"Perpetual Force" Air Molecule Motor Part IV

Statistical analysis

In order to determine the significance of the silux modelling results discussed in my last couple of posts, I'll have to get into statistical decision theory. It's a long time since I've done anything in that area, so I'll use a worked textbook example as a basis, adapting it to suit. For the record, I'll use solved problem no. 5, p173 of Theory and Problems of Statistics, by Murray R. Spiegel, Schaum Publishing Co., 1961.

Adapting this, we have:—

A. Combined cases

    For the combined set of 22 cases modelled, the foil moved fully upwards in 20 cases. Determine whether the results are significant at the (a) 0.05 and (b) 0.01 level of significance.

Solution:
    If p is the probability of the foil moving fully upwards, then we have to decide between the following two hypotheses:

    Ho: p = 0.5, and the foil is only moving randomly, i.e. the results are due to chance.

    Hi: p > 0.5, and the effect is genuine, i.e. there really is an overall imbalance in air molecule impacts which causes the foil to move upwards.

    We choose a one-tailed test, since we are not interested in the foil moving predominantly against the predicted direction of motion.

    If the hypothesis Ho is true, the mean (μ) and standard deviation (σ) of the results are given by

    μ = Np = 22 × 0.5 = 11   and

    σ = √(Npq) = √(22 × 0.5 × 0.5) = 2.3452

(a) For a one-tailed test at the 0.05 significance level, we must choose the ordinate z1 for the standard normal curve (i.e. the bell curve) so that the area of the curve to the right of the ordinate is 0.05. Then the area between 0 (at the center of the curve) and z1 is 0.4500, and so z1 = 1.645.

(b) For a one-tailed test at the 0.01 significance level, we must choose the ordinate z1 so that the area of the curve to the right of the ordinate is 0.01. Then the area between 0 and z1 is 0.4900, and so z1 = 2.330.

    The 20 upwards-movement cases in standard units = (20 - 11)/2.3452 = 3.8376, which is greater than both 1.645 and 2.330. This means that the results obtained are significant at both the 0.05 and the 0.01 levels, i.e. they are "highly significant" and we can conclude that the effect is genuine.

B. Second set of cases only

    If we now look only at the second set of 11 cases modelled, the foil moved fully upwards in 9 cases.

    With a similar analysis, we get μ = 5.5 and σ = 1.6583

    The 9 upwards-movement cases in standard units = 2.1106, which is greater than 1.645, but less than 2.330. So these cases alone would be considered "probably significant" implying that further investigation is warranted.

Further investigation, and final comments

The results obtained so far do look promising, and certainly support the basic idea of a force imbalance, arising from the impacts of air molecules on a device that has "structure" at an appropriate microscopic scale.

One of several questions yet to be resolved is:— Do some more subtle boundary effects play a part in the complete success of the original small-container model? If so, what exactly are they, and could they also be exploited in a real physical model?

I have already mentioned some reasons against placing too much faith yet in these results, i.e. the unrealistic modelling of the air molecules, boundary effects, and the 2D rather than 3D analysis.

The silux analysis done so far has to be 2D. A 3D analysis would be greatly preferable. (But I have to admit I don't yet know enough about 3D finite elements analysis programs to be confident in doing such an analysis for cases like these, where large numbers of individual molecules are involved. For example: how are molecule joints to be dealt with?)

I know of only one good 3D finite differences program, "sonar-code," which unfortunately is not for sale to the public [but see Update below]. The animation shown at http://www.realphysics.ch/sonar.htm indicates that this program would most probably be well-adapted to modelling large numbers of individual air molecules.

There is another problem that I think will apply to any realistic computer modelling of air molecule impacts: on a normal "laboratory" scale, the molecules are extremely small, and there will be a huge number of them, nearly all travelling at very high speeds. This will place very high demands on the computer, to achieve correct elastic impacts in the extremely small time durations for each impact. Probably supercomputers would be required for such modelling.

[Update April 2018: There is another website for sonar simulation, at https://sonarsimulation.com/. From there (under "Products > download"), it is now possible to download and install sonar-3D-LAB for free, and to build models with it. However sonar-3D-SIM will also be needed to run simulations on these, and there is currently no information on when that will be available for download, or its pricing.]