Brownian motion: from wandering to a measurable law

On the molecular-kinetic theory of heat, Einstein writes, bodies just visible in a microscope must move enough to be seen, and this may be the Brownian motion, though the reports he could obtain were too imprecise for him to judge. If the motion and its laws are observed, classical thermodynamics is not exactly valid for spaces a microscope can resolve and the true size of atoms can be determined; if the prediction fails, that is a weighty argument against the molecular-kinetic view of heat.

The paper opens with a claim about what a theory requires. The molecular-kinetic theory of heat takes the heat of a liquid to be the irregular motion of its molecules. From that theory, Einstein announces, it will be shown that bodies suspended in a liquid and large enough to see in a microscope must, because of the molecules' thermal motion, move by amounts large enough to be detected easily with a microscope.

Open the foundation: Temperature and thermal energy

He does not claim to have explained a motion already seen. The motions to be treated, he writes, may be the same as the so-called Brownian molecular motion; but the information he could obtain about it was 'so ungenau, daß ich mir hierüber kein Urteil bilden konnte': so imprecise that he could form no judgment. The identification is left open. The paper predicts a motion and derives its laws; whether the reported motion is that one is a question for observation.

The second paragraph says what depends on the prediction, in both directions. If the motion, together with the laws the paper expects it to follow, can really be observed, then classical thermodynamics can no longer be regarded as exactly valid even for spaces that can be told apart in a microscope, and an exact determination of the true size of atoms becomes possible. If instead the prediction proves false, that would be a weighty argument against the molecular-kinetic view of heat.

Why would a visible motion limit thermodynamics? Classical thermodynamics describes a liquid in equilibrium at one temperature by a few quantities, such as its pressure and temperature, which then stay fixed. It has no place for a part of the liquid that keeps moving of its own accord, now one way and now another. On the molecular view such restless departures from the average are always present. For a body of ordinary size they are far too small to notice; the paper argues that for a grain about a thousandth of a millimetre across, the size §5 uses, they are large enough to see.

Why would it give the size of atoms? §§4 and 5 tie how far such a grain wanders in a given time to N, the number of real molecules in a gram-molecule. With N known, the mass of a single molecule is the mass of a gram-molecule divided by N, and its size can be estimated from the volume a gram-molecule fills. The introduction announces this; the later sections carry it out.

Open the foundation: Orders of magnitude

The road there: §1 argues that suspended bodies should exert osmotic pressure just as dissolved molecules do; §2 derives that from the molecular-kinetic theory; §3 turns it into a diffusion coefficient; §§4 and 5 give the spread of a grain over time and the displacement to look for.

A dissolved substance held behind a wall that passes only the solvent presses on it with the osmotic pressure. Classical thermodynamics, as Einstein characterizes it, expects small suspended bodies to exert no such force, because the free energy seems not to depend on where they are; on the molecular-kinetic view they differ from dissolved molecules only in size, so equally many of either, far enough apart, should press alike.

Take a liquid of total volume V. In part of it, a volume $V^*$, dissolve z gram-molecules of a non-electrolyte, a substance that does not split into ions, and separate $V^*$ from the pure solvent by a wall that lets the solvent through but not the dissolved substance. The dissolved molecules push on that wall. The push on each unit of its area is the osmotic pressure p, and when $V^*/z$ is large enough, that is, when the solution is dilute, it obeys the law van 't Hoff found, which has the form of the gas law:

R is the gas constant and T the absolute temperature. Pressure is force per unit area: the force on the whole wall is p times the wall's area.

Now put small bodies suspended in the liquid into $V^*$ in place of the dissolved substance, bodies that also cannot pass through the wall. What does classical thermodynamics expect? Einstein states its answer together with its reason. At a fixed temperature the force on the wall follows from how the free energy of the system changes when the wall is moved. On the usual view the free energy depends on the total masses and kinds of the suspended substance, the liquid and the wall, and on pressure and temperature, but not on where the wall and the suspended bodies are. If moving the wall does not change the free energy, the wall feels no force. So, gravity aside, classical thermodynamics does not expect the suspended bodies to exert any force on the wall.

Open the foundation: Osmotic pressure and free energy

Einstein sets two effects aside so that the comparison is fair. Gravity, which would pull the bodies down, does not concern him here. The energy and entropy of the surfaces where the bodies meet the liquid (capillary forces) would also enter the free energy, but he assumes the moves considered do not change the size or nature of those surfaces, so they drop out.

The classical expectation is a coherent position, and for ordinary bodies it agrees with experience: a few pebbles held behind a sieve do not push on it measurably. The question is whether it still holds for bodies small enough to be jostled by the molecules of the liquid.

