Ranking Countries in Order of Athletic Level

I never attach importance to the number of gold medals won by different countries in Olympic Games, nor want to find fault with that attained by China in Beijing Olympic Games held this summer. I would just like to point out that the ranks of countries are quite different when the number per population is considered instead of the actual number.

The ranks of countries by the number of gold medals (more than five) in Beijing Olympic Games are as follows (Ref. 1):

1	China	51
2	United States	36
3	Russian Fed.	23
4	Great Britain	19
5	Germany	16
6	Australia	14
7	Korea	13
8	Japan	9
9	Italy	8
10	France	7
10	Ukraine	7
10	Netherlands	7
13	Jamaica	6

On the other hand, the ranks by the number of gold medals per population among the above thirteen countries are as shown below. The first number for each country is the population taken from Ref. 2 in units of one hundred million; and the second number is the number of medals per population multiplied by one hundred million.

1	Jamaica	0.0271	221
2	Australia	0.214	65
3	Netherlands	0.164	43
4	Great Britain	0.610	31
5	Korea	0.482	27
6	Germany	0.822	19
7	Russian Fed.	1.42	16
8	Ukraine	0.461	15
9	Italy	0.596	13
10	United States	3.05	12
11	France	0.645	11
12	Japan	1.28	7
13	China	13.3	4

If we include the countries and regions that won at least one gold medal, the ranks would be different much more. Anyway, the second method of ranking clearly shows that a good fight exhibited by Jamaican athletes including the sprinter Usain Bolt was wonderful in Beijing Olympic Games.

Let us consider two countries A and B, A having much larger population than B, and assume that these countries have the same athletic level. Then, country A is statistically expected to produce a larger number of excellent athletes. This makes the method of ranking by the number of medals favorable to A. In country A, however, the selection of the delegation for Olympic Games would be severer because of the limited number of medals expected, making the method of ranking by the number of medals per population unfavorable to A. Therefore, the method that combines these two methods with certain weights might be adequate to know the proper rankings of countries in order of athletic level.

1. Overall Medal Standings, The Official Website of Beijing Olympic Games August 8-24, 2008.
2. List of countries by population, Wikipedia, the free encyclopedia (26 August 2008, at 07:09).

World Records for Men's 100 m Defy Simple Curve Fitting

World record progression for men's 100 m. Data, from Re. 1; curve, least-squares fit of exponential function to data. (You can see the real size image by clicking on the image).

Jamaican sprinter Usain Bolt won the gold medal for the 100-meter race of the 2008 Olympic Games on Saturday, August 16, establishing the new world record of 9.69 seconds. We find world records for this race since 1912 at the Wikipedia site [1]. A few data points of the latest world records show rapid decrease (see the figure above). This trend seems to defy simple curve fitting.

However, I dared to try fitting of an exponential function, y = a + b exp(−cx), to the data. The asymptotic value a, i.e., the limit of the world record, has been found to be 9.43 seconds with a probable error of 0.17 seconds, namely, to lie between 9.26 to 9.60 seconds. During many years, unexpected factors might come to affect the making of records, so that the result of curve fitting should not be much relied upon.

Duffy [2] also considered the limit to 100-m sprinting. He fitted a logistic function to the data up to 2002, and estimated a limit of 9.48. This value is rather in good agreement with the present result.*

A logistic function is useful to model the S-curve of growth [3]. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops. To use this function for the decay phenomenon that reaches a limit, it is necessary to make the function upside-down by making the coefficient in the exponential function negative. Further, one more coefficient should be added to give a finite limit. The function thus obtained has the properties of a slow initial decrease and a final decrease of approximately exponential type. To model a data set without a slow initial decrease, the exponential function of the form I used suffices.

