Microstates

Microstates are the different possible ways a system can achieve a particular macrostate.

To illustrate this, we build a game with coins (heads and tails) where for a set number of coins, macrostate is defined by the number of coins that are tails. If we have several coins, there is more than one way to get a macrostate, and each of these ways is a microstate. When making this game, we have a requirement that for all microstates, the probability of getting any microstate is equal to the probability of getting any other microstate. With this being true, the probabilities of the different macrostates correspond to the numbers of microstates. For example, if there are four microstates for a system and two of them correspond to macrostate “A”, then we know macrostate “A” has half of the probability. You will see this in the example below.

For the flips of coin (H for heads, T for tails) where we have two coins, we have three macrostates:

  • H,H (zero of T)
  • H,T (one of T)
  • T,T (two of T)

We have four microstates (H,H), (H,T), (T,H), (T,T)

  • H,H has one microstate (H,H) (25% of probability)
  • H,T has two microstates (H,T) (T,H)(50% of probability)
  • T,T has one microstate (T,T) (25% of probability

Doing it with three flips might be a good homework problem. Let’s do it next with four flips:

  • H,H,H,H (zero of T)
    • one microstate (H,H,H,H) (6.25% of probability)
  • H,H,H,T (one of T)
    • four microstates (H,H,H,T) (H,H,T,H) (H,T,H,H) (T,H,H,H) (25% of probability)
  • H,H,T,T (two of T)
    • six microstates (H,H,T,T) (H,T,H,T) (H,T,T,H) (T,H,H,T) (T,H,T,H) (T,T,H,H) (37.5% of probability)
  • H,T,T,T (three of T)
    • four microstates (H,T,T,T) (T,H,T,T) (T,T,H,T) (T,T,T,H) (25% of probability)
  • T,T,T,T (four of T)
    • one microstate (T,T,T,T) (6.25 % of the probability) (6.26% of probability)

We could do this another way, instead of using H and T, use 0 and 1 and let our combinations come from binary numbers. We can then use a computer program to make all the microstates and count which ones are in which macrostate. We can have the computer scan through each number and count the number of 1’s to determine which macrostate that particular microstate is in.

For example, 0110 and 0101 are two different microstates in the macrostate that gets a score of “2”.

  1. 0000 “0”
  2. 0001 “1”
  3. 0010 “1”
  4. 0011 “2”
  5. 0100 “1”
  6. 0101 “2”
  7. 0110 “2”
  8. 0111 “3”
  9. 1000 “1”
  10. 1001 “2”
  11. 1010 “2”
  12. 1011 “3”
  13. 1100 “2”
  14. 1101 “3”
  15. 1110 “3”
  16. 1111 “4”

“0” – 1 time; “1” – 4 times; “2” – 6 times; “3” – 4 times; “4” – 1 time

You can probably figure out the answers for 3 flips fairly quickly. When it comes times to test the computer program, we might agree that if it correctly answers the scenarios for 2, 3 and 4 flips, we will trust what it says for scenarios with even more flips.

We have a reason for counting “1” for the scoring: the computer might convert “00000101” to “101”, but it will still correctly score it as a “2”.