BROWNIAN I

Bertrand Duplantier

Service de Physique Theorique,´ Saclay, France

DNA & CHROMOSOMES 2006 Physical & Biological Approaches Institut d’Etudes´ Scientifiques de Cargese`

19 June - 1st July 2006 “Thus, let us assume, for example, that someone jots down a number of points at random on a piece of paper. [...] I maintain that it is possible to find a geometric line whose notion is constant and uniform, following a certain rule, such that this line passes through all the points in the same order in which the hand jotted them down [...] But, when the rule is extremely complex, what is in conformity with it passes for irregular.” G. W. LEIBNIZ, Discourse on Metaphysics, 1786. A Possible Precursor to Brown

L’astronome,1668, Musee´ du Louvre, Paris; Der Geographer, 1668-69, Stadelsches¨ Kunstinstitut am Main, Frankfurt Johannes VERMEER (1632-1675) Who is in the guise of the Scientist? The Scientist

Antony van Leeuwenhoek (1632-1723), ROBERT BROWN Robert Brown (1773-1858) Robert Brown (1827) Edinburgh New Phil. J. 5, 358 (1828): “A Brief Account of Microscopical Observations Made in the Months of June, July and August, 1827, on the Contained in the of Plants; and on the General Existence of ACTIVE in Organic and Inorganic Bodies” The Scottish Botanist Robert Brown reports on the ceaseless random motion of various particles small enough to be suspended in water. Adolphe Brongniart (1827) Ann. Sci. Naturelles (Paris) 12, note B, pp. 44-46 (1827).

Aspect of the Brownian motion of a pollen grain in suspension. The motion is extremely erratic, and apparently never stops; it is universal, applying to organic and inorganic particles as well (Sphinx of Gizeh).` Charles Darwin (1809-1882) about the 1830’s: “I saw a good deal of Robert Brown, “facile Princeps Botanicorum,” as he was called by Humboldt. He seemed to me to be chiefly remarkable by the minuteness of his observations and their perfect accuracy. [...] On one occasion he asked me to look through a microscope and describe what I saw. This I did [...]. I then asked him what I had seen; but he answered me, “That is my little secret.” THE KINETIC THEORY OF HEAT Ignace Carbonelle (1877-80) “In the case of a surface having a certain area, the molecular collisions of the , which cause the , would not produce any perturbation of the suspended particles, because these, as a whole, urge the particles equally in all directions. But if the surface is of area less than necessary to insure the compensation of irregularities, there is no longer any ground for considering the mean pressure; the unequal pressure, continually varying from place to place, must be recognised, as the law of large numbers no longer leads to uniformity; and the resultant will not now be zero but will change continually in intensity and direction. Further, the inequalities will become more and more apparent the smaller the body is supposed to be, and in consequence the oscillations will at the same time become more and more brisk...” THE THEORY OF BROWNIAN MOTION

Mens agitat molem. VIRGIL, AEneid. lib. VI , , 17, 549-560 (1905):

“On the Motion of Small Particles Suspended in at Rest Required by the Molecular-Kinetic Theory of Heat”

Aim: to determine Avogadro’s number and the size of

“Let us hope that a researcher will soon succeed in solving the problem presented here, which is so important for the theory of heat.” “Mog¨ e es bald einem Forscher gelingen, die hier aufgeworfene, fur¨ die Theorie der Warme¨ wichtige Frage zu entscheiden !” Brownian Formula

Sutherland, Einstein (1905), Smoluchowski (1906): Mean square displacement ¥ ¡ £ ¤ RT 1 x2 ¢ 2Dt ¢ 64 27 t t N 3πηa Fluctuation-Dissipation

¡ RT 1 2 ¢ ¢ x 2Dt D t N 6πηa

¡ D: diffusion coefficient ¡ t: diffusion time

¡ T: absolute ¡ R: ideal molar constant ¥ £ ¤ ¢ ¡ 23 AVOGADRO’S NUMBER N 6 £ 02 10

¡ η: liquid’s ¡ a: ’s radius It is the first appearance of a fluctuation-dissipation relation, here linking position fluctuations to a dissipation (viscosity) property. x2 ¡ , t, a and η are measurable, so Avogadro’s number can be determined. Prepare a set of small spheres which are nevertheless huge compared with simple molecules, use a stopwatch and a microscope and measure N ! (A. Pais, in “...”)

Einstein gives an example: for water at 17oC, a ¢ 1µm, ¡ ¡ ¤ 1 2 23 2 ¢ N 6 10 , x 6µm for t 1mn £ W. SUTHERLAND, Phil. Mag. 9, 781 (June 1905):

“A Dynamical Theory of Diffusion for Non-Electrolytes and the Molecular Mass of Albumin” ¤ RT 1 3η βa ¢ ¤

D 6πηaN 1 2η βa β: sliding coefficient Van ’t Hoff’s law, 1884 The perfect gas law applies to the of molecules in dilute .

