Entropy Explained, With Sheep

From rearrangements to the 2nd law of Thermodynamics

By an idea of Aatish Bhatia, adapted by Leonardo Salicari

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Each section is delimited by an horizontal line

Why does this gif looks natural?

Ice melting in a glass — Image: Moussa / Public Domain

While this one seems to be off?

Ice melting in a glass, played in reverse

Thermodynamic approach: 2nd Law

\inline \Delta S \geq 0 \quad\text{for an Isolated system}

Learning objectives

> Entropy interpretation
> Explain why Entropy always increases

Sheep grazing in lands and the Einstein model

Say we’ve got three sheep. These sheep are shuffling about in a farm, pretty much at random. And this farm is split into three plots of lands.

Did you find all 10 arrangements? If you missed any, click here to see them all.

Ok, here are all 10 arrangments:

But how many arrangements could we obtain increasing the numbers of sheep (energy packets) and/or the number of lands (energy levels)?

The parallelism:

molecules inside a balloon — The entropy of the air inside a balloon counts all the ways that the air molecules can be arranged while maintaining the same overall temperature, pressure, and volume.

Entropy is the measure of many ways you can rearrange the microscopic variables of a system while keeping its macroscopic state unchanged

What happens when you place together two solid, one warmer than the other?

As sheep move across lands with no driving force, energy packets have no preferential direction on which energy level to occupy

Each energy level is equally probable

Question:

What will it be the distribution of arrangements of sheep (energy packets) in the upper land (warmer solid)?

Give your answer following this link. Code: SAMUAI

Simulation

Final distribution

Comparing a low and high entropy state — There are fewer arrangements where the sheep (energy) are concentrated, and more arrangements where the sheep (energy) are spread out.

Since energy packets has driving force, there is an higher probability to find the system in the state with more possible arrangements i.e. the one where energy is equipartitioned between the two solids

States with highest number of arrangements (Entropy) are the most probable

Recap

> Entropy measure the number of ways you can rearrange the microscopic variables of a system while keeping its macroscopic state unchanged
> The macrostate obtainable from the highest number of possible microscopic arrangements is the most probable one
> Therefore, Entropy is far more likely to increase

Addendum: the Thermodynamic Limit

Or: is really the highest entropy state that far more probable than the others?

Energy Packets:

Atoms:

The odds of finding the system in the least likely state are about 1 in

graphs of entropy as number of particles increase — As you add more pieces to your system, its entropy graph becomes steeper and steeper. So you're increasingly likely to find it at a state near the peak.

Thank you for listening!

Please have a look to the amazing, original, blog post by Aatish Bhatia. His post is the self consistent explanation of the topic for a general, non-expert, audience. Conversely, this page (which source code can be found on my repo) is meant to be experience together with an oral presentation of the subject. Moreover, it demands some prior mathematical knowledge because is meant for undergraduates or final high-school students.

Hope you had fun in this micro-lecture, and now a message from the original author together with some additional, and very interesting, resources:

Thanks for checking out this post!

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Learn More

The model of a solid described in this post was first proposed by none other than Albert Einstein in 1907.

If you're interested in learning about entropy at a more mathematical level, I recommend the excellent textbook Thermal Physics by Daniel Schroeder. This post is my take on the ideas in Chapter 2 of his book (and also summarized in this paper).

It's a short step to go from understanding entropy to understanding temperature at a more fundamental level. I explored that link in my 2013 post, What the Dalai Lama can teach us about temperatures below absolute zero.

If you want to learn more about the arrow of time, pick up Sean Carroll's 2010 book From Eternity To Here, or watch one of his many excellent talks on the subject. His latest book is The Big Picture.

Huge thanks to Yusra Naqvi, Upasana Roy, Jonathan Zong and Nicky Case for their helpful feedback.

Remix

This post is released in the public domain, under the Creative Commons Zero license. That means you can distribute, remix and adapt it however you like, without asking for permission. (Although if you do something cool with it, I'd love to hear from you.) Attribution isn't legally required, but it's always appreciated.

Spot a typo? You can fix it here.

Here's the source code, if you're into that. The interactives were built using p5.js and plotly.js