Ω_n).
5. Create a table showing the entropy for each of the three macro-states.

   State	Multiplicity Ω	Entropy ln[Ω]	Probability Ω/Ωtotal
   S(2H, 0T)
   S(1H, 1T)
   S(0H, 2T)

6. Create another table for the case of three coins. There are four possible micro-states.
7. Create another table for the case of four coins. There are five possible micro-states.
8. Lay out 20 coins heads up, five coins in each of four rows. Number the positions 1-5, 6-10, 11-15, 16-20.
9. Roll the 20-sided die and flip the penny that corresponds to the position indicated by the number on the die. Record the number of tails.
10. Continue until you've done 80 flips (depending on the die rolls, you may not flip every coin).
11. Plot the number of heads as a function of the number of flips. What is the shape of your curve?
12. Assume that you have a macrostate of (15H, 5T).
    1. How many different results from the die roll will lead to a macrostate of 14H, 6T?
    2. How many different results from the die roll will lead to a macrostate of 16H, 4T?
    3. Which outcome is more likely?

Report

1. Write up your results, including your tables.

Post your report to the Lab assignment link at the Moodle.

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