Tuesday, March 24, 2009

Joseph Stefan: anniversary

Jožef Stefan was born on March 24th, 1835, in Austrian Carinthia, near the Slovenian borders, to Slovene parents: Aleš Stefan (*1905) was a milling assistant while Marija Startinik (*1915) was a maidservant.

You could be surprised by these jobs but they were somewhat typical for the ethnic Slavs in the Austrian empire.

As we know today, this background didn't hurt the boy much. Jožef started as the best student in the class. In his college years, he wrote many poems in Slovenian: he is included among his nation's poets. Pretty quickly, he began to teach in Vienna. Among other topics, he studied interfaces of phases in phase transitions.

The Stefan-Boltzmann law is the most famous discovery that originated in his head and we will discuss it in some detail. But Stefan also had three well-known students of his own:
  • Ludwig Boltzmann, an Austrian giant of statistical physics and thermodynamics
  • Marian Smoluchowski, a Polish pioneering statistical mechanician who co-discovered the explanation of the Brownian motion independently of Albert Einstein
  • Johann Loschmidt, an Austrian chemician and physicist, born in Greater Carlsbad (Czech Kingdom: he also spent some time at Charles University in Prague)
You may remember that Johann Josef Loschmidt also misunderstood Boltzmann's precious derivations of irreversible phenomena (e.g. the second law itself) from statistical mechanics applied to the reversible laws contraining the microscopic building blocks (atoms).

This profound misunderstanding of Loschmidt (and others) is misleadingly referred to as Loschmidt's paradox.

Of course, much like with other "paradoxes" in physics, there's no paradox here. I always find the contrast between different "peers" amazing: Boltzmann was able to derive all these difficult things - out of nothing - while his classmate couldn't even understand these things when they were clearly written down. It's just like with the pompous fools who can't understand even basic results of string theory today - despite the ability of others to discover many things about this aspect of Nature.

But Loschmidt has done some correct, things, too.

The Stefan-Boltzmann law

In 1879, Stefan analyzed the experiments by John Tyndall, an Irish physicist, and deduced that the total radiation from a black body was proportional to the fourth power of the absolute temperature (in Kelvins). Five years later, Stefan's student Boltzmann derived the law theoretically.

The previous sentence should sound surprising to you because the coefficient depends on Planck's constant i.e. quantum mechanics that they couldn't know in the 1880s. The full classical calculation leads to an infinite result - the ultraviolet catastrophe - because arbitrarily high frequencies seem to contribute. But Boltzmann was able to derive the power law for the correct finite result from thermodynamics, anyway.

In fact, the term "ultraviolet catastrophe" was coined and appreciated later, in the 20th century. Planck himself didn't want to cure this "catastrophe", although popular books often present this childish oversimplification of the history. Planck wanted to find a theoretical derivation of a more quantitative law, Wien's displacement law.

Today, the fourth power may be explained as a simple consequence of dimensional analysis. In the quantum relativistic statistically mechanical units, where "hbar=c=k(Boltzmann)=1", the energy density scales like "mass^4" in a four-dimensional spacetime: the inverse 3-volume adds "mass cubed" to the dimension. In the vacuum, the energy density (or radiation) can only depend on the temperature (whose dimension is that of mass) because the photons are massless etc., so the density (and the radiation per unit area and unit time) must scale like "T^4".

But one should appreciate how many non-trivial insights about Nature and about the observables, their relationships, and their symmetries are being inserted into the assumptions that make this dimensional analysis possible.

The coefficient "sigma" in the Stefan-Boltzmann law
dE/(dA dt) = sigma T4
is equal to "pi^2/60" in the quantum relativistic statically mechanical units. It becomes "pi^2.k^4/60 hbar^3 c^2" in the SI units: a lot of cultural garbage has to be added when units are chosen unnaturally.

Today, we can derive the law (including the coefficient "sigma") from Planck's black body curve, by a simple integration. You integrate the density proportional to "f^3/[exp(hf/kT)-1]" over frequencies: don't forget about the volumes of spherical shells and the two physical polarizations of the photon. To determine the coefficient analytically, write "1/[exp(hf/kT)-1]" as "exp(hf/kT)/[1-exp(-hf/kT)]" and use the Taylor expansion for geometric series to rewrite the fraction. You obtain factors like "exp(-n hf/kT)" in the integral and the integral in the sum can be easily calculated.

One can reduce the integrals inside the sum to integrals over {t,0,infinity} of "t^3 exp(-t)" which is the Euler integral producing "3!=6". Why? If you integrate this integral by parts, you obtain the recursive relation for the factorial, "n!=n.(n-1)!".

The final sum is proportional to "sum 1/n^4" which is called "zeta(4)". And this zeta function can be calculated to be exactly "pi^4/90". For example, you can construct a piecewise quadratic, continuous, periodic function whose Fourier coefficients scale like "1/n^2". The squared norm of this function can be calculated either by summing the squared Fourier coefficients; or by integrating the squared function (still a simple polynomial!) over the interval. By realizing that the results are universally equal, you get "zeta(4)=pi^4/90".

Multiply "pi^4/90" by "6" from the Euler integral, and divide by "4 pi^2", from various conversions of frequencies to angular frequencies, from the area of the sphere, and from 2 polarizations, and you obtain "pi^2/60" for the Stefan-Boltzmann constant (in the natural units). Stefan would surely be happy about this complete, modern calculation but unfortunately he died in 1893 in Vienna, at age 57.

No comments:

Post a Comment