# Evolving statistical systems: application to academic courses

## TL;DR

This paper models the evolution of academic course enrollments using statistical mechanics principles, comparing expected and random evolutions, and applies the model to real data to analyze course dropout patterns.

## Contribution

It introduces a novel application of statistical mechanics to academic course data, modeling student dropout as an evolving statistical system similar to radioactive decay.

## Key findings

- Evolving statistical systems of courses resemble radioactive decay models.
- Estimated mean lifetimes range from 0.37 to 27.73 years.
- Half-life times vary from 0.25 to 19.22 years, influenced by data completeness.

## Abstract

Statistical systems are conceived from the standpoint of statistical mechanics, as made of a (generally large) number of identical units and exhibiting a (generally large) number of different configurations (microstates), among which only equivalence classes (macrostates) are accessible to observations. Further attention is devoted to evolving statistical systems, and a simple case including only a possible event, E, and related opposite event, $\neg$E, is examined in detail. In particular, the expected evolution is determined and compared to the random evolution inferred from a sequence of random numbers, for different sample populations. The special case of radioactive decay is considered and results are expressed in terms of the fractional time, $t/\Delta t$, where the time step, $\Delta t$, is related to the decay probability, $p=p(\Delta t)$. An application is made to data collections from selected academic courses, focusing on the extent to which expected evolutions and model random evolutions fit to empirical random evolutions inferred from data collections. Results could be biased by the assumed number of students who abandoned their course, defined as suitable impostors (SI). Extreme cases related to a lower and an upper limit of the SI number are considered for a time step, $\Delta t=(1/12)$y, where fitting expected evolutions relate to $0.003\le p\le0.200$. In conclusion, evolving statistical systems made of academic courses are similar to poorly populated samples of radioactive nuclides exhibiting equal probabilities, $p$, and time steps, $\Delta t$, where inferred mean lifetimes, $\tau$, and half-life times, $t_{1/2}$, range within $0.37<\tau/{\rm y}<27.73$ and $0.25<t_{1/2}/{\rm y}<19.22$, respectively, and upper limits are related to incomplete data collections.