10 ms·

This is such a fun observation. Physicists originally used differential equations to describe large scale, yet fundamentally atomistic, behaviour because, I ass

by tw1010 9y ago

This is such a fun observation. Physicists originally used differential equations to describe large scale, yet fundamentally atomistic, behaviour because, I assume, that was easier to work with using pen and paper than by using some more discrete, computational, atom-by-atom model. And then computers were invented, but we're still stuck with these continuous, legacy, models. Almost no-one questions the fact that the original motivation for using the differential equation technique is no longer appropriate given that the underlying technology has shifted underneath the feet of the modellers.

openasocket 9y ago
Well, we still don't have the computational power to do things atom-by-atom, even in small things like modeling cells, let alone something like simulating wind shear on a plane or predicting weather patterns. It's not just computational power but memory. There's something like 1E14 atoms in a cell. Storing the x,y,z position with 4 bytes for each dimension (though you'll probably want 8 bytes) comes out to 1.2E15 bytes = 1.2 petabytes of information. Add in more fields for velocity and other pieces of state and you're talking about something that would only fit in working memory in some of our largest machines.
- tw1010 9y ago
  Sure, but you'd think we would shift our focus to using models explicitly tailored with that aim in mind. Using something that explicitly approximates a desired, precise, foundation. Perhaps parameterized by model accuracy. I would think taking inspiration from approximation algorithms would be a much more appropriate lens with which to model physics.
  - auntienomen 9y ago
    In the case of Navier-Stokes, this was done in 1934 by Jean Leray, who showed that the so-called 'weak' solutions always exist. (A weak solution is not a function, but a more general object called a distribution which can be fuzzed out on very short distance scales.) It's a very reasonable idea, and useful enough for computations, but it lacks the essential tension that makes Navier-Stokes a mathematical Everest.
  - btilly 9y ago
    The mental model of a bunch of hard little atoms flying around, colliding, rebounding, etc is actually NOT the desired, precise, foundation for understanding how gases behave. The better foundation is quantum mechanics. An example of a macroscopically visible difference between these foundations are the van der Waals forces.
- Retric 9y ago
  Here is a billion particle model (e9) which is a long way from atomic scale simulation (e14+), but not outside the realm of the possible. https://www.youtube.com/watch?v=B8mP9E75D08 https://www.youtube.com/watch?v=B8mP9E75D08
  - openasocket 9y ago
    We are definitely making progress. But another factor to consider, in additional to the number of particles, is the energy of the system. When going through one step of a simulation you have to compute the effects that every particle has on every other particle, but generally a particle only affects the other particles near it. You can take advantage of this fact to speed up computation with the right data structures. However, as particles start moving around faster and faster, the distance they can travel in one time step increases, and the neighborhood of effect increases, limiting the speedup of this optimization. This matters less in larger scale simulations like of weather patterns, because you don't have to worry about air molecules zipping to the other side of the continent in one second. But it matters a lot in small scale simulations, especially in cells. As an example, an average glucose molecule in one of your cells is bouncing around at around 250 miles per hour! That's not 250 miles when scaled up, that's really 250 miles per hour. A molecule in your body is colliding with another billions of times every second. (Source: http://www.righto.com/2011/07/cells-are-very-fast-and-crowded-places.html http://www.righto.com/2011/07/cells-are-very-fast-and-crowde...). At that scale and that level of activity it becomes much harder to simulate each time step.
    - Retric 9y ago
      The other side of this is cells are also really fast. Modeling even 0.01 seconds could be very useful.
      - SAI_Peregrinus 9y ago
        Much of the action inside cells is done via proteins. It's taken a huge amount of distributed computing power just to figure out a reasonable hypothesis for how they fold (only one part of the movement) and €1.22 billion to build the XFEL and have a way to image them on the required timescales: 1-2 femtoseconds. 0.000000000000001 seconds. 0.01 seconds is an eternity, calculating that long would take an utterly enormous amount of computer time.
- electricslpnsld 9y ago
  > Well, we still don't have the computational power to do things atom-by-atom People do atom-by-atom molecular dynamics simulations of proteins and such.
  - openasocket 9y ago
    Certainly, but not at the level of an entire cell
  - effie 9y ago
    Yes, but those are simulations of classical mechanics models of molecules which are known to be very limited when it comes to atom-scale objects. We have another model that was much more useful there (Schroedinger's equation) but it is only computationally tractable for systems of few atoms. We do not have the computational power to work out the consequences of the model for proteins.
- ctdonath 9y ago
  we still don't have the computational power to do things atom-by-atom That's what puzzles me about this discussion. Of course we don't. The goal here is to model more atoms than we have atoms to model with. Until we get to quantum computers that can represent more information per atom, than information per atom we want to represent, it's simply a matter of objectively & obviously inadequate resources. You can't simulate wind shear on a plane when there's more atoms in that wind & plane than there are in the computer simulating them: every single atom being simulated contains the complete information about its state (information independent of any other atom), so simulating it will require at least one atom per atom simulated - if you don't have at least as many atoms to simulate as you are simulating, you can't achieve a complete simulation.