Ingredients used by Einstein & Sutherland

¡ Van ’t Hoff’s law for suspended particles.

¡ Stokes’ relation: viscous friction force F of a liquid on a sphere of radius a at speed V: ¢ ¢ F µV 6πηaV £ External Force

Particles under external force F (x axis); ΠF ¡ nF per unit volume

Gradient of concentration n Osmotic force per unit volume: ∂p R ¡ ¡ Π ¢ ¢ Tgradn ∂x N Equilibrium:

£ R ¡ ¡ Π Π 0 ¤ nF Tgradn F N Flux Equilibrium ¥ At stationary equilibrium, the flux linked to the force ΦF ¡ nV ¡ nF µ ¦ ¡ ¢ (viscous friction coefficient µ ¤ and the diffusion flux ΦD D gradn (Fick’s law) compensate each other: ¥ £ § 1 RT ¡ ¡ ¡ ¨ Φ Φ 0 ¤ nF µ D gradn D F D µ N Gaussian Probability (L. Bachelier, 1900)

2 ∞ ∂P ∂ P ¡ ¢ £ ¢ 2 ¢ 2 ¢ D 2 x t x P x t dx 2Dt ∂t ∂x ∞¡

x 2 e 4Dt 4 π Dt 0.5

0.4

0.3

0.28Dt

0.1 -4 -2 2 4 x Einstein (1905), Smoluchowski (1906) : probability to be within dx of x at time t, starting from x ¤ 0 at t ¤ 0. Diffusion Equation and Probability Density

∂P ∂2P ¡ D ∂t ∂x2 ¦

The probability density, P x ¤ t , for a Brownian particle, starting at 0 at time t ¡ 0, to be within dx of x at time t, is the Gaussian distribution x 2 e 4Dt 4 π Dt 0.5

0.4

0.3

0.28Dt

0.1 -4 -2 2 4 x ¥ ¡ ¢ ¤ ∞ § © 2 £ 2 £ ¨ x t ∞¦ x P x t dx 2Dt [Fourier (1807), Kelvin (1854), Fick (1855), Bachelier (1900), Einstein (1905), Smoluchowski (1906)] M. V. SMOLUCHOWSKI

r

0

a

“ in 1906 when Smoluchowski (influenced by the appearance of Einstein’s two papers [on Brownian motion]) finally published his results, random phenomena would not come readily to mind. It required therefore, I think, an intellectual tour de force, to bring games of chance to bear upon understanding of physical phenomena.” MARK KAC SMOLUCHOWSKI: each collision of a Brownian particle in a liquid is treated as a random event, the elementary probabilities of which are determined by elastic collision mechanics. (P. Langevin, 1908.)

r

0

a

Random walk on a square lattice of mesh size a with probability 1 ¥ 4 along each direction. A very long will look from far away like Brownian motion. Mean Square Displacement

¡

¢ N 2 ¢ 2 ¢

∑ ¡ r i 1 ai in a nomber N of steps ai ai a ai 0 :

2 N N N N ¡ ¡ ¡ ¡

¢ ¢ ¢

2 r N ∑ ai ∑ ai ∑ a j ∑ ai a j ¡ ¡ ¡ ¡ i 1 i 1 j 1 i ¢ j 1 N ¡ ¡ ¡

¢

2 ∑ ai ∑ ai a j £ ¡ ¡ i j 1 i ¡ j

¢ 2 Na £ General Theory, Einstein, December 1905 Einstein considers a variable α with a Boltzmann distribution ¡ ¢ ¦ N Φ α ¡ ¡ dn Ae RT dα F α dα ¤

A a normalisation coefficient, Φ α ¦ the potential energy associated with the parameter α. Here dn, proportional to the probability density of occupation, gives the elementary number of systems in states within dα of α in a system ensemble a` la Gibbs. Einstein states that the function F α ¦ stays stationary in an infinitely small time interval t under the combined effect of the force associated with potential Φ and of the irregular thermal phenomena. Consider the real line representing the variable α, at an arbitrary value α0. During time t, as many systems must pass through point α0 in both directions. ∂Φ The force ¢ ∂α derived from potential Φ gives a change of α per unit of time: dα ∂Φ ¡ ¢ B ¤ dt ∂α where B is, according to Einstein, the “system’s mobility with respect to ¥ α”; given by B ¡ 1 µ in terms of viscous friction. Whence the signed variation of the number of systems passing through α0 in time t:

∂Φ ¦ ¡ ¡ ¢

α ¨ n1 B ∂α tF 0 α α0 Brownian Fluctuations Assume the probability for the parameter α to change, in the same time t and under irregular thermal processes, by an amount ∆ is equal to ¦ ¦ ¡ ϕt ∆ ϕt ¢ ∆ , independent of α. (Intrinsic nature of thermal motion).

