Brownian motion: from wandering to a measurable law: Introduction · A motion the theory requires

Fractions and ratios

What does dividing one quantity by another tell you?

The fraction 6/2 is 3, because 2 fits into 6 three times; it also says that 6 is three times 2. Every fraction a/b works the same way. The bottom number can never be zero, since no count of zeros adds up to 6.

When the two quantities have different units, their ratio is a new quantity with a unit of its own: 12 micrometres travelled in 4 seconds is 3 micrometres per second. When they have the same unit, the units cancel and the ratio is a plain number: 2 particles out of 8 is 0.25 of the group.

Worked example: One in four, two in eight

1. In a group of 4 particles, 1 moves to the right: the fraction is 1/4 = 0.25.
2. In a group of 8, 2 move to the right: the fraction is 2/8 = 0.25.
3. Doubling both the number that moved and the size of the group leaves the fraction at 0.25.

Where this lesson stops

A ratio of two quantities in the same unit is a plain number. A ratio of different units, such as micrometres per second, is a new quantity and keeps its unit.

This lesson builds on

• Adding and averaging

A sign records direction

How can movement add up to zero?

Put the starting point at zero on a ruler. A final mark three units to the right has displacement +3; one three units to the left has displacement −3. Adding the signed displacements gives zero. Adding the distances from the start gives six.

Displacement compares where something ends with where it started, whatever it did in between. A walker who goes three units right and comes back has displacement zero after walking six.

Worked example: Two walkers, opposite directions, average zero

Two walkers both finish one unit from their start, one at −1 and the other at +1. The signed average is zero. The average distance is one.

Where this lesson stops

The sign names a direction relative to a chosen axis; it does not mean a negative distance.

This lesson builds on

• Adding and averaging

Powers of ten and physical units

How do you write a very small length without counting zeros?

The Brownian paper works its example, in §5, for particles 0.001 mm across. In metres that is 0.000001 m: easy to miscount by one zero, which is a factor of ten. Written as 10⁻⁶ m, the −6 says the decimal point has moved six places to the left, and that length has a name of its own, the micrometre (μm).

The unit goes through the arithmetic with the number. A micrometre is 10⁻⁶ m, so a square micrometre is 10⁻¹² m², not 10⁻⁶ m²: squaring a length squares its unit, power of ten included.

Worked example: Squaring six micrometres

1. 1 micrometre (1 μm) is 10⁻⁶ metres, or 0.000001 m.
2. The paper's displacement after one minute, about 6 μm, is 6 × 10⁻⁶ m.
3. Squaring it squares the number and the unit: (6 × 10⁻⁶ m)² = 36 × 10⁻¹² m² = 3.6 × 10⁻¹¹ m².

Where this lesson stops

A measurement is a number and a unit together. Squaring it squares both: (6 × 10⁻⁶ m)² is 3.6 × 10⁻¹¹ m².

This lesson builds on

• Fractions and ratios

Squares and square roots

If the mean square grows four times, why does the typical distance only double?

Both +3 and −3 have square 9, so squaring removes the sign, and it weights a large distance more than a small one. A square root returns to the original unit: the square root of a square micrometre is a micrometre.

The root mean square, or RMS, uses both steps: square each displacement, average the squares, then take the square root. The result is a typical distance from the start, in the original unit.

The square root of four q is twice the square root of q, for nonnegative q.

Worked example: Doubling every displacement, and what the RMS does

1. The displacements −3, −1, +1, +3 have squares 9, 1, 1, 9. Their average is 5, so the RMS is √5, about 2.236.
2. Double each displacement: −6, −2, +2, +6. The squares are 36, 4, 4, 36, and their average is 20.
3. The mean square went from 5 to 20, four times as large. The RMS went from about 2.236 to √20, about 4.472: twice as large.

Where this lesson stops

Four times the mean square is twice the RMS. That is why, in the Brownian paper, waiting four times as long doubles the typical distance.

This lesson builds on

• Adding and averaging

Adding and averaging

What does an average keep, and what does it lose?

Four jars hold 3, 1, 1 and 3 marbles. Pour them together and there are 8. Share the 8 equally between the four jars and each holds 2. Two is the average, although no jar held 2 to begin with.

The average keeps the total and the count and nothing else, so 3, 1, 1, 3 and 2, 2, 2, 2 have the same average. Listing every value twice doubles both the total and the count, and the average stays at 2.

