14 ms·

What Makes the Hardest Equations in Physics So Difficult? (2018)

119 points by c89X 7y ago

HenryKissinger 7y ago
"When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first."
- amelius 7y ago
  Meaning that the latter is an emergent property, while the former is not?
  - pretzell 7y ago
    Possibly, but I either interpreted, or heard this somewhere before, that turbulence is way harder to make sense of and predict (?). Not a physicist, but essentially that both are nuts, but that turbulence is even nuttier
Causality1 7y ago
Because we're drilling down into finer and finer descriptions. If there was a black hole in earth orbit where it could be closely observed I'm sure we'd find small details of its behavior and the behavior of dust crossing its event horizon that require difficult math to explain.
quantum_state 7y ago
From the perspective of trying to get to the actual physical behavior of such systems, the exercise is purely theoretical since we know there are other mechanisms at work to prevent the “ blow up” ...
- tom-thistime 7y ago
  (I'm not a fluid flow person and this is probably not a very good comment.) Yeah, that seems likely. But sometimes it's difficult to draw the line between issues that are "purely theoretical" and issues that make a practical difference. Conceivably at some scale the Navier-Stokes equations match up with molecular dynamics simulations in a way that is illuminating. Then in that situation maybe the problems with the N-S equations become nice indicators of what's going on.
- steve76 7y ago
  If you are really good at the circumference of the flow at the boundary layer, you should be able to interface with it and have control all up and down the line. Instead of pressurizing an entire water network at 15 psi, just detect when a home faucet is turn on, and hit a tank or pipe at the right spot with a hammer. Take that concept to the human circulatory system, and start pinpoint drug treatments at just the right place. If you can say what's inside that little mass cube, either matter or empty space, throw EM out the window. Just acoustics and matter waves, and to some degree everything is just the Sun. No more little tiny cans brought to us by Boltzmann. No more dimples on a golf ball stretching and squeezing with everything inside having all paths as the fastest path. No more film strips at the speed of light, looking for symmetry between two frames that we can flip around with nothing knowing the difference. No more thermodynamic engine with an infinite thermal reservoir and the smallest interface to it resulting in something mechanical. You would give incredible confidence to humans and effect directly our collective knowledge, likely close any future empirical pursuit. As big as the universe is now, you will shrink it and close it down, effectively saying we are equally inside a singularity as we living in sunshine. You will fix the Planck constant from all future debate and even fix the natural units, closing down science. New technologies and medical treatments would be hasten with no ethical question. All efforts on thought and discovery would end and would only focus on human comfort and decency.
ocfnash 7y ago
Note that finite-time singularities for the good old Newtonian n-body problem have been known to exist for quite a while. (See for example [1].) It's curious to think that a mathematical phenomenon like this can hint at new physics. [1] Saari, D. and Xia, Z., "Off to Infinity in Finite Time", https://www.ams.org/notices/199505/saari-2.pdf https://www.ams.org/notices/199505/saari-2.pdf
- TheFiend7 7y ago
  Honestly I feel as though that's one of the most incredible aspects of theoretical physics I've seen. The fact that often times these underlying physics are danced around and even though they're unknown, we can tell something is not quite right. I believe this guy even talks about it a bit in this video made by the Royal Institution (which I really enjoy watching), discussing the possibility of different fundamental building blocks of nature. https://www.youtube.com/watch?v=zNVQfWC_evg https://www.youtube.com/watch?v=zNVQfWC_evg
- xelxebar 7y ago
  That paper is from 1993. Hard to believe that it took such a long time to find a concrete pathology in a Newtonian system. Now the question is whether these blowups only occur on initial condition sets of zero measure. John Baez also has a nice, accessible series of articles called "Stuggles with the Continuum" [0]. As a side product, it gives a nice perspective on the development of modern physics as a series of attempts to fix these infinities (only to create more subtle one). [0]:https://johncarlosbaez.wordpress.com/2016/09/08/struggles-with-the-continuum-part-1/ https://johncarlosbaez.wordpress.com/2016/09/08/struggles-wi...
bnjmn 7y ago
One of my favorite takes on this topic: "From... elementary theories we build up descriptions of more and more complex systems. But in all these efforts we take for granted that we may use any language we wish and as many [languages] as necessary. That is, we choose whatever mathematical formalism is most useful and then interpret the symbols and measurement operations in very highly developed natural language. To a large degree, the simplicity of natural laws arises through the complexities of the languages we use for their expression." – H. H. Pattee
- mr_overalls 7y ago
  I did an undergrad in physics decades ago, and it pleases me to no end to see that someone else has made this observation. Especially when dealing with quantum weirdness, the advice of professors to focus "follow the math" always seem to gloss over the extensive interpretation (Copenhagen and otherwise) that gave context and meaning to the math & experimental results.
  - guygurari 7y ago
    In my experience, the reason to “follow the math” is that it is the best and perhaps only way to truly understand a physical theory, especially one as strange as quantum mechanics. One can understand the math first, and then develop an understanding of the context and meaning. But this abstract understanding is anchored in math. This is important because when the meaning gets too obscure, one can return to the math to resolve any confusion. The other way around does not work.
    - hutzlibu 7y ago
      "This is important because when the meaning gets too obscure, one can return to the math to resolve any confusion. The other way around does not work." Note: my math is not very advanced, at least not good enough to understand quantum mechanics But why does it not work the other way? When the math tells me something very wrong, can't the conctext and meaning show where the math modell is wrong? As far as I understood, every physical modell is only a limited model of reality, so they all have flaws. Meaning the math can be wrong when applied to reality, which one could spot, with the understanding of reality?
