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.
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-crowde...). At that scale and that level of activity it becomes much harder to simulate each time step.
PS: Also of note protean folding simulations don't even simulate the water surrounding the protean and are again very simplified.
Unfortunately, a useful cell simulation would be far more complex than simple particle collisions and a much shorter step size. But, that's not to say different kinds of simulations can't be useful. And ASIC's or general improvement in computing power can also boost things.
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.
We still don't have the computational power to do certain classes of continuous Navier--Stokes calculations (i.e. not atom based).
The better foundation is quantum mechanics. An example of a macroscopically visible difference between these foundations are the van der Waals forces.
It's a very reasonable idea, and useful enough for computations, but it lacks the essential tension that makes Navier-Stokes a mathematical Everest.
People do atom-by-atom molecular dynamics simulations of proteins and such.
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.
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.
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.
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.
Sure, our awareness of how much we don't know has grown over time.
The point I was trying to make is that theoretical models (like N-S) not only don't have to be perfect to be useful, but more, are useful precisely because they are not complete. By ignoring irrelevant details we get theory, not just simulation.
Understanding that a model or theory is useful even when we ignore certain aspects of reality is quite different to the often displayed belief that a specific theory is "gospel" even in the face of anomalies and discrepancies of the real world compared with prediction. Too much of the "theoretical physics" genre (word specifically chosen) is based on the idea that mathematics is the means of finding the "truth".
As I said above, mathematics is a wonderful and useful tool, but it is not a good master. It provides a possible insight into what is going on. However, those insights are not "truth" as such. I have been doing a review of my old mathematics texts for scientists and engineers, as well as other resources. It is interesting that all of them talk of and demonstrate that all the mathematical models are simplified and incomplete. Yet, if one raises the various problems with the various models in use today, one is shouted down. This does not bode well for our advancement in understanding of the universe around us.
In my understanding the big shift was the understanding of renormalisation, Kadanoff and Wilson, around 1970. This took airy ideas about useful approximation and turned them into serious tools, which are both useful for everyday things and illuminating about why any of it works.
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.
The “Any” is the value here for the analytical solution.
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.