Worked example: Adding four numbers and dividing by four

1. Add 3 + 1 + 1 + 3 to get 8.
2. Count the values: there are four.
3. Divide 8 by 4 to get 2.
4. List the values twice: 16 divided by 8 is still 2.

Where this lesson stops

An average is the total shared equally among the observations, and two very different lists can have the same one.

Mean, variance and RMS

If particles wander both ways, which average shows how far they got?

Four particles move −3, −1, +1 and +3 micrometres. Their average position has not changed, yet they have spread out. Three averages answer three different questions.

Add the M signed displacements and divide by M.

Square each displacement before adding and dividing by M.

Variance is the mean square minus the square of the mean.

RMS means root mean square: average the squares, then take the square root. The standard deviation is the square root of the variance, so it measures spread around the mean, where the RMS measures distance from zero. The two are equal only when the mean is zero. The average of the distances, ignoring sign, is a third statistic and gives a different number.

Worked example: Four displacements, four different averages

1. For −3, −1, +1, +3 the signed sum is 0, so the mean is 0.
2. The distances, ignoring sign, are 3, 1, 1, 3; their mean is 2.
3. The squares are 9, 1, 1, 9; their mean is 5.
4. The RMS is the square root of 5, about 2.236. With a mean of 0, the variance is also 5 and the standard deviation is also about 2.236.
5. A small random sample need not have a signed mean of exactly zero, even when the model's mean is zero.

Where this lesson stops

Say which average you mean. Squaring then averaging, averaging then squaring, and averaging the distances give three different numbers.

This lesson builds on

• A sign records direction
• Squares and square roots

Orders of magnitude

How big, how fast and how energetic are the things in the 1905 papers, measured against each other?

Two numbers of the same order differ by less than a factor of ten. Numbers several orders apart belong to different worlds, and comparing them usually tells you which effect can be ignored.

Sizes. A water molecule is about 0.3 nanometres across and a Brownian grain about 1 micrometre: the grain is roughly 3,000 times larger. The light-quanta paper prints the mass of a hydrogen atom as 1/N gram, 1.62 × 10⁻²⁴ g.

Speeds against light. The ratio v/c is about 3.3 × 10⁻⁶ for a rifle bullet at 1,000 m/s, 8.3 × 10⁻⁷ for a jet at 250 m/s, and 9.9 × 10⁻⁵ for the Earth in its orbit at 29.8 km/s. The corrections of relativity go as the square of v/c, so for the Earth they are about 10⁻⁸. Only Kaufmann's fast electrons, a large fraction of the speed of light, reached speeds where they are not small.

Energies of light. One quantum hν of red light (650 nm) carries about 1.91 electron volts, green (530 nm) 2.34 eV, and ultraviolet (250 nm) 4.96 eV. The light paper's photoelectric estimate, about 4.3 volts, is of the same order.

Molecular kicks. An estimate of how often water molecules strike a grain 1 μm across gives about 1.6 × 10¹⁹ blows a second, and about 6 × 10¹⁹ for a grain of 1 μm radius. The assumptions are rough, so the honest statement is 10¹⁹ to 10²⁰ blows a second: far too many for any single blow to be seen.

Worked example: Two estimates, done with powers of ten

1. The Earth: v/c = 29,800 ÷ 299,792,458 ≈ 9.9 × 10⁻⁵. Squared, that is about 9.9 × 10⁻⁹, near 10⁻⁸.
2. Water's molecules: 1,000 kg per cubic metre divided by 0.018015 kg per gram-molecule, times 6.02 × 10²³, is 3.34 × 10²⁸ molecules per cubic metre.
3. Molecules near 600 m/s crossing a surface at a quarter of n times their mean speed give about 5 × 10³⁰ strikes per square metre each second.
4. A sphere 1 μm across has a surface of 3.1 × 10⁻¹² m², so it takes about 1.6 × 10¹⁹ strikes a second. The inputs are rough, so the result is an order of magnitude, 10¹⁹ to 10²⁰.

Where this lesson stops

This lesson places numbers on a scale of powers of ten. Converting the papers' printed units into SI units is a separate step, with its own conventions.

This lesson builds on

• Powers of ten and physical units
• Fractions and ratios

Probability and independence

When you square a sum of random steps, why do the mixed terms average to zero?

Flip a coin twice and take one step for each flip: a metre to the right for heads, a metre to the left for tails. How far from the start do you end up, on average?

