Friday, March 21, 2008

Joseph Fourier: 240th birthday

Exactly 240 years ago, on March 21st, 1768, Joseph Fourier, an 18th century politician and 19th century string theorist, was born in Auxerre, France.

Parents and Napoleon

The boy's father, who was a tailor, as well as his mother were dead when the kid was nine years old. The local bishop arranged him to be educated by the Benedictine monks in a military school at the Convent of St Mark, probably a similar school attended by the Bogdanov brothers that they were showing me. ;-) The kid was happy there and fell in love with maths (and wrote some good poems). Jean-Baptiste-Joseph supported the French revolution and was given a job in École Normale Supérieure mainly for his political achievements (of course, he was jailed several times, too). Later he became a boss at École Polytechnique.

In 1798, Joseph went with Napoleon Bonaparte to Egypt and even became the governor of Lower Egypt - Fourier the Lower Pharaoh, if you wish. You wouldn't believe it but Napoleon actually established a mathematical institute in Cairo (of course, the motivation was to weaken the English influence in the region) and Fourier wrote a few papers for the institute.

The French loss in the conflict didn't destroy Fourier who became the prefect of Isère and started to do experiments with heat. After a few years in England, he became the permanent secretary of the French Academy of Sciences. Quite a career for this orphan, right? ;-)

Science

We haven't really started with his science. Some people like to say that big minds do their first revolutionary work before they are 40 (or even less). Fourier is one of many examples showing that the rule is complete rubbish.

In 1822, at the age of 54, Fourier finally publishes "Théorie analytique de la chaleur," i.e. "Analytical Theory of Heat", where the heat flux is argued to be proportional to the temperature gradients times a negative constant, the so-called Fourier's law. More importantly, he also discovers the Fourier analysis: any function may be written as a continuous combination of sines and cosines. And any periodic function may be written as a sum of multiples of sines and cosines whose arguments are multiples of "x" times the period over two pi.

This includes un-smooth and even discontinuous functions.

When I was 14 or so and I read an encyclopedia of mathematics, Fourier analysis was discussed in one of the chapters and it was a complete shock for me. At that time, I was kind of obsessed with analytical functions of complex variables that can be differentiated infinitely many times and are beautifully smooth: their values in the whole complex plane can also be predicted from their behavior in an open set. Suddenly, someone could draw an "artificial" function with discontinuities and circles and piecewise linear segments and write it as a sum of the beautifully smooth sines and cosines.

It was very shocking for me that such a thing was possible but of course, after some time, I checked that it really worked and that it is the infinite number of terms - the limit - that could violate my wrong intuition that a sum of smooth functions must be smooth.

Fourier claimed that every periodic function could be expanded in sines and cosines. Rigorously speaking, that was wrong but there exists a very mild, physically natural refinement of his statement that is correct. Consider, for example, L2-integrable equivalence classes of periodic functions to see that Fourier's statement was "morally true" as long as the morality is dictated by quantum physicists. Two centuries later, the Fourier series are famous well beyond Napoleon's empire, even among English-speaking Asians:

Joseph Lagrange's and Dirichlet's later analyses of the question which functions may be Fourier-transformed do not look particularly fundamental to me.

Needless to say, the Fourier expansion is the first step that you must do with coordinates on a string in perturbative string theory, so I count Fourier to be a string theorist, much like other giants whom we discussed previously.

Fourier has also found some theorems about various functions of the roots of algebraic equations that can be calculated in new ways. More importantly, he introduced dimensional analysis as a tool to verify that an equation has a chance to make sense. It is kind of amazing but it seems that he was the first man to do so.

Greenhouse effect

In 1824, at the age of 56, Joseph Fourier wrote the paper "MEMOIRE sur les temperatures du globe terrestre et des espaces planetaires" and established the concept of planetary energy balance and derived the phenomenon that we nowadays call the greenhouse effect. He realized that planets radiate energy by the dark heat ("chaleur obscure"): yes, it is nothing else than the thermal infrared radiation. Fourier referred to an experiment by M. de Saussure (glass on a black box under the Sun heats up the box) to justify the existence of dark heat.

The quantitative formula describing the energy content of his "dark heat" was found 50 years later by Stefan and Boltzmann.

Fourier understood that the real task was to calculate the point (temperature) where equilibrium is reached: such a point should exist because, as he knew, warmer bodies emit more dark heat. Gases that can absorb "dark heat" in the atmosphere reduce the energy losses and increase the equilibrium temperature, he correctly stated.

His energy budget correctly contained the solar radiation and the dark heat but it had one additional, unusual term: interplanetary radiation. While it is more than plausible that cosmic radiation plays a role for the terrestrial climate, it is not its direct energy but its ability to catalyze cloud formation that matters.

He died of heart attack in 1830 so he couldn't see the breathtaking application of his innocent effect in extremist environmentalist politics.