farisjarrah 7y ago
As a non-mathematician, I found the Numberphile video on YouTube very helpful for breaking down the basics of Navier-Stokes equations in a very easy to comprehend way: https://www.youtube.com/watch?v=ERBVFcutl3M https://www.youtube.com/watch?v=ERBVFcutl3M
RhysU 7y ago
Big whorls have little whorls Which feed on their velocity, And little whorls have lesser whorls And so on to viscosity. - Lewis Fry Richardson, 1922
mannykannot 7y ago
Given that the equations involve continuous functions of some bulk properties of a fluid, while the fluid itself is made up of discrete particles, is there anything fundamentally or metaphysically problematical if these blow-ups are unavoidable? Or is it about finding practical ways to avoid them when modeling real-world scenarios?
- lgeorget 7y ago
  I guess that it's just that it would be super convenient if the equation that work for a fluid would work for any part of the fluid, however small it is.
- steve76 7y ago
  I don't think the fluid is made up of discrete particles as far as pressure goes. The discrete particles are covalent bonded. If they blew up, I think you could put an elbow on a copper pipe, rotate it just the right way with fluid flowing through it and get a voltage. Instead you get cross flow and cavitation. Perhaps it does happen, and it can't be detected due to matter waves or not being able to accurately trace the path of "something thermal".
- treeman79 7y ago
  Sounds sort of like how they found the plank constant
mkagenius 7y ago
Is it like purely random event - akin to butterfly effect and its mostly indeterminate ahead of time? (I am not an expert)
- mrob 7y ago
  "Butterfly effect" refers to chaotic systems, i.e. those where the outcome is highly sensitive to the starting position. They are deterministic, not random, but they are hard to predict because this sensitivity means any small errors in simulation can give completely wrong results.
dang 7y ago
Discussed at the time: https://news.ycombinator.com/item?id=16159744 https://news.ycombinator.com/item?id=16159744
davidw 7y ago
Since these equations are something I've never studied and certainly don't hear about on a day to day basis, the thing they bring to mind is the scene in Cryptonomicon when Waterhouse doesn't get that a simple question on an army test is actually a simple question and starts in on some complex math. It's an entertaining read.
- waterhouse 7y ago
  [The Navy] gave [Lawrence Waterhouse] an intelligence test. The first question on the math part had to do with boats on a river: Port Smith is 100 miles upstream of Port Jones. The river flows at 5 miles per hour. The boat goes through water at 10 miles per hour. How long does it take to go from Port Smith to Port Jones? How long to come back? Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier-Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn’t prove his intelligence, what would? Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal. Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.
  - umvi 7y ago
peteradio 7y ago
Doesn't this point to some renormalization strategy being necessary? Or at least some correction to how renormalization is being applied if its leading to subspace infinities. Not sure how mathematicians view renormalization when you are trying to ask questions about indefinitely far distances into the future but it seems like there needs to be some necessary coarseness which suppresses the infinities, not sure why those terms wouldn't naturally arise.
Koshkin 7y ago
I wonder what a good textbook on theoretical physics would look like if it assumed all the math involved as a prerequisite. (Say, a 10-volume set "compressed" into, what, 300 pages still containing a complete treatment of the subject?)
- yummypaint 7y ago
  Jackson's electrodynamics is probably the closest thing to this I've seen. He often mentions offhand that it's trivial to show one thing or another, but when you actually do the derivation it's many pages.
ncmncm 7y ago
If you think the Navier-Stokes equations are hard, try the version that plasma physicists have to use, instead. Rather than confront those, cosmologists have chosen to pretend that, while every single thing they can see is plasma (excepting, uniquely, planets), none of it does anything plasma-ish. Since plasma physics is scale-invariant, freaky phenomena seen in labs should be playing out at stellar, galactic, and super-cluster scale. If they don't, it needs explanation why not. Huge props to solar physicists, who confront plasma physics, face-to-face, daily.
- xelxebar 7y ago
  I picked up a copy of Chandrasekar's "Hydrodynamic and Magnetodynamic Stability" a while back and started working through it. The basics are relatively approachable, but dang the differential equations can get gnarly. It feels like a large portion of the art is in asking the right question that can aviod toppling into a mess of non-linearities. Really good book, as far as I got through it. The beginning has a nice derivation of Navier-Stokes, which was mew to me.
  - ncmncm 7y ago
    The mess of non-linearities is where the action is, at least in nature. Designing systems to stay out of that mess is, as you say, art.
aSplash0fDerp 7y ago
From a laymens POV, it seems like half of the equation would represent the energy used in the bonds of each molecule representing smoothness (almost moving as a solid), with the other half laying out a dynamic list of finite variables that disrupt or exceed the energy forming the bonds. Its almost like they would have to define a theoretic limit similar to 9.808175174 m/s^2 to limit the finite possibilities of outside forces acting on the molecules before the molecules themselves become the outside force, with those that exceed the limits be classified with a custom equation, and those below the threshold fitting nicely in a bow wrapped package. And just to add more SWAG, is it going to be easier to produce these finding/understandings in space, where theoretic limits are more well defined and we'll be forced to use custom equations within our atmosphere looking for an answer from a perspective that is environmentally more complex? Its all greek to me... https://en.m.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations https://en.m.wikipedia.org/wiki/Navier%E2%80%93Stokes_equati...
- aSplash0fDerp 7y ago
  OK. I was thinking more of a Newtons apple moment in space, where someone with a syringe and fluid may make an elegant observation on blow-up or collapse. Eddys in a river with more eddys creating eddys is a dizzying theory though.
bawana 7y ago
Physicists who like to preserve their jobs. If everyone taught like grant Sanderson there would be too much understanding.

