Paul Langevin

Brownian Motion and the Appearance of White Noise


The Langevin Equation is the first differential equation of which a term is random. This random term is now called 'white noise'. Today, Brownian Motion and the stochastic differential equation are tools used currently in all the sciences.

Paul Langevin was a celebrated physicist and a committed scientist of his era.

But who knows... the mathematician? Henri Lebesgue, president of the Société mathematique de France (SMF), wrote to him in 1919 : "My dear friend, it is awful that you do not take part in the SMF. Apart from general physics, you interest yourself in a theoretical physics which is mathematics. As evidence, the problems that you submitted to Montel and to me, at the start of the war, and which you hurried to solve because it amused you. At this title you belong to us, you are even a breed apart and very rare in the mathematical genre.

Brownian Movement

In 1908, Paul Langevin submitted a report in three parts to the Academy of Sciences, entitled On Brownian Movement. The reports were then a means of rapid communication : one could lodge a report on Monday afternoon (at this time, a day of weekly briefings), correct the proofs at Gunthier-Villars on Wednesday morning, and it was published the following Monday, thanks to the work of the typographers.

Robert Brown (1773-1858) was an eminent Scottish botanist. A mystery in the 18th century was the disorderly and perpetual motion of organic particles in suspension in a liquid, observed by microscope. What could be the energy responsible for this perpetual motion? The contemporary hypothesis was that it concerned a vital energy contained in the organic particles, which were generally the pollen of plants. And Robert Brown stepped up the experiments, and made some tests with inorganic particles. His discovery, published in Movement of organic and inorganic particles in suspension in a liquid, was that the same type of perpetual motion could be observed with inorganic particles. The hypothesis of the vital force therefore had to be abandoned. Brown's role was to transfer the problem of irregular and perpetual motion of particles in suspension in a liquid from botany to physics.

Those physicists who believed in the existence of atoms and molecules were faced with strong opposition in the 19th century; did it concern the effect of the impact of molecules, rapid and very small, on particles relatively bigger?

Paul Langevin reported to Louis Georges Gouy as bearer of this idea and director of supporting experiments. Nevertheless, it is solely with Albert Einstein and Marian Smoluchowski that the physics theory of Brownian Movement was established on this basis, at the same time that the qualitative results were being furnished via microscope observation having attained the dimensions of molecules, inaccessible by direct observation.

The Einstein theory was inspired by the thermodynamics of gas, explained by the motion of molecules : if it was thus for liquids, this motion must create a movement much slower, and observable, of particles in suspension in the liquid. At the end of his article, Einstein hopes that an experiment would decide the question. At this moment, curiously, he did not know the Brownian Movement. In a second article, he meets it, the experiment is thus made and it is a means to attain molecular dimensions. It is a nice program, which will be realized by Jean Perrin.

Smoluchowski had started a study which was purely kinetic and had arrived at the same result, except for a multiplicative factor (equal to $\frac{64}{27}$). It is a source of the note of Paul Langevin. Here is the formula obtained by Einstein \[ \bar{\left(\Delta x\right)^2}= \frac{R.T}{N}\frac{1}{3\pi \mu a}\tau\] and that obtained by Smoluchowski \[ \bar{\left(\Delta x\right)^2}= \frac{64}{27}\frac{R.T}{N}\frac{1}{3\pi \mu a}\tau\] where R is the ideal gas constant, T is the absolute temperature, N is the number of molecules in a mole (Avagadro's Number), a is the radius of a particle (on supposing all particles have the same radius), $\mu$ is the viscosity of the liquid, $\tau$ is the temperature ecoule, and $\Delta x$ represents the displacement of the particle in the direction of the axis of R (in the formulas, one takes its square, and the mean of this square). This could be the mean over a large number of molecules, ou bien the mean over a great number of equal intervals of time

Jean Perrin realised this experiment : the Brownian Motion, permitting the determination of the Avagadro Number, is the validation, par excellence, of the atomic hypothesis. The description that he gives of the Brownian Motion of a particle is signifying the trajectories are madly irregular, apparently it does not have a tangent at any point, and it is a case where it is truly natural to think of those continuous functions without derivatives which mathematicians have imagined, and which one regarding them wrongly like simple mathematical curiosities since the experiments can suggest them.

That's thus which intervene mathematicians. First Norbert Wiener (1894-1964), then Paul Levy (1886-1971). The physics suggest the existence of a random process such that the means of squares of the displacements are equal to the time expended, or \[ (\Delta x)^2 = \Delta t \] or still, more mathematically \[ E \left(\left(x_t-x_s\right)^2\right) = |t-s| \] Yet more, the displacements during a given time must be independent of what has occured beforehand.

During the course of the 1920s, Norbert Wiener managed to construct this process and it called it the fundamental random function. It ratifies the intuition of Jean Perrin, therefore it is a continuous function nowhere differentiable. One caled it also the process of Wiener, written more commonly $W(t)$. Its fundamental role in probability has only made assert since its creation, and today one calls it usually the Brownian Motion. It was Paul Levy, who following on from Wiener conducted the study in a masterly fashion, which he has named also. Thus the Brownian Motion has passed from the botanical sphere to the physical, then to mathematical physics.