  - evandijk70 9y ago
    It is not necessarily the goal to model more atoms than we have atoms to model with. The goal can be, for example, to model a folding protein, which has fewer atoms than a CPU.
    - njarboe 9y ago
      And one might be happy to spend a month computing the folding while in the cell it takes a few seconds.
  - pcvarmint 9y ago
    That may be true of digital computers, but it's not necessarily true of analog ones.
- RhysU 9y ago
  > Well, we still don't have the computational power to do things atom-by-atom... We still don't have the computational power to do certain classes of continuous Navier--Stokes calculations (i.e. not atom based).
al2o3cr 9y ago
It's unclear whether atomic-level data would be useful in these sorts of calculations - the input data isn't known to sufficient precision. For more, check out the last two paragraphs from the Feynman Lectures Vol III Chapter 2: http://www.feynmanlectures.caltech.edu/III_02.html http://www.feynmanlectures.caltech.edu/III_02.html A similar sort of mechanism led to the discover of chaotic behavior in weather models - checkpointing results at a precision slightly lower than the machine's internal precision caused simulations that were resumed from the checkpoint to deviate rapidly.
srirachahot 9y ago
You are missing the distinction between a numerical solution to a set of equations and an analytical one. Usually, the analytical solution gives you the dynamics of a system at any point in time and space given a set of parameters. For numerical solutions you have to run each individual set of parameters to find the corresponding values in time and space (not even considering stochastic equations). This is very computationally expensive. This is the value in solving these things analytical — hence the prize.
- srirachahot 9y ago
  Sorry this should read: “Usually, the analytical solution gives you the dynamics of a system at any point in time and space given ANY set of parameters.” The “Any” is the value here for the analytical solution.
improbable22 9y ago
Human limitations is one reason, but there's something much deeper going on here. Navier and Stokes worked before we were sure that atoms existed, certainly before we had any idea of how many there were. Nevertheless they were able to write down useful theories for describing fluids. This is how all of science works. The things about which we are totally ignorant are much smaller today, of course... but useful theories of any set of phenomena always omit a great many things we do in fact know about. From these theories, we can understand what's going on, and use this to extrapolate to things we have not seen yet. An atom-by-atom computational model would (in some sense) be no more useful than what we had before N-S, just blind experiment. To try out any given swirl of smoke etc. we can equally well walk next door to the lab and videotape it... but this doesn't help us imagine what else might be possible. The "blowup scenario" discussed is an example of this kind of imagining.
- oldandtired 9y ago
  > The things about which we are totally ignorant are much smaller today, of course One would have to disagree with this idea. The more we study the universe around us, the less we actually know. As a somewhat philosophical point, there are many times when our mental (mathematical) models get in the road of understanding. Very often, people believe that because we have a model that works and appears to give good predictive results about some phenomena then we understand the how and the what (and even the why) of those phenomena. When this happens, we get into a situation where alternative models are actually discouraged. If one looks the the history of the 19th, 20th and 21st centuries, one can see that more and vaster avenues of investigation have arisen as time passes. Our increasing knowledge is continuing to be shown as ever smaller in relation to what we are now seeing. No theory is ever complete, nor is it ever accurate to the extent that it describes the reality of the universe around us. All theories make those simplifying assumptions that when taken too far lead into inaccurately describing and predicting what we should see. Too often people get enamoured by the beauty of the mathematics and forget it is only an attempt at reflecting reality. Mathematics is a magnificent and useful tool, but it is a foolish master. Too often we forget that. Theories and models help us gain some understanding of the nature of the universe around us. This understanding, however, is always subject to change, no matter how "perfect" the theory may appear to be. There are too many scientists, both theoretical and practical, who are so infatuated and enamoured with their current models that they have forgotten that the models are approximations only and are subject to change or even overturning.
contact_fusion 9y ago
Your assumption is frankly incorrect. Physics is generally expressed in terms of differential equations. This is not due to their analytical tractability - as anyone who has attempted to solve PDEs before will know, most (nearly all) differential equations do not yield to analytical solution. Perhaps you think that quantum mechanics demands a discretized view of reality. This would be a complete misunderstanding of quantum mechanics, and physics in general.
adrianratnapala 9y ago
> computational, atom-by-atom model Remember that as long as computers continue to be made atoms, they aren't going to do atom-by-atom simulations, unless these simulated systems are much smaller than the computer itself or other are simplifying assumptions that can be made.