The numbers of systems n passing through point α0 during time t in the positive or negative direction are respectively

¢ ∞ ¦ ¦ ¡ £ ¡

n F α0 ∆ χt ∆ d∆ ¤ 0 ¦ where χt ∆ represents the cumulated probability that the system makes a step of at least ∆ in the positive or negative directions in time t: ¢ ¢ ∞ ∞ ¤ ¤ ¤ ¤ ¦

¦ ¦ ¡ ¡ χt ∆ ϕt ∆ d∆ ϕt ¢ ∆ d∆ ∆ ∆ Stationarity at Equilibrium Invariance of the equilibrium distribution F α ¦ : Algebraic conservation law for the ensemble number £ £ ¡ ¢ ¨ n1 n n 0

Using the expressions of n1 and n , and for infinitesimally small t, hence ¦ infinitesimally small ∆’s for which ϕt ∆ differs from 0, one finds at first order: ¤

¦ ¡ ∂Φ ¦ £

¡ 1 2 ¡

α α ∆ ¨ B ∂α tF 0 F 0 t 0 2 α α0

¢ ∞ ¦ ¡

2 ¡ 2 ∆ t ∆ ϕt ∆ d∆

∞ represents the mean square fluctuation of α induced by thermal agitation over time t. Boltzmann Brown The exponential form of the Boltzmann-Gibbs distribution ¡ ¢

¦ N ¦ F α ∝ exp ¢ Φ α RT automatically satisfies equation (1), for any potential Φ. Mean Square Fluctuations

¡ RT 2 ¡ ∆ 2B t ¨ t N Einstein’s study shows that Boltzmann’s equilibrium distribution, in a dynamical interpretation, implies the existence of a Brownian diffusion movement for any physical quantity α for which the system possesses a mobility. Inverting the point of view, one can consider the equilibrium ¢ £ 2 ¡ equation as an equation in F α , where ∆ t ∝ t is independent of α, and where t is arbitrary. The solution then takes on the exponential form of the Boltzmann distribution, ∆2 ¡ where the ratio RT ¢ t then appears as a time-independent N 2Bt universal parameter tied to Brownian diffusion. The general dynamical study of Brownian motion by Einstein thus implies the Boltzmann-Gibbs equilibrium distribution. Einstein’s Unpublished Lecture, Zurich,¨ 1910

“ON BOLTZMANN’S PRINCIPLE AND SOME IMMEDIATE CONSEQUENCES THEREOF” A particle suspended in a fluid, and slightly denser than the latter.

¡ : the particle sinks to the bottom.

¡ Boltzmann: the particle changes its height z ceaselessly and in an irregular way, and a probability density n is associated to each height z : ¢ £ £ n z ¢

¢ N

£ ¢ £ exp m m gz n 0 RT 0 Statics & Dynamics

¡ Mean height z : ¡ ¢

N¡ m m¡ gz ze RT 0 dz RT 1 ¢ ¢ £ ¢ £

z ¡ ¢

N¡ m m¡ gz e RT 0 dz N g m m0

¡ Stokes’ law: the spherical particle falls in time τ by £ ¢ g m m

¢ 0

δ τ 6πηa with η fluid’s viscosity, a particle’s radius. Nota Bene:

¤ RT 1 ¢ z δ τ £ N 6πηa

£

z

¢

0 ¢

with

∆

¡ ∆ mean

∆

δ

z ¢ the

z ¡ amount

displaced

∆ z is

τ

a £ height η 2 1 particles, z π

at ¢ motion. random 3 τ of

a ¡ T particle N 2

R ¥

by ¢ time ∆

ution

ownian Fluctuations δ wnian 2 Br z δ

z

after

2 ¢ of equally Bro of distrib is

z

£

¡

z 2 the the ards that

∆

τ : gligeable; of

0 ¢ ne height wnw 2

time ¡ δ at equals do

∆

displacement

2

the or z

δ £

of ¢ 0

of ¢ small,

stationarity ¡ particle ards τ

law ¡ ∆ During A By or

δ

¡ ¡ ¡

∆

upw F the value P. LANGEVIN, 1908

dV ¢ ¢

m µV X µ 6πηa dt The Langevin force X sustains the agitation of the particle against viscous damping:

¢ £ ¢ £ ¢ £ ¡ ¢ X t X t 2µkBTδ t t ¡

1 2 ¢ 1 RT ¢ 1 Equipartition of energy: m V k T equal to 2 2 N 2 B that of a gaseous (Maxwell). Implies Einstein-Sutherland’s formula. PERRIN'S EXPERIMENTS Height Distribution, 1908

Atoms: “An EXPERIMENTUM CRUCIS could be obtained, deciding for or against the molecular theory of Brownian motion.”