1. The four outcomes, right-right, right-left, left-right and left-left, are equally likely: one chance in four each.
2. They leave you 2 m to the right, back at the start, back at the start, or 2 m to the left.
3. Square those distances so that left and right count alike: 4, 0, 0 and 4. Their average is 2, one square metre for each step.

Combining the two steps added nothing beyond one square metre each. In symbols, with Δ₁ and Δ₂ for the two steps, squaring the sum gives each step squared and a mixed term, twice their product:

Delta one plus delta two, squared, equals delta one squared plus two delta one delta two plus delta two squared.

In the coin walk the product of the two steps is +1 for right-right and left-left and −1 for the other two, so on average it is zero and the mixed term adds nothing. Two conditions make that happen. Independence: learning the first step does not change the chances for the second, so the chance of a pair is the product of the two chances. Centring: each step averages zero, left as likely as right. Then the average of the product is the product of the averages, zero times zero:

The average of delta one times delta two equals the average of delta one times the average of delta two, which is zero.

Both conditions matter. If each step drifts, averaging +1, the product of the averages is 1, not 0. If the second step tends to copy the first, the product is positive more often than negative. The Brownian paper's §4 assumes that a particle's motions in successive intervals are independent, as long as the intervals are not chosen too small.

Worked example: What changes when the second step copies the first

1. Now let the second flip always copy the first. Only right-right and left-left can happen, one chance in two each.
2. The product of the two steps is +1 every time, so the mixed term no longer averages zero.
3. You always end 2 m from the start, so the squared distance is 4 every time: the average is 4, not 2. Independence removed that extra 2.

Where this lesson stops

Before dropping a mixed term, ask two things: does learning one step change the chances for the next, and does each step average zero?

This lesson builds on

• Adding and averaging
• A sign records direction

From steps to spread

Why does typical displacement grow as the square root of the step count?

A walker flips a coin before every step: heads, one metre to the right; tails, one metre to the left. After four steps, how far from the start is the walker likely to be? Not four metres: the steps partly cancel. Count the walks, and square each final distance so that left and right count alike:

1. After one step the walker is 1 m away, every time. The squared distance is 1.
2. After two steps there are four equally likely walks, ending 2 m right, at the start, at the start, or 2 m left. The squared distances 4, 0, 0 and 4 average 2.
3. After four steps there are sixteen equally likely walks. Two end 4 m out, eight end 2 m out and six end at the start. The squared distances average (2 × 16 + 8 × 4) ÷ 16 = 4.

The average squared distance grows by one square metre per step: after n steps it is n. The typical distance, its square root, grows only as the square root of n: 1 m after one step, about 1.4 m after two, 2 m after four, 10 m after a hundred. To get twice as far takes four times as many steps.

In symbols, call the steps Δ₁, Δ₂ and so on, and the position after n steps X, their sum:

Position after n steps is the sum of the n signed increments.

Suppose each step averages zero and has the same mean square l², where l is the typical size of one step (1 m in the coin walk). Squaring the sum gives the n squared steps and many products of two different steps. When the steps are independent and each averages zero, every such product averages zero, as the lesson on independence shows for two coin steps. Only the squares are left to add up:

The mean square is n times ell squared, and RMS displacement is ell times the square root of n.

Worked example: A hundred steps, then four hundred

A hundred steps of typical size 1 m leave a typical distance of 10 m from the start, because the square root of 100 is 10. Four hundred steps leave 20 m: four times the steps, twice the distance. No single walker has to end exactly 10 or 20 m away; these are typical sizes over many walkers. Nor is the distance from the start the length of the path: to end 10 m from home, the walker has walked 100 m.

Where this lesson stops

The coin walk is a picture, not a claim that molecules push a particle in fixed jumps. The Brownian paper needs only that the particle's moves in successive intervals are independent of one another.

This lesson builds on

• Probability and independence
• Mean, variance and RMS

Temperature and thermal energy

What does a temperature say about the jostling of molecules and grains?

The air in a room at 20 °C is a crowd of molecules moving in every direction and colliding billions of times a second. Warm the room and, on average, they move faster. The kinetic theory of heat reads temperature this way: through the average energy of motion of the particles, taken over very many of them.

For any particle in equilibrium with its surroundings, the average energy of motion along each direction is the same:

The average of one half m v x squared equals one half k B T.

