The Heart of Statistical Mechanics Probability By: Jodi Schmelz

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

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

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”

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.

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)

The Odds of Rolling A Number With Fair Dice

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

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:

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)

Fair Dice vs. Loaded Dice

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: http://hyperphysics.phy-astr.gsu.edu/hbase/math/dice.html