by tw1010 9y ago

openasocket 9y ago

Well, we still don't have the computational power to do things atom-by-atom, even in small things like modeling cells, let alone something like simulating wind shear on a plane or predicting weather patterns. It's not just computational power but memory. There's something like 1E14 atoms in a cell. Storing the x,y,z position with 4 bytes for each dimension (though you'll probably want 8 bytes) comes out to 1.2E15 bytes = 1.2 petabytes of information. Add in more fields for velocity and other pieces of state and you're talking about something that would only fit in working memory in some of our largest machines.

tw1010 9y ago
Sure, but you'd think we would shift our focus to using models explicitly tailored with that aim in mind. Using something that explicitly approximates a desired, precise, foundation. Perhaps parameterized by model accuracy. I would think taking inspiration from approximation algorithms would be a much more appropriate lens with which to model physics.
- auntienomen 9y ago
  In the case of Navier-Stokes, this was done in 1934 by Jean Leray, who showed that the so-called 'weak' solutions always exist. (A weak solution is not a function, but a more general object called a distribution which can be fuzzed out on very short distance scales.) It's a very reasonable idea, and useful enough for computations, but it lacks the essential tension that makes Navier-Stokes a mathematical Everest.
- btilly 9y ago
  The mental model of a bunch of hard little atoms flying around, colliding, rebounding, etc is actually NOT the desired, precise, foundation for understanding how gases behave. The better foundation is quantum mechanics. An example of a macroscopically visible difference between these foundations are the van der Waals forces.
Retric 9y ago
Here is a billion particle model (e9) which is a long way from atomic scale simulation (e14+), but not outside the realm of the possible. https://www.youtube.com/watch?v=B8mP9E75D08 https://www.youtube.com/watch?v=B8mP9E75D08
- openasocket 9y ago
  We are definitely making progress. But another factor to consider, in additional to the number of particles, is the energy of the system. When going through one step of a simulation you have to compute the effects that every particle has on every other particle, but generally a particle only affects the other particles near it. You can take advantage of this fact to speed up computation with the right data structures. However, as particles start moving around faster and faster, the distance they can travel in one time step increases, and the neighborhood of effect increases, limiting the speedup of this optimization. This matters less in larger scale simulations like of weather patterns, because you don't have to worry about air molecules zipping to the other side of the continent in one second. But it matters a lot in small scale simulations, especially in cells. As an example, an average glucose molecule in one of your cells is bouncing around at around 250 miles per hour! That's not 250 miles when scaled up, that's really 250 miles per hour. A molecule in your body is colliding with another billions of times every second. (Source: http://www.righto.com/2011/07/cells-are-very-fast-and-crowded-places.html http://www.righto.com/2011/07/cells-are-very-fast-and-crowde...). At that scale and that level of activity it becomes much harder to simulate each time step.
  - Retric 9y ago
    The other side of this is cells are also really fast. Modeling even 0.01 seconds could be very useful.
    - SAI_Peregrinus 9y ago
      Much of the action inside cells is done via proteins. It's taken a huge amount of distributed computing power just to figure out a reasonable hypothesis for how they fold (only one part of the movement) and €1.22 billion to build the XFEL and have a way to image them on the required timescales: 1-2 femtoseconds. 0.000000000000001 seconds. 0.01 seconds is an eternity, calculating that long would take an utterly enormous amount of computer time.
electricslpnsld 9y ago
> Well, we still don't have the computational power to do things atom-by-atom People do atom-by-atom molecular dynamics simulations of proteins and such.
- openasocket 9y ago
  Certainly, but not at the level of an entire cell
- effie 9y ago
  Yes, but those are simulations of classical mechanics models of molecules which are known to be very limited when it comes to atom-scale objects. We have another model that was much more useful there (Schroedinger's equation) but it is only computationally tractable for systems of few atoms. We do not have the computational power to work out the consequences of the model for proteins.
ctdonath 9y ago
we still don't have the computational power to do things atom-by-atom That's what puzzles me about this discussion. Of course we don't. The goal here is to model more atoms than we have atoms to model with. Until we get to quantum computers that can represent more information per atom, than information per atom we want to represent, it's simply a matter of objectively & obviously inadequate resources. You can't simulate wind shear on a plane when there's more atoms in that wind & plane than there are in the computer simulating them: every single atom being simulated contains the complete information about its state (information independent of any other atom), so simulating it will require at least one atom per atom simulated - if you don't have at least as many atoms to simulate as you are simulating, you can't achieve a complete simulation.
- evandijk70 9y ago
  It is not necessarily the goal to model more atoms than we have atoms to model with. The goal can be, for example, to model a folding protein, which has fewer atoms than a CPU.
  - njarboe 9y ago
    And one might be happy to spend a month computing the folding while in the cell it takes a few seconds.
- pcvarmint 9y ago
  That may be true of digital computers, but it's not necessarily true of analog ones.
RhysU 9y ago
> Well, we still don't have the computational power to do things atom-by-atom... We still don't have the computational power to do certain classes of continuous Navier--Stokes calculations (i.e. not atom based).