Here m is the particle's mass, $v_x$ its speed along x, T the absolute temperature, and $k_B$ = 1.38 × 10⁻²³ joules per kelvin is Boltzmann's constant. It equals R/N, the gas constant shared out per molecule. The Brownian paper writes RT/N where a modern text writes $k_BT$; the paper's own letter k means the viscosity, not this constant.

The average is the point. One molecule's energy of motion changes at every collision, from almost nothing to several times the average. The temperature fixes how energy is shared out over many particles. It is not the energy of any one of them.

Nothing in the rule mentions size. A grain ten billion times heavier than a molecule has the same average energy of motion, so it moves more slowly, by the square root of the mass ratio. That is why a suspended grain belongs to the same thermal story as a dissolved molecule, as §1 of the paper insists.

Worked example: A nitrogen molecule and a half-micrometre grain at 20 °C

1. At T = 293 K, $k_BT$ = 1.381 × 10⁻²³ × 293 = 4.05 × 10⁻²¹ J, so the average energy of motion along one direction is half of that, 2.02 × 10⁻²¹ J.
2. A nitrogen molecule has a mass of 4.65 × 10⁻²⁶ kg. Setting $\frac{1}{2} m v_x^2$ equal to $\frac{1}{2} k_BT$ gives a typical speed along one direction of √(4.05 × 10⁻²¹ ÷ 4.65 × 10⁻²⁶), about 295 metres a second.
3. A grain of radius 0.5 μm and density 1200 kg/m³ has a mass of 6.3 × 10⁻¹⁶ kg, about 1.4 × 10¹⁰ times the molecule's. The same average energy gives it about 2.5 millimetres a second.
4. In water that motion is turned about within some 70 nanoseconds: the grain's mass divided by its drag coefficient 6πηa. That is far too brief to follow, so what a microscope sees is the net result of the zigzag.

This is why the Brownian paper asks for a displacement over a time rather than a speed: the thermal speed is real, but it turns about far faster than any observer can watch.

Where this lesson stops

This lesson stops at the average energy of motion. How the jostling of many molecules makes a grain wander, and how far, is §4 of the Brownian paper.

This lesson builds on

• Energy of motion and inertia
• Powers of ten and physical units
• Squares and square roots

Energy of motion and inertia

If a body moves at the same speed but carries less energy of motion, what has changed?

A force that pushes a body through a distance does work on it, and the work becomes the body's energy of motion, its kinetic energy. That is separate from energy stored inside the body, as heat or in its chemistry.

For one body at everyday speeds, doubling the speed multiplies the energy of motion by four. At a fixed speed, doubling the mass doubles it. The mass-energy paper's title asks about a body's Trägheit, its inertia: its resistance to a change in its motion, which is measured by pushing it, not by weighing it.

In Newtonian mechanics, kinetic energy is one half times inertial mass times speed squared.

This formula holds at speeds small compared with the speed of light, and it is not exact close to that speed. So the mass-energy argument works with the exact expression and reads the mass from its low-speed limit, where it takes this form.

Worked example: Two bodies at the same speed, different masses

1. Two bodies move at 2 metres per second.
2. The 3 kg body has ½ × 3 × 2² = 6 joules of energy of motion; the 2 kg body has ½ × 2 × 2² = 4 joules.
3. The speed is the same, and the energies differ by 2 joules because the masses differ by 1 kg.

At the same low speed, the kinetic-energy drop equals one half times the mass decrease times the squared speed.

Here the two masses are given, so the example proves nothing about mass and energy. The 1905 argument runs the other way: it computes the energy difference from the light the body gives off, finds that it has this form, and reads the mass difference from the coefficient of ½v².

Where this lesson stops

At the same low speed, less energy of motion means less mass. The comparison gives only a difference in mass, never the total energy stored inside the body.

This lesson builds on

• Squares and square roots
• Fractions and ratios

bm-01: a built-in worked result

Static host-calculated example, not a running experiment or an observation. Canonical units and owner-supplied statuses are retained. Open bm-01 online

18 array outputs are not reproduced. This table does not claim to reproduce the interactive plot.

Settings (canonical units)

H

10

M

400

T

293.15

a

5e-7

axis

0

d

1

eta

0.001

h

0.02

interval

1

seed

1905

statistic

rms

Evaluator: source:sha256:5cf1023395b9ece25ce1bb1d33a03d9f305254e5c66cb2c5c6a73921897d54f2