* Originally I wrote here the following sentence put in parentheses: Among the legends in his Figure 2, "exponential fit" should read "straight-line fit". However, this was wrong. Duffy actually used the function of the form y = b exp(−cx) [4]. Reference 2 now includes new results (see sections of [2] entitled "The Performance Enhancing Drugs (PEDs)" and "100 Years - The Jamaicans"). (This note written August 17, 2012)

References

1. "World record progression 100 metres men," Wikipedia, The Free Encyclopedia (18 August 2008, at 12:42).
2. Kevin Duffy, 100 m sprinting: Is there a limit? (September 21, 2002; last revision, January 15, 2008).
3. "Logistic function," Wikipedia, The Free Encyclopedia (August 11, 2008, at 12:38).
4. Kevin Duffy, private communication (August 14, 2012)

Note added later:

• Read part 2 of this article.
• Read a similar analysis made on August 18, 2009, by taking Bolt's record of 9.58 seconds into account.

Backscattering of Antiprotons

The figure shows the energy and number backscattering coefficients of hydrogen ions incident on thick layer of aluminum as a function of incident kinetic energy [1], and can be used to estimate the backscattering coefficients of antiprotons. (You can see the real size image by clicking on the image).

In the previous essay I wrote about the backscattering (also called reflection) of antiprotons from aluminum wall found by Italian physicists. The ratio of the number of backscattered particles to the number of particles incident on a layer of material is called number backscattering coefficient R_N.

R_N of antiprotons are almost the same as that of protons, because the basic formula of the phenomena related to the passage of particles through matter, i.e., the energy loss per unit pass length and the Rutherford scattering cross section, are independent of the sign of the incident particles. Therefore, R_N of antiprotons can be estimated by the use of a universal empirical equation we published for R_N of light ions, which include the hydrogen ion, i.e., the proton, incident on different absorber materials [1].

The equation for the hydrogen ion incident on the thick aluminum layer is given by the upper curve in the above figure. The energies of the antiprotons referred to by the Italian physicists are from 1 to 10 keV. For these energies, R_N of the hydrogen ion and accordingly that of the antiproton can be seen to decrease from about 13%, a considerable fraction, at 1 keV to 2.5% at 10 keV.

Reference

1. R. Ito, T. Tabata, N. Itoh, K. Morita, T. Kato and H. Tawara, "Data on the backscattering coefficients of light ions from solids (A revision)," Institute of Plasma Physics Nagoya University Report IPPJ-AM-41 (1985).

No Surprise to a Radiation Physicist

The e-mail note for "Physical Review Focus 11 August 2008" carried the news story entitled "Antimatter Bounces off Matter" (online Focus story is given in [1]). The story tells that in the August Physical Review A, a team of Italian researchers reports that a good fraction of a low energy antimatter beam directed at a normal matter wall will bounce right back.

Fans of science fiction know that the meeting of matter and antimatter results in the annihilation of both, accompanied by a release of a tremendous amount of energy. Therefore, the above result, based on a new analysis of 12-year-old data, is reported to be "surprising even to most physicists, though it is explained by basic textbook principles."

I have read the full story [1] and the abstract of the paper [2] written by the Italian physicists. The antimatter in their experiment was a beam of low energy antiprotons, and they observed a large fraction of the beam was reflected by an aluminum wall at the end of the apparatus. They made a Monte Carlo simulation of the antiproton path in aluminum, and found that the observed reflection occurred primarily via a multiple Rutherford scattering on Al nuclei.

The phenomenon is the backscattering of antiprotons from a thick layer of matter. Radiation physicists well know the same phenomenon for the beams of electrons, protons, ions and positrons (note that positrons are the antiparticles of electrons). I'm one of those physicists, and especially studied the backscattering of electrons and ions. Thus the story came as no surprise to me.

The abstract concludes with the sentence, "These results contradict the common belief according to which the interactions between matter and antimatter are dominated by the reciprocally destructive phenomenon of annihilation." I would like to say, "It is rather surprising that such a common belief has been held not only by fans of science fiction but also by high-energy physicists."

Antiprotons having some energy and passing far from nuclei behave like ordinary particles with a unit negative charge, and suffer the same amount of Rutherford scattering as that protons do; they annihilate with protons only when they come quite close to nuclei. This should be a common belief about antiproton beams instead of the one described by the Italian team.

1. Antimatter Bounces Off Matter, Physical Review Focus, 11 August 2008.
2. A. Bianconi, et al., Experimental evidence of antiproton reflection by a solid surface, Phys. Rev. A Vol. 78 (issue of August 2008).