BROWNIAN MOTION 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 Particles Contained in the Pollen of Plants; and on the General Existence of ACTIVE MOLECULES 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 liquid, which cause the pressure, 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 ALBERT EINSTEIN, Annalen der Physik, 17, 549-560 (1905): “On the Motion of Small Particles Suspended in Liquids at Rest Required by the Molecular-Kinetic Theory of Heat” Aim: to determine Avogadro’s number and the size of atoms “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 Diffusion 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 temperature ¡ R: ideal gas molar constant ¥ £ ¤ ¢ ¡ 23 AVOGADRO’S NUMBER N 6 £ 02 10 ¡ η: liquid’s viscosity ¡ a: particle’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 “Subtle is the Lord...”) 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 friction coefficient Van ’t Hoff’s law, 1884 The perfect gas law applies to the osmotic pressure of molecules in dilute solution. 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 random walk 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.