PHY3703 Assessment 2 Solutions Year 2026

UNIVERSITY OF SOUTH AFRICA (UNISA)
College of Science, Engineering and Technology



⋄



ASSESSMENT 2
PHY3703: Statistical Mechanics


⋄




Module Code: PHY3703

Module Name: Statistical Mechanics

Assignment No.: Assessment 2

Due Date: 2026




Submitted in partial fulfilment of the requirements for PHY3703
at the University of South Africa.

, UNISA | PHY3703 Assessment 2



Problem 3.20: Uncertainty (Entropy)

The Shannon entropy (or uncertainty) of a probability distribution is a fundamental quan-
tity in statistical mechanics, measuring the degree of ignorance about the state of a system
(Mandl, 1988). It is defined as:


X
S=− Pn ln Pn
n


where Pn is the probability of the n-th outcome.


1
3.20(a) Uncertainty for P1 = P2 = 2


When both outcomes are equally probable, the entropy is calculated as follows:



   
1 1 1 1
S=− ln + ln
2 2 2 2


 
1
S = − ln
2



S = ln 2 ≈ 0.693


This result represents the maximum uncertainty for a two-outcome system, since neither out-
come is preferred over the other (Kittel and Kroemer, 1980:28).


3.20(b) Uncertainty for P1 = 15 , P2 = 4
5


With unequal probabilities, the entropy becomes:



   
1 1 4 4
S=− ln + ln
5 5 5 5

Expanding each term:


  
1 4 4
S=− · (− ln 5) + · ln
5 5 5

Page 1 of 12