119 points by c89X 7y ago

"When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first."

amelius 7y ago
Meaning that the latter is an emergent property, while the former is not?
- pretzell 7y ago
  Possibly, but I either interpreted, or heard this somewhere before, that turbulence is way harder to make sense of and predict (?). Not a physicist, but essentially that both are nuts, but that turbulence is even nuttier

Causality1 7y ago

Because we're drilling down into finer and finer descriptions. If there was a black hole in earth orbit where it could be closely observed I'm sure we'd find small details of its behavior and the behavior of dust crossing its event horizon that require difficult math to explain.

quantum_state 7y ago

From the perspective of trying to get to the actual physical behavior of such systems, the exercise is purely theoretical since we know there are other mechanisms at work to prevent the “ blow up” ...

tom-thistime 7y ago
(I'm not a fluid flow person and this is probably not a very good comment.) Yeah, that seems likely. But sometimes it's difficult to draw the line between issues that are "purely theoretical" and issues that make a practical difference. Conceivably at some scale the Navier-Stokes equations match up with molecular dynamics simulations in a way that is illuminating. Then in that situation maybe the problems with the N-S equations become nice indicators of what's going on.
steve76 7y ago
If you are really good at the circumference of the flow at the boundary layer, you should be able to interface with it and have control all up and down the line. Instead of pressurizing an entire water network at 15 psi, just detect when a home faucet is turn on, and hit a tank or pipe at the right spot with a hammer. Take that concept to the human circulatory system, and start pinpoint drug treatments at just the right place. If you can say what's inside that little mass cube, either matter or empty space, throw EM out the window. Just acoustics and matter waves, and to some degree everything is just the Sun. No more little tiny cans brought to us by Boltzmann. No more dimples on a golf ball stretching and squeezing with everything inside having all paths as the fastest path. No more film strips at the speed of light, looking for symmetry between two frames that we can flip around with nothing knowing the difference. No more thermodynamic engine with an infinite thermal reservoir and the smallest interface to it resulting in something mechanical. You would give incredible confidence to humans and effect directly our collective knowledge, likely close any future empirical pursuit. As big as the universe is now, you will shrink it and close it down, effectively saying we are equally inside a singularity as we living in sunshine. You will fix the Planck constant from all future debate and even fix the natural units, closing down science. New technologies and medical treatments would be hasten with no ethical question. All efforts on thought and discovery would end and would only focus on human comfort and decency.