The Brownian Motion of mathematicians is thus a random function, designated traditionally nowadays by B, such as for all values of time the increments to come are independent of the past, and such that for all pairs of time $(s,t)$ with $s But to what resembles the Brownian Motion? One has a first idea of it with a random walk.

Now let us imagine a random walk viewed from far away, from so far that one does not see any more the elementary displacements. The trajectory appears therefore as a continuous curve, very irregular, which went and came and returned backwards in a fantastic manner and unpredictable, apparently without tangent at any point. This recalls in effect the continuous functions without derivatives constructed by mathematicians. Very irregularly, these functions are, in a certain fashion, regularly irregularly : they present a crowd of properties. Langevin gave a new demonstration much more simple and very interesting, of the formula of Einstein, stemming from considerations related to statistical mechanics. But his support for the theory of Brownian Motion did not stop there. There is a paradox : the Brownian Motion of mathematicians does not have derivatives at any point, whereas Paul Langevin makes a theory of the derivatives of Brownian Motion, from the equation \[ \frac{d\zeta}{dt} = - \lambda \zeta (t) + X(t)\] It is a random differential equation, called the Langevin Equation. It is the first example of a diferential equation containing a random term. In place of a random differential equation, one says rather stochastic differential equation.

But how to treat this equation if one does not know what is X. It's there that appeasr the idea of white noise.

White Noise

White light is decomposed by a prism into rays of all colors. These colors represent different wavelengths, different frequencies. A similar situation exists in acoustics. Waves represent pure sound and a superposition of pure sounds produce a compound sound. When all the frequencies mix independently one with the other and with same weighting for each, one obtains the white noise. Noise because it is a mixture of varied pure sounds and white in recalling the decomposition of white light. A peculiarity of white noise is that if one considers it over the line of time, and over disjointed intervals of time, the effects are independent : the white noise operating over an interval gives a random Gaussian variable, of which the variance is the length of the interval. And over disjoint intervals, one obtains independent random variables. This suggests that the white noise, formally, is nothing more than a way of viewing the derivative of Brownian Motion. And effectively that's the case, and one can write dB a sort of "differential of Brownian Motion", as a representation of white noise. Thus the Langevin Equation \[ ***** \] becomes, on taking from X the white noise : $d\zeta (t) =-\lambda \zeta (t) + dB(t)$.

This form is due to Joseph Leo Doob (1910-2004). One arrives at a result somewhat little curious : one can give two interpretations of the speed of Brownian Motion, which does not exist in the *** sense. The bb of a side and the function $\zeta (t)$ of the other, calculable as a function of Brownian Motion. That is a little troubling : the Brownian Motion does not admit of of derivatives at any point, but at the same time it is on considering the derivative that one makes rapidly the theory, and on using the Brownian Motion anew, one obtains the derivative as a new random function. That is not at all a trick, it is just that there exist several possible idealizations of Brownian Motion of the physicists, and that of the mathematicians is one of them. The two approaches are at the time perfectly incompatible and perfectly complimentary!!!




 

  • A Great Physicist

    In the popular miliue to which his family belonged, it was expected that Paul Langevin (1872-1946) would follow classical studies at school. Happily, the School of Physics and Chemistry of the City of paris came to be opened, he passed the entrance exam and received top marks. On leaving this school he became an engineer with a solid foundation in physics and in chemistry. His teachers advised him to go further and to pass the entrance exam for the École normal supérieure. He was not yet 20 years old. He studied by himself two hours a day Latin, and the rest of the time a la taupe. he had as comrades of promotion the future mathematicians Henri Lebesgue and paul montel. He left the l'école with a diploma in physics. He was par ailleurs by Jean Perrin, his elder by several years, a good knowledge of X-rays, discovered a few years beforehand. With a scholarship from the City of Paris, he left to work at Cambridge, the main location for X-rays at the time. The X-rays are located by the ionisation which it causes in gas and Langevin became a theoretician of the ionisation and a promoter of its application. His work, related to statistical mechanics, to magnetisn, to ultrasound, brought him at a very young age a considerable reputation. He became professor at the College of France at the age of 37 years.

  • Langevin and Relativity

    The articles of Albert Einstein of 1905 are celebrated as the major advance of physics. The theory of relativity eliminated the notion of an absolute time, and gives new perspectives on mass and energy, the famous $E=mc^2$. Langevin adhered immediately to the concepts of Einstein. He had independently established the equivalence of mass and energy; a striking proof of this equivalence is to be found, if you look well, in the Periodic table of Mendeleyev and the atomic masses of different elements. One knows better the bolide of Langevin or the twins of Langevin : the two twins, the one is embarked on an interstellar bolide of very great speed. When he returns to Earth, he will be less old than his twin! This imaginary experiment illustrates well the failure of absolute time.

  • Robert Brown, as in Brownian Motion
    Robert Brown