z

5

4

3

2

(m m )g z 1 n(z) 0 = e kB T n(0) 0.2 0.4 0.6 0.8 1 ¤ ¢ 23 N 7 £ 05 10 Gaussian Distribution of Displacements

Grains diffusing from Counting particles. a centre in a plane orthogonal to ¤ ¢ 23 Second determination: N 7 £ 15 10 Brownian Trajectories, 1908

Positions of grains of radius 0 5µm, taken every 30s. Brownian Trajectories, 1908

Bottom Left: Positions of a grain of radius 0 £ 5µm, as recorded every 30s by Perrin’s team. Top Right: The continuum limit. Insert: ad infinitum of Brownian motion, non-differentiability. Non-Differentiable Functions

“The direction of the tangent is found to vary absolutely irregularly as the time between two instants is decreased. An unprejudiced observer would therefore come to the conclusion that he was dealing with a function without derivative, instead of a curve to which a tangent could be drawn.”

These remarks stimulated the young , and prefigured the concept of of Benoˆıt MANDELBROT. Brownian Rotational Motion

¡ RT 1 ϑ2 ¢ t t N 4πηa3 Angular motion’s variation faster (1 ¤ a3) than translational motion’s (1 ¤ a).

Perrin prepared 50 µm spherules that could be put in suspension in with 27% urea, with a Brownian angular velocity of a few angular degrees per mn. Refringent inclusions made their rotation observable! Whence a spectacular verification of Einstein’s second formula. Einstein wrote to Perrin (11 November 1909): “I would not have considered a measurement of rotations as feasible. In my eyes it was only a pretty triffle”. Perrin received the Nobel Prize in in 1926 for his experiments on Brownian motion. Conclusion of Atoms: “La theorie´ atomique a triomphe.´ Nombreux encore naguer` e, ses adversaires enfin conquis renoncent l’un apres` l’autre aux defiances´ qui longtemps furent legitimes´ et sans doute utiles.” “The has triumphed. Its opponents, who until recently were numerous, have been convinced and have abandoned one after the other the sceptical position that was for a long time legitimate and likely useful. Equilibrium between the instincts towards caution and towards boldness is necessary to the slow progress of human ; the conflict between them will henceforth be waged in other realms of thought. [...] Nature reveals the same wide grandeur in the and the nebula, and each new aid to knowledge shows her vaster and more diverse, more fruitful and more unexpected, and, above all, unfathomably immense.” EINSTEIN, 1910: “If to conclude we ask once more the question: “Are the observable physical facts correlated one to another in an entirely causal way?”, we must surely answer this question in the negative. The positions of a particle engaged in a Brownian motion at two instants separated by one second must always appear, even to the most conscientious observer, as independent from each other, and the greatest mathematician would never succeed in any determined case to compute in advance, even approximately, the path covered in a second by such a particle. According to the theory, to be able to do so one should know the position and speed of each molecule exactly, which appears in principle excluded. However, the laws of mean values, which proved themselves everywhere, as well as the statistical laws of fluctuations, valid in these domains of finest effects, lead us to the conviction that in theory we must firmly hold onto the hypothesis of a complete causal connection of events, even if we should not hope to ever obtain by improved observations of Nature the direct confirmation of such a concept. EINSTEIN, 1949 “The agreement of these considerations with experience together with Planck’s determination of the true molecular size from the law of radiation (for high ) convinced the sceptics, who were quite numerous at that time (Ostwald, Mach) of the reality of atoms. The antipathy of these scholars towards atomic theory can undubitably be traced back to their positivistic philosophical attitude. This is an interesting example of the fact that even scholars of audacious spirit and fine instinct can be obstructed in the interpretation of facts by philosophical prejudices. The prejudice–which has by no means died out in the meantime–consists in the faith that facts by themselves can and should yield scientific knowledge without free conceptual construction. Such a misconception is possible only because one does not easily become aware of the free choice of such concepts, which, through verification and long usage, appear to be immediately connected with the empirical material.” Einstein 09.03.2005 9:07 Uhr Seite 1

Institut Henri Poincar · Amphi Hermite 11, rue Pierre et Marie Curie · 75005 Paris

O. Darrigol : La gense de la Relativit · 10h00 C. M. Will : Tests of · 11h00 B. Duplantier : Le mouvement brownien · 14h00 Ph. Grangier : Expriences un seul photon · 15h00 T. Damour : Einstein pistmologue · 16h00

Einstein, 1905-2005 Sminaire Poincar Samedi 9 avril 2005

Crdit photos : Julie Mehretu, excerpt (suprematist evasion), 2003 ink and acrylic on canvas, 32"x54" Fondation Archives Lotte Jacobi, Universit du New Hampshire Iagolnitzer www.lpthe.jussieu.fr/poincare BIRKHA¨ USER, 2006