Brownian motion

Brownian motion (or Brownian movement) is the ceaseless, irregular motion of small particles suspended in a liquid or a gas. It is produced by the collisions of the particles with the fast-moving molecules of the fluid. The botanist Robert Brown described it in 1828 for particles from pollen, and Albert Einstein derived its quantitative law in 1905 from the molecular-kinetic theory of heat.^[1][2] Between 1908 and 1913 Jean Perrin measured the motion in carefully prepared emulsions and used it to obtain the value of Avogadro's constant, which persuaded most of the remaining sceptics that molecules exist; that story is told in Molecular reality (history of science).^[3] The same random walk describes how molecules diffuse to the opening of a nanopore, and Louis Bachelier's 1900 random-walk model of prices is the ancestor of the mathematics used for stock markets.^[4]

Perrin said in his Nobel lecture that the phenomenon was "predicted by Lucretius, suspected by Buffon, and established with certainty by Brown".^[5] Lucretius, in the second book of On the Nature of Things, asks the reader to watch the specks in a sunbeam crossing a dark room: "many a speck, impelled by viewless blows, to change its little course, and beaten backwards to return again".^[6] Buffon and Spallanzani knew of the motion, Perrin notes, but possibly for lack of good microscopes took the dancing particles for rudimentary animalcules.^[3]

Robert Brown began the work in June 1827. The first plant he examined was Clarkia pulchella, whose pollen grains held particles about 1/4000th to 1/5000th of an inch (5 to 6 µm) long, and he saw many of them moving in water. He made all the observations with a simple microscope, "indeed with one and the same lens, the focal length of which is about 1/32nd of an inch", the single-lens kind that Antonie van Leeuwenhoek had used.^[1] He concluded after repeated observation that the motion arose neither from currents in the fluid nor from its evaporation.^[1]

The motion persisted in pollen from specimens that had been dried in a herbarium for more than twenty years, and in some that were not less than a century old. Brown then bruised plant and animal tissues in water and obtained the same moving molecules, as he called them, from all of them, and he went on to minerals. Window glass gave them, as did rocks of every age. He found them "in each of the constituent minerals of granite, a fragment of the Sphinx being one of the specimens examined", and in meteorites, lava and pumice, and in the soot of London.^[1] The following year he published a supplement to correct a misreading. Several writers had said he took the active molecules to be alive, and he wrote that this was an erroneous assertion. The smallest of them, he reported, were between 1/20,000th and 1/30,000th of an inch in diameter. To exclude evaporation he shook water containing the particles into almond oil, which produced droplets as small as 1/2000th of an inch, and the particles moved in those too with undiminished activity.^[7]

For decades the motion attracted little notice, and the physicists who mentioned it likened it to dust dancing in a sunbeam under the push of air currents. Perrin pointed out that neighboring specks in a sunbeam move together, whereas neighboring Brownian particles move independently even when closer together than their own diameters.^[3] The proposal that the cause lies in the liquid itself goes back to Christian Wiener in 1863, who ruled out the particles themselves and causes outside the liquid. William Ramsay in 1876, Delsaulx and Carbonelle, and Georges Gouy in 1888 saw how the impacts of molecules could do it, and Perrin calls Gouy's exposition particularly brilliant.^[3][4] Perrin added that the motion never stops. It can be watched for days, months and years in a closed cell, and it is seen in liquid inclusions that have been shut up in quartz for thousands of years.^[3]

The explanation was a hypothesis, and in Perrin's words, "it is nevertheless a hypothesis only."^[3] A particle a micrometer across is hit from all sides by very many molecules, and the impacts nearly cancel. Delsaulx and Carbonelle argued that when a surface is small enough, the pressures on it are unequal and shift from point to point, so that the resultant is no longer zero. Smaller particles are knocked about more violently than larger ones, as Brown had pointed out at his original discovery.^[3]

Einstein's paper, received by the Annalen der Physik on May 11, 1905, derived the motion from theory without knowing whether it was the Brownian motion of the literature. "It is possible that the movements to be discussed here are identical with the so-called 'Brownian molecular motion'," he began, but the information about that was "so lacking in precision, that I can form no judgment in the matter." He added that if the predicted movement could not be observed, "a weighty argument would be provided against the molecular-kinetic conception of heat."^[2][8]

The result was a diffusion law for a sphere of radius a in a liquid of viscosity η at absolute temperature T: D = k_BT / (6πηa), now called the Stokes-Einstein relation, where k_B is Boltzmann's constant (the gas constant divided by Avogadro's constant). The mean square displacement along one axis after a time t is 2Dt, so the distance travelled grows as the square root of the time and not in proportion to it. For a particle 1 µm across in water at 20 °C, D is about 0.43 µm² per second and the typical displacement along one axis in a second is about 0.9 µm.^[a]

William Sutherland in Australia reached the same formula at almost the same time, and Marian Smoluchowski published a kinetic theory of the motion in 1906.^[9][10] In 1908 Paul Langevin introduced a stochastic equation of motion for the particle, now called the Langevin equation.^[4]

Perrin, who learned of Einstein's paper through Langevin, spent months centrifuging a kilogram of resin to obtain a few decigrams of grains of equal size. He counted the grains at different heights in a sealed cell and tracked single grains with a camera lucida, drawing each grain's position every 30 seconds. The tracks led him to remark that if the intervals were 100 times shorter each segment would be replaced by a polygon as complicated as the whole figure, so that it is meaningless to speak of a tangent to the trajectory.^[3] Values of Avogadro's constant from these experiments, and from nine other phenomena, fell between 6.0 and 7.5 × 10²³ per mole.^[3] Perrin received the Nobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter, and especially for his discovery of sedimentation equilibrium".^[11]

The first mathematical definition of Brownian motion was given by Norbert Wiener in 1923, and the process is called the Wiener process.^[4] Bachelier had used a random-walk model of price changes in his 1900 thesis Théorie de la spéculation, before Einstein, and his work was rediscovered by Andrey Kolmogorov in 1931.^[4] Almost every path of the process is continuous and has no tangent at any point, the property that Perrin described in words in 1913.

A molecule in solution reaches a nanopore by diffusion before the electric field takes over, and once it is in the pore its motion is the sum of drift in the field and diffusion. In the model of Wanunu and colleagues, DNA motion is almost purely diffusive beyond a capture radius r* and dominated by drift in the field inside it. If the arrival of molecules limits the rate, Smoluchowski's theory of absorption by a hemisphere gives a capture rate of 2πDr* per unit concentration, where r* = d²μΔV / (8lD), d and l are the diameter and length of the pore, μ the mobility of the molecule and ΔV the applied voltage. The diffusion coefficient D cancels from the product, so in that regime the capture rate does not depend on the length of the DNA.^[12] Capture rate (nanopore) treats the subject in full.

Molecular Reality Corporation describes this motion in its own words. An overlay on its website titled "The Science of Molecular Streaming", dated February 2026, headlines "the molecular mosh pit" and says that molecules "spin, vibrate, and bounce like 'crusty punks in a mosh pit.'"^[13]

• Molecular reality (history of science)
• Jean Perrin
• Capture rate (nanopore)
• Maxwell's demon
• Johnson-Nyquist noise