# Brownian Motion

Laplace’s Mini World · Plate 78 · Matter, Light & Sound · Matter

> Under a microscope, specks in water never keep still. Unseen water molecules push them about; measure how far they wander and you can count the molecules.

- Interactive model: https://world.lapla.si/en/brownian/
- Embed: https://world.lapla.si/embed/en/brownian/
- 中文: https://world.lapla.si/zh/brownian/
- Updated: 2026-11-02

## 01 Particles from Pollen Grains

In the summer of 1827 the botanist Robert Brown looked at the pollen of a plant called Clarkia pulchella. Its grains were full of particles about 5 to 6 µm long. He put them in water and saw many of them plainly moving: changing place, and now and then turning on their long axis.

After watching again and again he was sure it was neither a current in the water nor its evaporation. The motion belonged to the particles.

The pollen grains themselves were not moving. A grain is about 100 µm across, far too big to show it; Brown watched the small particles that came out of the grains, adrift in the water. They are now known to be tiny bodies holding starch (the long ones) and oil (the round ones).

*Pollen grain about 100 µm · long particles about 5⁠–⁠6 µm · round ones about 1 µm*

## 02 Not Alive

The motion looks very like tiny creatures swimming. So Brown looked at pollen from plants kept in a herbarium for more than twenty years, some for at least a century: its particles still moved. So did those of mosses dried for over a hundred years.

Then he tried things that were never alive. A splinter of window glass, crushed on the microscope’s stage, at once gave plenty of particles of the same size and the same motion. Rocks of every age, granite (one piece from the Sphinx), lava, meteorites, several metals and the soot that settled everywhere in London all did the same.

In 1829 he made a point of saying that he had never called the particles alive. Why they moved he could not account for, and he would not guess.

*Century-⁠old pollen · window glass · granite · soot · metals: the particles all move*

## 03 Pushed from Every Side

In 1888 the French physicist Louis-⁠Georges Gouy noted that the smaller the particles, the livelier the motion; it grows with temperature and differs from one liquid to another. It owes nothing to currents, vibrations or differences of temperature: it comes from the nature of liquids. Perrin later wrote that cutting the light a thousandfold changed nothing, and that the motion is the same at night in a cellar in the country as by day beside a street of heavy vehicles.

Today’s explanation: the water molecules round a grain are always jostling it. A mastic grain about 1 µm across is some 20 billion times heavier than a water molecule, so a single hit cannot move it. But at every instant the pushes from all sides fail to cancel exactly, and what is left over keeps changing in size and direction. The jiggling we see is the sum of a vast number of uneven pushes.

The smaller the grain, the less the liquid holds it back and the further it goes; the warmer the liquid, the harder its molecules move; the thicker the liquid, the harder the grain’s way.

*Smaller, warmer, thinner: the livelier the motion*

## 04 A Rule in the Randomness

Each step goes in a random direction, a random distance: a random walk. Watching one grain, you cannot say where it will be next; watching many, a rule appears.

Start a crowd of grains at one point and they spread into a cloud. The cloud grows, but more and more slowly: the square of the distance from the start, averaged, grows in proportion to time. Four times the time gives only twice the distance.

A drop of ink spreading through still water is the same kind of diffusion. Diffusion through a cell’s membrane is on the page Across the Cell Membrane.

*Red ring: the measured spread of 64 grains · dashed: the law · the page’s model*

## 05 A Prediction, 1905

In 1905 Albert Einstein worked out, from the molecular theory of heat, that grains suspended in a liquid must move like this. In a time t, the square of a grain’s displacement along one direction averages ⟨x²⟩ = 2Dt.

D, the diffusion coefficient, is D = RT/(N·6πηr): T is the temperature, η the liquid’s viscosity, r the grain’s radius, R the gas constant and N Avogadro’s number, the number of molecules in a mole. Measure how far grains wander and you can work out N.

He gave an example: in water at 17 °C a grain 0.001 mm across moves about 0.8 µm in a second and about 6 µm in a minute. With today’s constants and water’s viscosity it comes to 0.89 µm and 6.9 µm. He was not sure this was the “Brownian molecular motion” others had described, because the accounts he had were too imprecise; the paper ends with the hope that someone would soon settle the question.

*⟨x²⟩ = 2Dt · water at 17 °C, a 1 µm grain: about 6 µm in a minute*

## 06 A Dot Every 30 Seconds

From 1908 the French physicist Jean Perrin set out to test the prediction. First he needed grains all of one size: he made gamboge and mastic, two resins, into tiny spheres in water and sorted them by size in a centrifuge, again and again. Once he worked through a kilogram of gamboge, daily for months, to get a few decigrams of grains about 0.75 µm across.

The microscope stood upright with a camera lucida on its eyepiece, so the eye saw the field and the squared paper beside it at once. He and a colleague, Dabrowski, took turns at the microscope: one called the time, the other dotted the same grain’s position on the paper every 30 seconds.

His three published tracings follow mastic grains about 0.5 µm in radius; one has 50 positions in a row. On the scale given in his book of 1913, 16 divisions stand for 50 µm.

*A dot every 30 s · 16 divisions = 50 µm · mastic grains, radius about 0.52 µm*

## 07 Jagged at Every Scale

Each straight segment in a tracing only joins two positions 30 seconds apart. Perrin warned that they give a very faint idea of how tangled the real path is: dotted every second, each segment would become a path of 30 segments as complicated as the whole drawing, and so on as you divide again.

So no point of the path has a tangent you could draw, and a measured “speed” has no definite value: the closer the dots, the faster the average comes out.

In the page’s model, one mastic grain dotted every 30 seconds averages about 0.21 µm a second; dotted every second, about 1.1 µm; 30 times a second, about 6.2 µm.

*Red dots: every 30 s · grey line: every second · the page’s model*

## 08 An Atmosphere in a Drop

Air thins with height: the molecules’ motion spreads them out, gravity pulls them down, and in balance the air halves about every 6 km up. Perrin saw that grains in water should settle into layers the same way, only much more steeply, since they are far heavier.

In a cell 100 µm deep he put gamboge grains of radius 0.212 µm and, through a pinhole, counted them at four heights, 5, 35, 65 and 95 µm above the floor: 13,⁠000 grains in all. The counts at the four heights stood as 100 : 47 : 22.6 : 12, halving about every 30 µm.

The halving height, with the grains’ size and density, gives Avogadro’s number; he got 70.5 × 10²². With today’s constants these grains should halve about every 34 µm (computed here): his height came out a little short, so his number came out a little high.

*5, 35, 65, 95 µm · 100 : 47 : 22.6 : 12 · halving about every 30 µm; air about every 6 km*

## 09 Counting Molecules

Run Einstein’s formula backwards and a measured displacement gives N. From about 3,⁠000 displacements Perrin got 71.5 × 10²², close to the 70.5 × 10²² from the layers.

In 1913 he set side by side more than a dozen quite different methods: the viscosity of gases, Brownian motion, the blue of the sky, black-⁠body radiation, radioactivity and more. Every value fell between 60 × 10²² and 75 × 10²². He wrote that one is seized with admiration at such precise agreement from such different phenomena.

One of the best-⁠known doubters of atoms, the chemist Wilhelm Ostwald, wrote in a preface of November 1908 that the counting of gas ions by J. J. Thomson and Perrin’s confirmation of Brownian motion now justified the most cautious scientist in speaking of the experimental proof of the atomic nature of matter. Perrin won the Nobel Prize in Physics in 1926. Today the Avogadro constant is fixed at 6.02214076 × 10²³; Perrin’s values were about 6 % to 19 % high.

*1909: 70.5 and 71.5 × 10²² · 1926: crude average 64 × 10²² · today: 60.22 × 10²²*

## 10 Straight, for an Instant

Perrin’s point that the path has no tangent anywhere holds at the scales a microscope shows. Over very short times a grain coasts on its own momentum in a straight line before its direction is scrambled. In a simple model, a mastic grain of radius 0.52 µm in water loses track of its direction in under 0.1 µs, after a straight stretch only about 0.2 nm long.

Its true speed over that stretch is about 2.5 mm a second along one direction, more than ten thousand times the “speed” read off a tracing dotted every 30 seconds. Einstein held that for such small grains this speed could not be found by observation. In 2010 physicists holding glass beads in air with optical tweezers measured the instantaneous velocity of a Brownian particle for the first time; in 2014 they measured it in water.

*Under 0.1 µs · about 0.2 nm · instantaneous velocity measured in air in 2010, in water in 2014*

## 11 The Microscope, Taken Apart

The microscope on the board comes apart in seven pieces: the camera lucida, the eyepiece, the body tube, the nosepiece, the slide, the condenser and the mirror.

Rest the pointer on one and it is lifted out and named; left alone, all seven are named beside it.

*Camera lucida · eyepiece · body tube · nosepiece · slide · condenser · mirror*

## 12 Size, Heat and Liquid

This is the page’s model: each grain moves at random by Einstein’s formula, with D from the grain’s radius, the temperature and the liquid’s viscosity (water by the IAPWS formula; glycerol and water, 50 % or 80 % glycerol by mass, by Cheng’s formula of 2008). The clock runs in real time.

Each fresh start, the model follows 30 grains from that moment, averages the squares of their displacements and sets the result beside 2Dt. With only 30 grains the two never quite agree; the longer the run, the closer they come.

*The page’s model · real time · 30 grains*

**Try this**: In the lab, pick water at 20 °C and a radius of 0.52 µm: about how far does a grain go along one direction in 30 seconds? Halve the radius, or warm the water to 80 °C: what does D become each time? And in 80 % glycerol?

## What the model leaves out

The grains in the eyepiece are this page’s model, not a recording: at every small step each grain moves a random distance along each of three directions, drawn from Einstein’s formula for its size, the temperature and the liquid. Grains do not touch each other and feel no walls (near a wall they really move more slowly). A grain drifting above or below focus turns paler and softer to look as it would in a microscope; this is not an optical calculation. On the cover the red dots come every 5 s; in Perrin’s chapter the clock runs 60 times fast, a dot every 30 s. The layers chapter is a side view drawn for clarity, its grains shown larger than life and far fewer. In the pollen chapter a long particle wanders as far as a sphere of the same volume would, an approximation. The tracings on the paper come from the same model, not from Perrin’s drawings. The microscope on the board is a model after brass microscopes of about 1905 and an Abbe-⁠pattern camera lucida, in proportions from museum examples; it is not the instrument Perrin used. “Under 0.1 µs” and “about 2.5 mm a second” come from the simplest model, which leaves out the water the grain drags with it; in water that part matters.

## Easy to get wrong

Brown did not see pollen grains move: at about 100 µm they are far too big; it was the small particles from inside them. Nor did Brown discover atoms or explain the motion. The jiggle is not one molecule striking once but very many pushes failing to cancel; a single water molecule is far too small to see. It is not a current, not convection and not living things swimming; dust dancing in a sunbeam is carried mostly by air currents, and Perrin said it should not be compared. The zigzag in a tracing is not the path the grain took but straight lines between its positions every 30 seconds. Perrin’s values of Avogadro’s number were about 6 % to 19 % above today’s, not right to two figures from the start.

## Common questions

### Did Brown see pollen grains moving?

A grain is about 100 µm across, far too big to show it; Brown watched the small particles that came out of the grains, adrift in the water.

See: [01 Particles from Pollen Grains](https://world.lapla.si/en/brownian/#pollen)

### What drives Brownian motion?

Today’s explanation: the water molecules round a grain are always jostling it.

See: [03 Pushed from Every Side](https://world.lapla.si/en/brownian/#kicks)

### How fast does a random walk spread?

The cloud grows, but more and more slowly: the square of the distance from the start, averaged, grows in proportion to time.

See: [04 A Rule in the Randomness](https://world.lapla.si/en/brownian/#walk)

### What did Einstein predict in 1905?

In 1905 Albert Einstein worked out, from the molecular theory of heat, that grains suspended in a liquid must move like this.

See: [05 A Prediction, 1905](https://world.lapla.si/en/brownian/#einstein)

### How did Perrin count molecules?

Run Einstein’s formula backwards and a measured displacement gives N.

See: [09 Counting Molecules](https://world.lapla.si/en/brownian/#count)

## Sources

1. [R. Brown (1828): A brief Account of Microscopical Observations … on the Particles contained in the Pollen of Plants, Philosophical Magazine 4 (the particles in the pollen, their size, currents and evaporation ruled out, old specimens, glass, rocks and a fragment of the Sphinx)](https://doi.org/10.1080/14786442808674769)
2. [The same, reprinted in The Miscellaneous Botanical Works of Robert Brown, vol. 1 (1866)](https://archive.org/details/miscellaneousbot01brow)
3. [R. Brown (1829): Additional Remarks on Active Molecules, Philosophical Magazine 6 (he never said they were alive)](https://doi.org/10.1080/14786442908675115)
4. [P. Pearle et al. (2010): What Brown saw and you can too, American Journal of Physics 78 (pollen grains about 100 µm, the particles are amyloplasts and spherosomes, Brown did not see the pollen move)](https://arxiv.org/abs/1008.0039)
5. [A. Einstein (1905): Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Annalen der Physik 17](https://doi.org/10.1002/andp.19053220806)
6. [The same in English: Investigations on the Theory of the Brownian Movement (transl. A. D. Cowper, 1926)](https://archive.org/details/investigationson0000albe_e1q8)
7. [A. Einstein (1906): Zur Theorie der Brownschen Bewegung, Annalen der Physik 19 (Siedentopf and Gouy)](https://doi.org/10.1002/andp.19063240208)
8. [L.-G. Gouy (1888): Note sur le mouvement brownien, Journal de Physique 7](https://hal.science/jpa-00238904)
9. [J. Perrin (1909): Mouvement brownien et réalité moléculaire, Annales de chimie et de physique 18 (camera lucida and squared paper, every 30 s, the grains’ layers, Avogadro’s number)](https://fr.wikisource.org/wiki/Mouvement_brownien_et_r%C3%A9alit%C3%A9_mol%C3%A9culaire)
10. [The same in English: Brownian Movement and Molecular Reality (transl. F. Soddy, 1910)](https://archive.org/details/brownianmovement00perr)
11. [J. Perrin (1913): Les Atomes, chapter IV (the tracings’ scale: 16 divisions = 50 µm)](https://fr.wikisource.org/wiki/Les_Atomes/IV)
12. [J. Perrin (1913): Les Atomes, conclusions (a dozen methods agreeing)](https://fr.wikisource.org/wiki/Les_Atomes/Conclusions)
13. [J. Perrin (1926): Nobel lecture, Discontinuous Structure of Matter](https://www.nobelprize.org/prizes/physics/1926/perrin/lecture/)
14. [nobelprize.org: the Nobel Prize in Physics 1926](https://www.nobelprize.org/prizes/physics/1926/perrin/facts/)
15. [W. Ostwald: Outlines of General Chemistry (English of the 4th German edition, 1912), preface of November 1908](https://archive.org/details/outlinesgeneral00taylgoog)
16. [NIST: Avogadro constant (CODATA 2022, exact)](https://physics.nist.gov/cgi-bin/cuu/Value?na)
17. [NIST: Boltzmann constant (CODATA 2022, exact)](https://physics.nist.gov/cgi-bin/cuu/Value?k)
18. [NIST: the 2019 redefinition of the SI](https://www.nist.gov/si-redefinition)
19. [IAPWS: viscosity of ordinary water substance (2008)](https://www.iapws.org/relguide/visc.pdf)
20. [M. L. Huber et al. (2009): New International Formulation for the Viscosity of H₂O, J. Phys. Chem. Ref. Data 38 (Eq. 37)](https://doi.org/10.1063/1.3088050)
21. [N.-S. Cheng (2008): Formula for the Viscosity of a Glycerol−Water Mixture, Ind. Eng. Chem. Res. 47](https://doi.org/10.1021/ie071349z)
22. [J. B. Segur and H. E. Oberstar (1951): Viscosity of Glycerol and Its Aqueous Solutions, Ind. Eng. Chem. 43](https://doi.org/10.1021/ie50501a040)
23. [T. Li, S. Kheifets, D. Medellin and M. G. Raizen (2010): Measurement of the Instantaneous Velocity of a Brownian Particle, Science 328](https://doi.org/10.1126/science.1189403)
24. [S. Kheifets et al. (2014): Observation of Brownian Motion in Liquids at Short Times, Science 343](https://doi.org/10.1126/science.1248091)
25. [R. Huang et al. (2011): Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid, Nature Physics 7](https://doi.org/10.1038/nphys1953)
26. [T. Li and M. G. Raizen (2013): Brownian motion at short time scales, Annalen der Physik 525 (Einstein held the instantaneous velocity unobservable)](https://arxiv.org/abs/1211.1458)
27. [Smithsonian: a Zeiss microscope of about 1910 (its size and parts, for the model’s proportions)](https://www.si.edu/object/nmah_1349170)
28. [Science Museum Group: Abbe-⁠type camera lucida, 1910⁠–⁠1930](https://collection.sciencemuseumgroup.org.uk/objects/co120062/abbe-type-camera-lucida)

## Data

History after R. Brown (Philosophical Magazine, 1828 and 1829), A. Einstein (Annalen der Physik 17, 1905, and 19, 1906; English by A. D. Cowper, 1926), L.-G. Gouy (Journal de Physique, 1888), J. Perrin (Annales de chimie et de physique, 1909; English by F. Soddy, 1910; Les Atomes, 1913; Nobel lecture, 1926) and W. Ostwald’s preface of November 1908 (English, 1912); what Brown saw after P. Pearle and colleagues (American Journal of Physics, 2010). The constants k and N\_A are the exact SI values (CODATA 2022); water’s viscosity from the formula of M. L. Huber and colleagues (J. Phys. Chem. Ref. Data, 2009) for the IAPWS 2008 formulation; glycerol–water mixtures from N.-S. Cheng (Ind. Eng. Chem. Res., 2008). The short-⁠time measurements after T. Li and colleagues (Science, 2010) and S. Kheifets and colleagues (Science, 2014). Every grain in the eyepiece is this page’s model: each moves by random steps whose spread follows Einstein’s formula for its size, the temperature and the liquid; the tracings on the paper are drawn from the same model, not copied from Perrin’s figures. The microscope on the board is this page’s model of a brass stand of about 1905 with an Abbe-⁠pattern drawing apparatus, in proportions from museum examples (Smithsonian, Science Museum Group), not a copy of any instrument Perrin used. Material presets and the model kit as on the site’s other model plates (public/looks/CREDITS.md).

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## Cite this plate

Laplace’s Mini World, Plate 78, “Brownian Motion”, https://world.lapla.si/en/brownian/, CC BY-NC 4.0, updated 2026-11-02.