## 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.

• Robert Brown