1. The Heart of Statistical Mechanics
Probability
By: Jodi Schmelz

2. Statistical Mechanics
Key link between Quantum Mechanics and Thermodynamics
Quantum Mechanics-microscopic level
Thermodynamics-macroscopic level
Statistical mechanics relates the microscopic and macroscopic properties through the Boltzmann equation
Probability and statistics is the heart of statistical mechanics

3. The Link The Boltzmann Equation is the link S=k ln W S is entropy
k is Boltzmann’s constant
W is the number of configurations of the ensemble

4. Statistical Mechanics and Dice
If two dice are rolled and there is no distinction between them the outcome is called a combination
A permutation is when the two dice are distinct from one another
is different from
A microstate is one arrangement of the dice or one permutation
A configuration is the collection of all equivalent microstates and is defined as “W”

5. Probability
Probability (P): is a number between zero and one which tells the chance of a certain result or outcome occurring
Probability is determined by the ratio of how many times an outcome can occur and how many possible outcomes there are.

6. Rolling Dice
Let’s compare the probability of rolling a 2 to the probability of rolling a 7
There are 36 possible outcomes when two dice are rolled.
There is only one way to roll a 2
P = 1/36 = 0.028
There are 6 possible ways to roll a 7
P = 6/36 = 0.167
(graph on next slide)

7. The Odds of Rolling A Number With Fair Dice

8. Loaded Die
There is no longer an equal likelihood for each of the six outcomes
One of the outcomes is favored more then the others
A weight function is used to describe how each outcome is favored

9. Loaded Die (cont.) Example of a Weight Function w(n) = 1/n
If one die was being rolled the probability of rolling a 6 would be determined by:
The probability of rolling a one would be:

10. Yes That’s Cheating
When the loaded die is characterized by the weight function w(n)=1/n, the probability of rolling a 1 is six times greater than rolling a six.
0.408/0.068=6
Depending on how the dice is loaded the probability of rolling a specific number increases
(Graph on next slide)

11. Fair Dice vs. Loaded Dice

12. References
Dr. Darin J. Ulness. “Physical Chemistry 351 Notes.” (Concordia College, Moorhead, MN 2005)
Dr. Darin J. Ulness. “Honors Chemistry 137 Notes.” (Concordia College, Moorhead, MN 2005)
Statistics of Dice Throw at: