The more familiar transitions we are familiar with like liquid water into solid water happen at constant temperature.
However, other types of phase transitions we are less familiar in our daily lives happen across a continuum of such "state variables", notably:
- superfluidity and its related manifestation, superconductivity
- ferromagnetism
Reaches 2 mK[ref]. youtu.be/upw9nkjawdy?t=487 from Video "Building a quantum computer with superconducting qubits by Daniel Sank (2019)" mentions that 15 mK are widely available.
Used for example in some times of quantum computers, notably superconducting quantum computers. As mentioned at: youtu.be/uPw9nkJAwDY?t=487, in that case we need to go so low to reduce thermal noise.
For scales from absolute 0 like Kelvin, is proportional to the total kinetic energy of the material.
This section is more precisely about classical mechanics.
Boooooring. Except for Navier-Stokes existence and smoothness. That's OK.
A report on the Navier-Stokes Problem by Vladimir Šverak
. Source. 2025.As mentioned on the Wikipedia page en.wikipedia.org/w/index.php?title=Stationary_Action_Principle&oldid=1020413171, "principle of least action" is not accurate since it could not necessarily be a minima, we could just be in a saddle-point.
Calculus of variations is the field that searches for maxima and minima of Functionals, rather than the more elementary case of functions from to .
These are the final equations that you derive from the Lagrangian via the Euler-Lagrange equation which specify how the system evolves with time.
In the case of field however, we can expand the Lagrangian out further, to also integrate over the space coordinates and their derivatives.
Since we are now working with something that gets integrated over space to obtain the total action, much like density would be integrated over space to obtain a total mass, the name "Lagrangian density" is fitting.
Of course, if we were to write it like that all the time we would go mad, so we can just write a much more condensed vectorized version using the gradient with :
And in the context of special relativity, people condense that even further by adding to the spacetime Four-vector as well, so you don't even need to write that separate pesky .
The main point of talking about the Lagrangian density instead of a Lagrangian for fields is likely that it treats space and time in a more uniform way, which is a basic requirement of special relativity: we have to be able to mix them up somehow to do Lorentz transformations. Notably, this is a key ingredient in a/the formulation of quantum field theory.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact