Actually a quite nice article. After also spending years as a professional physicist not understanding entropy, I finally decided that I was not necessarily the problem, and spent the last 5 years or so trying to understand it better by rewording the foundations with my research group. (I'm one of the papers the author cites is part of a series from our group developing "observational entropy" in order to do so.)
A lot of what makes this topic confusing is just that there are the two basic definitions — Gibbs (\sum p_i log l_i) and "Boltzmann" (log \Omega) — entropy, and they're really rather different. There's usually some confusing handwaving about how to relate them, but the fact is that in a closed system one of them (generally) rises and the other doesn't, and one of them depends on a coarse-graining into macrostates and the other doesn't.
The better way to relate them, I've come to believe, is to consider them both as limits of a more general entropy (the one we developed — first in fact written down in some form by von Neumann but for some reason not pursued much over the years.) There's a brief version here: https://link.springer.com/article/10.1007/s10701-021-00498-x.
This entropy has Gibbs and Bolztmann entropy as limits, is good in and out of equilibrium, is defined in quantum theory and with a very nice classical-quantum correspondence, and has been shown to reproduce thermodynamic entropy in both our papers and the elegant one by Strasberg and Winter: https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuan...
After all this work I finally feel that entropy makes sense to me, which it never quite did before — so I hope this is helpful to others.
p.s. If you're not convinced a new definition of entropy is called for, ask a set of working physicists what it would mean to say "the entropy of the universe is increasing." Since von Neumann entropy is conserved in a closed system (which the universe is if anything is), and there really is no definition of a quantum Boltzmann entropy (until observational entropy), the answers you'll get will be either a mush or a properly furrowed brows.
Probably better not to use the term "steady state" here (even if pretty appropriate) in that the "steady state" cosmological model is/was one that is exponentially expanding, with all physical observables statistically time-independent. It solves Olber's paradox due to the radiation redshift. That model was observationally incorrect, but actually has pretty much been reborn in "eternal inflation" in which the Universe as a whole is in a quasi-exponential state with local regions expanding sub-exponentially like our observable universe.
In either classic steady-state or eternal inflation case, energy conservation is not necessarily a problem: you can have vacuum energy that converts steadily into radiation, while being generated by the expansion.
Quoting Wikipedia:
It is an intrinsic expansion whereby the scale of space itself changes. The universe does not expand "into" anything and does not require space to exist "outside" it. Technically, neither space nor objects in space move. Instead it is the metric governing the size and geometry of spacetime itself that changes in scale
https://en.m.wikipedia.org/wiki/Expansion_of_the_universe
Gravity is not the metric, but it does interact with the metric (well, gravity is what we call it when mass or energy affects space time). The presence of mass/energy causes the shortest distance between two points to no longer be a straight path through space. The path an object takes through space time is always the path with the shortest space time interval among all paths (in that sense the path is “straight” the same way that in flat space, a straight line is the shortest path between two points) — this distance is given by the metric tensor — but gravity makes this path appear curved in space.
When you have a space that isn't just normal globally euclidean space (such as: on the surface of a sphere), the idea of a vector as in like "a direction you can go in and an amount of how much or how fast or whatever" isn't something that makes sense as something independent of a base location. Instead, each point in the space has associated with it a space of "tangent vectors" at that point, and these spaces are related to each other.
The metric tensor associates to each point a "bilinear form" with some properties, essentially a way of doing something like a dot product of two tangent vectors at that same point.
This in turn allows for defining the notion of the length of some curve through the manifold.
The future, literally. To grossly oversimplify: if all of space is east-west, and time is north-south, the Big Bang is the north pole. Only the universe is the map rather than the globe, and the globe doesn’t have to exist for the map to exist and to have the same expansion in space (/longitude) with respect to time (/latitude).
Also there may or may not be a Big Crunch/south pole, this is all just a way to get into the nature of the geometry by way of a convenient frame of reference.
For those who happen to like both geometric algebra and general relativity, my former student Joey Schindler wrote this quite beautiful paper extending geometric algebra to calculus on manifolds. https://arxiv.org/pdf/1911.07145.pdf
This gets directly to my main question each time I see geometric algebra show up: how does it fit in with the "normal" notation of differential geometry? What assumptions does it make for a metric, what is the equivalent of parallel transport, Lie brackets, how does it represent gradients (and other things that are naturally 1-forms), etc., etc.?
All the treatments I've seen jump in to manipulation without really going in to the axioms used. That paper seems to do a lot better at fitting it together, so I'll certainly read it, thanks.
A real challenge for prediction markets (with real-money stakes) will be to get sufficient liquidity on relatively small-ticket items outside of sports and politics.
For such items platforms like metaculus.com and Good Judgement seem more likely to provide more useful predictions, but without the ability to hedge against events that prediction markets have.
The plan calls for ultra-reflective coatings that are tuned to the wavelength of the laser. Apparently these exist, though I have not checked the numbers. In my class I abandoned laser propulsion quickly for just this reason (incinerated the target), but with these coatings it might work.
This is a pretty cool initiative — I looked into beamed propulsion a bit while teaching a course this past fall, and it seems to me that if we (or human technologies) are going to reach a star in our lifetimes, this is by far the most likely way. Still very challenging though.
For a thorough, if somewhat outdated, treatment of the “starwisp” idea using microwaves rather than optical/IR lasers, see this paper: http://path-2.narod.ru/design/base_e/starwisp.pdf by Robert Forward.
To poll the success of this overall endeavor, as well as start to make predictions about which components will/won’t work, Metaculus is launching a series of questions —check it out if you have expertise or opinion: http://www.metaculus.com/questions/#/?order_by=-publish_time
I'm disappointed that more people don't mention Robert Forward when this idea is being covered in the media. (Probably because his Usenet posts, papers, and books predate the World Wide Web.)
This is a pretty cool initiative — I looked into beamed propulsion a bit while teaching a course this past fall, and it seems to me that if we (or human technologies) are going to reach a star in our lifetimes, this is by far the most likely way. Still very challenging though.
To poll the success of this overall endeavor, as well as start to make predictions about which components will/won’t work, [Metaculus](http://www.metaculus.com/questions/#/?order_by=-publish_time) is launching a series of questions —check it out if you have expertise or opinion.
A lot of what makes this topic confusing is just that there are the two basic definitions — Gibbs (\sum p_i log l_i) and "Boltzmann" (log \Omega) — entropy, and they're really rather different. There's usually some confusing handwaving about how to relate them, but the fact is that in a closed system one of them (generally) rises and the other doesn't, and one of them depends on a coarse-graining into macrostates and the other doesn't.
The better way to relate them, I've come to believe, is to consider them both as limits of a more general entropy (the one we developed — first in fact written down in some form by von Neumann but for some reason not pursued much over the years.) There's a brief version here: https://link.springer.com/article/10.1007/s10701-021-00498-x.
This entropy has Gibbs and Bolztmann entropy as limits, is good in and out of equilibrium, is defined in quantum theory and with a very nice classical-quantum correspondence, and has been shown to reproduce thermodynamic entropy in both our papers and the elegant one by Strasberg and Winter: https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuan...
After all this work I finally feel that entropy makes sense to me, which it never quite did before — so I hope this is helpful to others.
p.s. If you're not convinced a new definition of entropy is called for, ask a set of working physicists what it would mean to say "the entropy of the universe is increasing." Since von Neumann entropy is conserved in a closed system (which the universe is if anything is), and there really is no definition of a quantum Boltzmann entropy (until observational entropy), the answers you'll get will be either a mush or a properly furrowed brows.