ocfnash 7y ago

Note that finite-time singularities for the good old Newtonian n-body problem have been known to exist for quite a while. (See for example [1].) It's curious to think that a mathematical phenomenon like this can hint at new physics. [1] Saari, D. and Xia, Z., "Off to Infinity in Finite Time", https://www.ams.org/notices/199505/saari-2.pdf https://www.ams.org/notices/199505/saari-2.pdf

TheFiend7 7y ago
Honestly I feel as though that's one of the most incredible aspects of theoretical physics I've seen. The fact that often times these underlying physics are danced around and even though they're unknown, we can tell something is not quite right. I believe this guy even talks about it a bit in this video made by the Royal Institution (which I really enjoy watching), discussing the possibility of different fundamental building blocks of nature. https://www.youtube.com/watch?v=zNVQfWC_evg https://www.youtube.com/watch?v=zNVQfWC_evg
xelxebar 7y ago
That paper is from 1993. Hard to believe that it took such a long time to find a concrete pathology in a Newtonian system. Now the question is whether these blowups only occur on initial condition sets of zero measure. John Baez also has a nice, accessible series of articles called "Stuggles with the Continuum" [0]. As a side product, it gives a nice perspective on the development of modern physics as a series of attempts to fix these infinities (only to create more subtle one). [0]:https://johncarlosbaez.wordpress.com/2016/09/08/struggles-with-the-continuum-part-1/ https://johncarlosbaez.wordpress.com/2016/09/08/struggles-wi...

bnjmn 7y ago

One of my favorite takes on this topic: "From... elementary theories we build up descriptions of more and more complex systems. But in all these efforts we take for granted that we may use any language we wish and as many [languages] as necessary. That is, we choose whatever mathematical formalism is most useful and then interpret the symbols and measurement operations in very highly developed natural language. To a large degree, the simplicity of natural laws arises through the complexities of the languages we use for their expression." – H. H. Pattee

mr_overalls 7y ago
I did an undergrad in physics decades ago, and it pleases me to no end to see that someone else has made this observation. Especially when dealing with quantum weirdness, the advice of professors to focus "follow the math" always seem to gloss over the extensive interpretation (Copenhagen and otherwise) that gave context and meaning to the math & experimental results.
- guygurari 7y ago
  In my experience, the reason to “follow the math” is that it is the best and perhaps only way to truly understand a physical theory, especially one as strange as quantum mechanics. One can understand the math first, and then develop an understanding of the context and meaning. But this abstract understanding is anchored in math. This is important because when the meaning gets too obscure, one can return to the math to resolve any confusion. The other way around does not work.
  - hutzlibu 7y ago
    "This is important because when the meaning gets too obscure, one can return to the math to resolve any confusion. The other way around does not work." Note: my math is not very advanced, at least not good enough to understand quantum mechanics But why does it not work the other way? When the math tells me something very wrong, can't the conctext and meaning show where the math modell is wrong? As far as I understood, every physical modell is only a limited model of reality, so they all have flaws. Meaning the math can be wrong when applied to reality, which one could spot, with the understanding of reality?

farisjarrah 7y ago

As a non-mathematician, I found the Numberphile video on YouTube very helpful for breaking down the basics of Navier-Stokes equations in a very easy to comprehend way: https://www.youtube.com/watch?v=ERBVFcutl3M https://www.youtube.com/watch?v=ERBVFcutl3M

RhysU 7y ago

Big whorls have little whorls Which feed on their velocity, And little whorls have lesser whorls And so on to viscosity. - Lewis Fry Richardson, 1922

mannykannot 7y ago

Given that the equations involve continuous functions of some bulk properties of a fluid, while the fluid itself is made up of discrete particles, is there anything fundamentally or metaphysically problematical if these blow-ups are unavoidable? Or is it about finding practical ways to avoid them when modeling real-world scenarios?