al2o3cr 9y ago

It's unclear whether atomic-level data would be useful in these sorts of calculations - the input data isn't known to sufficient precision. For more, check out the last two paragraphs from the Feynman Lectures Vol III Chapter 2: http://www.feynmanlectures.caltech.edu/III_02.html http://www.feynmanlectures.caltech.edu/III_02.html A similar sort of mechanism led to the discover of chaotic behavior in weather models - checkpointing results at a precision slightly lower than the machine's internal precision caused simulations that were resumed from the checkpoint to deviate rapidly.

srirachahot 9y ago

You are missing the distinction between a numerical solution to a set of equations and an analytical one. Usually, the analytical solution gives you the dynamics of a system at any point in time and space given a set of parameters. For numerical solutions you have to run each individual set of parameters to find the corresponding values in time and space (not even considering stochastic equations). This is very computationally expensive. This is the value in solving these things analytical — hence the prize.

srirachahot 9y ago
Sorry this should read: “Usually, the analytical solution gives you the dynamics of a system at any point in time and space given ANY set of parameters.” The “Any” is the value here for the analytical solution.

improbable22 9y ago

Human limitations is one reason, but there's something much deeper going on here. Navier and Stokes worked before we were sure that atoms existed, certainly before we had any idea of how many there were. Nevertheless they were able to write down useful theories for describing fluids. This is how all of science works. The things about which we are totally ignorant are much smaller today, of course... but useful theories of any set of phenomena always omit a great many things we do in fact know about. From these theories, we can understand what's going on, and use this to extrapolate to things we have not seen yet. An atom-by-atom computational model would (in some sense) be no more useful than what we had before N-S, just blind experiment. To try out any given swirl of smoke etc. we can equally well walk next door to the lab and videotape it... but this doesn't help us imagine what else might be possible. The "blowup scenario" discussed is an example of this kind of imagining.

oldandtired 9y ago
> The things about which we are totally ignorant are much smaller today, of course One would have to disagree with this idea. The more we study the universe around us, the less we actually know. As a somewhat philosophical point, there are many times when our mental (mathematical) models get in the road of understanding. Very often, people believe that because we have a model that works and appears to give good predictive results about some phenomena then we understand the how and the what (and even the why) of those phenomena. When this happens, we get into a situation where alternative models are actually discouraged. If one looks the the history of the 19th, 20th and 21st centuries, one can see that more and vaster avenues of investigation have arisen as time passes. Our increasing knowledge is continuing to be shown as ever smaller in relation to what we are now seeing. No theory is ever complete, nor is it ever accurate to the extent that it describes the reality of the universe around us. All theories make those simplifying assumptions that when taken too far lead into inaccurately describing and predicting what we should see. Too often people get enamoured by the beauty of the mathematics and forget it is only an attempt at reflecting reality. Mathematics is a magnificent and useful tool, but it is a foolish master. Too often we forget that. Theories and models help us gain some understanding of the nature of the universe around us. This understanding, however, is always subject to change, no matter how "perfect" the theory may appear to be. There are too many scientists, both theoretical and practical, who are so infatuated and enamoured with their current models that they have forgotten that the models are approximations only and are subject to change or even overturning.

contact_fusion 9y ago

Your assumption is frankly incorrect. Physics is generally expressed in terms of differential equations. This is not due to their analytical tractability - as anyone who has attempted to solve PDEs before will know, most (nearly all) differential equations do not yield to analytical solution. Perhaps you think that quantum mechanics demands a discretized view of reality. This would be a complete misunderstanding of quantum mechanics, and physics in general.

adrianratnapala 9y ago

> computational, atom-by-atom model Remember that as long as computers continue to be made atoms, they aren't going to do atom-by-atom simulations, unless these simulated systems are much smaller than the computer itself or other are simplifying assumptions that can be made.