If you have entered the website here, then you may find computer simulations of the Kelvin-Helmholtz instability and the Von Karman vortex street on this page. The main theme of this website is the computer simulation of quantum mechanics, a subject related to fluid dynamics. You can find instructions on how to run the computer programs listed below on this page's mother page:
where you may also find simulations of other things which are of general interest.
In the author's opinion, quantum mechanical randomness is the agent of entropy production. A rival theory would have it that entropy production is an illusion of large numbers. Is there any way we can resolve the argument? For ordinary 'positive temperature' systems (in degrees Kelvin), probably not. However, there are 'negative temperature' systems where the predictions of these two theories may diverge.
Lars Onsager has predicted that if a mixture of point vortices is confined in a box, then this mixture may separate into clockwise and anti-clockwise components. This is apparent anti-kinetic behaviour. However, the addition of Chorin-style randomness may suppress such behaviour. Let us make a guess that if there are equal numbers of clockwise and anti-clockwise vortices, and every vortex has the same numerical strength, then the quantum of vorticity is equal to the magnitude of the Brownian motion which needs to be added to just suppress anti-kinetic behaviour.
This is something we can investigate by computer simulation (and it hasn't been done yet to the author's knowledge). Then we will need to do experiments to see how nature behaves. If the guess turns out to be correct, then we have a pointer to wave-particle duality. Quantum randomness must be present in nature in order to suppress anti-kinetic behaviour in all circumstances, and thus preserve the Second Law of Thermodynamics. The scale of the randomness must be equal to some characteristic scale of the system.
The computer simulation won't be an easy one. To gain experience, and the reader's confidence, we will first produce some other simulations based on vortex dynamics. Then if the guess turns out to be wrong, at least we are not left empty-handed. Simulations of vortex dynamics are pretty to look at, and it does no harm to have a computer simulation incorporating a random number generator available on the Internet as a reminder of our ambitions for quantum mechanics.
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The first simulation will be the Kelvin-Helmholtz instability. Here is an introduction to the subject which gives two alternative explanations of the instability ...
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Here is the main simulation which is best run on a 486 DX or a Pentium. It is both a real-time simulation and a virtual-time simulation. Press the f1 function key to see this. Two periods of a vortex sheet are shown, and one vortex has been given a tiny perturbation which will grow with time. You will see the mother of all chaotic dynamics simulations. Answer Y for Yes to the first question.
Now run the simulation again. Answer N for No to the first question. Instead of an artificial perturbation, you will see the simulation run with every vortex element in Chorin-style Brownian motion.
If QBasic is replaced with QuickBasic 4.5 or Visual Basic for DOS, you should get a better speed from this simulation. Try saving it on disk and compiling it to KHI.EXE for even better performance. QuickBasic 4.5 and Visual Basic for DOS are apparently able to take advantage of your maths co-processor, while QBasic does not (you get what you pay for). The program shares out the benefits of extra speed between more vortex elements and faster action. It is also suggested that you run under MS-DOS or DR-DOS (not Windows) and you get rid of device drivers such as EMM386.EXE.
If you do run the simulation under Windows, then you can run it in the background. We have put a reminder about Ctrl-Esc to help you. In Windows 95 you can set up a short cut to the program. Remember to enable program execution while in the background. Then you can have the program running all the time. You can look at it occasionally, and the virtual-time simulation will make sure you miss nothing.
If you have PKware's PKZIP available, then you can download KHI.ZIP, unzip it to KHI.EXE, and then run KHI.EXE. This will run much faster than KHI.BAS. Furthermore, the memory not now used by QBasic will be available for a longer replay of the animation.
After running this simulation, you will be able to see the author's thinking. One source of perturbations may be quantum mechanical randomness, the effect of which may be amplified up to macroscopic levels by instabilities like the Kelvin-Helmholtz Instability. This is how macroscopic entropy production has its origin in microscopic quantum randomness.
If you press f1 twice, you will find a control panel where you can press T for time reversal. Try doing this both for the artificial perturbation and for added randomness. For the artificial perturbation, you will find that the simulation is roughly reversible over small timescales. However, the interaction of numerical truncation errors with the violent instability of the vortex sheet means that reversibility soon breaks down. With added randomness, you will see that the system is definitely not reversible.
This is suggestive but not conclusive. The author would indeed like to draw one conclusion, but it is not well-supported by this simulation, and can never be, not even in principle, thanks to inevitable numerical errors. Maybe numerical truncation errors can be eliminated with a 'lattice-gas' simulation, but there the simulation is rather artificial. It would be better to find an experiment to resolve the argument, and that is why we are interested in Onsager's prediction about vortices in a box.
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The second simulation is the Von Karman vortex street. Here it is to run. Comments are welcome.
There has to be a spontaneous breaking of symmetry in order for a vortex street to form. The inclusion of a random number generator is essential to get this. Chorin's method is not just another Monte-Carlo method. Run the simulation several times to become familiar with the symmetry-breaking.
Some signal or other has to achieve this symmetry-breaking. This signal is able to travel at superluminal speed. It cannot be used for superluminal communication because all it does is to effect a correlation between random events (the situation is the same as in quantum nonlocality). The direction of travel of the signal is merely a matter of convention. Looking at things from Hugh Everett's 'relative state' point of view (which we are adopting as a commentary and not as an interpretation), all the signal does is to bring a particular relative state into existence, and the perceived sequence of initiation of the relative state is simply conventional.
Holism may be an over-worked concept, but here we have a genuine holistic phenomenon. Given two apparent events A and B, what matters is the joint correlation between these events. Whether we say that A causes B or that B causes A is merely a matter of convention, and so the separation between A and B may be spacelike. Given the spacelike separation, different observers may disagree about whether A precedes B or B precedes A, and the issue is apparently real enough in computer simulation where we take a slice through space-time. In the case of the vortex street, the symmetry may break with the upper eddy leading or the lower eddy leading. A and B are anti-correlated, but we are unable to say in principle whether the symmetry-breaking signal passes from A to B or B to A. There is therefore no objection to the proposition that the separation between A and B may be spacelike, and that the supposed direction of any signal that passes between them is conventional.
The price to pay for invoking holism is that the quantum many-body problem then requires an exponential-time algorithm (this is Baugh's Conjecture). We win one, but lose one.
This argument that the signal can travel at superluminal speed may be controversial, since a naive view of the argument is that it overturns established ideas of cause and effect (it doesn't). Anyway, this spontaneous symmetry-breaking is a very special and rare phenomenon and you are requested to make sure that everyone in your physics community knows about it. The argument becomes important when we tackle quantum mechanics, but before we do that, we should see to it that knowledge of the formation of the Von Karman vortex street by spontaneous symmetry-breaking is part of every physicist's general education.
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Then we will tackle the Onsager simulation. Here we have indeed a Second Front by which quantum mechanics may be attacked. The conclusion to draw will be that if you expect entropy production in your simulations, and you want something more than an ad hoc theory, then you must find a way to model quantum randomness. Chorin-style Brownian motion is an adequate stand-in for quantum mechanics in the case of vortex dynamics.
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