lgeorget 7y ago
I guess that it's just that it would be super convenient if the equation that work for a fluid would work for any part of the fluid, however small it is.
steve76 7y ago
I don't think the fluid is made up of discrete particles as far as pressure goes. The discrete particles are covalent bonded. If they blew up, I think you could put an elbow on a copper pipe, rotate it just the right way with fluid flowing through it and get a voltage. Instead you get cross flow and cavitation. Perhaps it does happen, and it can't be detected due to matter waves or not being able to accurately trace the path of "something thermal".
treeman79 7y ago
Sounds sort of like how they found the plank constant

mkagenius 7y ago

Is it like purely random event - akin to butterfly effect and its mostly indeterminate ahead of time? (I am not an expert)

mrob 7y ago
"Butterfly effect" refers to chaotic systems, i.e. those where the outcome is highly sensitive to the starting position. They are deterministic, not random, but they are hard to predict because this sensitivity means any small errors in simulation can give completely wrong results.

dang 7y ago

Discussed at the time: https://news.ycombinator.com/item?id=16159744 https://news.ycombinator.com/item?id=16159744

davidw 7y ago

Since these equations are something I've never studied and certainly don't hear about on a day to day basis, the thing they bring to mind is the scene in Cryptonomicon when Waterhouse doesn't get that a simple question on an army test is actually a simple question and starts in on some complex math. It's an entertaining read.

waterhouse 7y ago
[The Navy] gave [Lawrence Waterhouse] an intelligence test. The first question on the math part had to do with boats on a river: Port Smith is 100 miles upstream of Port Jones. The river flows at 5 miles per hour. The boat goes through water at 10 miles per hour. How long does it take to go from Port Smith to Port Jones? How long to come back? Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier-Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn’t prove his intelligence, what would? Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal. Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.
- umvi 7y ago

peteradio 7y ago

Doesn't this point to some renormalization strategy being necessary? Or at least some correction to how renormalization is being applied if its leading to subspace infinities. Not sure how mathematicians view renormalization when you are trying to ask questions about indefinitely far distances into the future but it seems like there needs to be some necessary coarseness which suppresses the infinities, not sure why those terms wouldn't naturally arise.

Koshkin 7y ago

I wonder what a good textbook on theoretical physics would look like if it assumed all the math involved as a prerequisite. (Say, a 10-volume set "compressed" into, what, 300 pages still containing a complete treatment of the subject?)

yummypaint 7y ago
Jackson's electrodynamics is probably the closest thing to this I've seen. He often mentions offhand that it's trivial to show one thing or another, but when you actually do the derivation it's many pages.

ncmncm 7y ago

If you think the Navier-Stokes equations are hard, try the version that plasma physicists have to use, instead. Rather than confront those, cosmologists have chosen to pretend that, while every single thing they can see is plasma (excepting, uniquely, planets), none of it does anything plasma-ish. Since plasma physics is scale-invariant, freaky phenomena seen in labs should be playing out at stellar, galactic, and super-cluster scale. If they don't, it needs explanation why not. Huge props to solar physicists, who confront plasma physics, face-to-face, daily.

xelxebar 7y ago
I picked up a copy of Chandrasekar's "Hydrodynamic and Magnetodynamic Stability" a while back and started working through it. The basics are relatively approachable, but dang the differential equations can get gnarly. It feels like a large portion of the art is in asking the right question that can aviod toppling into a mess of non-linearities. Really good book, as far as I got through it. The beginning has a nice derivation of Navier-Stokes, which was mew to me.
- ncmncm 7y ago
  The mess of non-linearities is where the action is, at least in nature. Designing systems to stay out of that mess is, as you say, art.

aSplash0fDerp 7y ago

From a laymens POV, it seems like half of the equation would represent the energy used in the bonds of each molecule representing smoothness (almost moving as a solid), with the other half laying out a dynamic list of finite variables that disrupt or exceed the energy forming the bonds. Its almost like they would have to define a theoretic limit similar to 9.808175174 m/s^2 to limit the finite possibilities of outside forces acting on the molecules before the molecules themselves become the outside force, with those that exceed the limits be classified with a custom equation, and those below the threshold fitting nicely in a bow wrapped package. And just to add more SWAG, is it going to be easier to produce these finding/understandings in space, where theoretic limits are more well defined and we'll be forced to use custom equations within our atmosphere looking for an answer from a perspective that is environmentally more complex? Its all greek to me... https://en.m.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations https://en.m.wikipedia.org/wiki/Navier%E2%80%93Stokes_equati...

aSplash0fDerp 7y ago
OK. I was thinking more of a Newtons apple moment in space, where someone with a syringe and fluid may make an elegant observation on blow-up or collapse. Eddys in a river with more eddys creating eddys is a dizzying theory though.

bawana 7y ago

Physicists who like to preserve their jobs. If everyone taught like grant Sanderson there would be too much understanding.