The molecular-kinetic theory of heat reaches a different view. On it, a dissolved molecule differs from a suspended body only in size (Einstein sets 'lediglich', only, in italics), and there is no reason why a number of suspended bodies should not give the same osmotic pressure as the same number of dissolved molecules. Jostled by the molecular motion of the liquid, the suspended bodies must perform an irregular motion in it, however slow; if the wall keeps them from leaving $V^*$, they exert forces on it, just as dissolved molecules do.

With n suspended bodies in $V^*$, so that there are $\nu = n/V^*$ of them in each unit of volume, and with neighbouring bodies far enough apart, the osmotic pressure should be

where N is the number of real molecules in a gram-molecule. The first form is van 't Hoff's law with $z = n/N$ gram-molecules; the second says that the pressure depends on the number of bodies per unit volume and on the temperature, and not on their size or mass.

Open the foundation: Temperature and thermal energy

The two views disagree about something that can be looked for. If suspended bodies exert osmotic pressure, then wherever their number per unit volume varies, so does the pressure, and it pushes them toward thinner regions; §3 shows that this appears as diffusion, and §§4 and 5 predict how far a grain wanders in a given time. On the classical expectation there is no osmotic pressure to drive such a spread. Observing the predicted wandering, with the predicted size, would count for the molecular-kinetic view, and its absence against it. First, §2 shows that the molecular-kinetic theory really leads to the extended law.

Einstein's statistical theory of heat gives the free energy as a logarithm of an integral over all the states of the system. He cannot compute that integral, but he needs only how it changes with the volume the particles are held in, and if they move independently in a uniform liquid with no forces on them, that dependence gives the osmotic pressure (RT/N)ν for dissolved molecules and suspended bodies alike. A footnote says the section is not needed for the paper's results.

A footnote to the heading says what this section assumes and what it is for. It takes as known Einstein's papers on the foundations of thermodynamics (Ann. d. Phys. 9, p. 417, 1902; 11, p. 170, 1903), and says that neither those papers nor this section is needed to understand the results of the present paper. The section is still worth reading, because it answers the question §1 leaves open: how can a theory of countless colliding molecules give a law as simple as van 't Hoff's, for dissolved molecules and suspended bodies alike, without anyone solving their motions?

Einstein describes the whole system, liquid, wall and particles, by state variables $p_1, \ldots, p_l$ that fix its momentary state completely, for example the coordinates and velocity components of all its atoms. How they change in time is given by equations of the form

with the condition $\sum \frac{\partial \varphi_\nu}{\partial p_\nu} = 0$. Two of these letters are used elsewhere in the paper for other things: these $p_\nu$ are not the osmotic pressure p of §1, and these $\varphi_\nu$, the rates of change of the state variables, are not the φ of §4.

For such a system his earlier theory gives the entropy S as an expression containing the logarithm, printed lg and meaning the natural logarithm, of an integral taken over every combination of the state variables that the conditions of the problem allow. In it T is the absolute temperature, Ē (printed with a bar) the energy of the system, and E the energy as a function of the $p_\nu$. The constant is printed as 2κ, and Einstein ties κ to N by 2κN = R, so 2κ is R/N. For the free energy F he obtains

and the integral is named B.

Open the foundation: Logarithms: turning products into sums

Even if the molecular picture were fixed in every detail, Einstein says, computing B would be so hard that an exact calculation of F is hardly conceivable. But the pressure needs only how F depends on the volume $V^*$ in which all the particles are held. (Particles, 'Teilchen', is his short word for dissolved molecules and suspended bodies alike.)

Put n particles in $V^*$, held there by a semipermeable wall, their total volume small compared with $V^*$. Where the wall stands limits the range of the integral B. Name the coordinates of the particles' centres of gravity $x_1, y_1, z_1$ through $x_n, y_n, z_n$, give each centre a tiny box inside $V^*$, and ask for the part of B that comes from states with every centre in its box. It has the form

where the factor J does not depend on the box sizes, nor on $V^*$, that is, on where the wall is. J does not depend on where the boxes are either. Take a second set of boxes, of the same sizes, in other places inside $V^*$; its part of B is $dB'$ with a factor $J'$. Since the sizes are equal,

Einstein's earlier theory gives these parts a meaning: dB/B is the probability that, at a moment chosen at random, the centres are in the given boxes. If the particles move independently of one another, to a sufficient approximation, the liquid is homogeneous and no forces act on the particles, then equal boxes are equally probable wherever they are, so

and with the previous equation, $J = J'$.

Open the foundation: Probability and independence

So J depends neither on $V^*$ nor on where the particles are. Integrating over all positions of the n centres, each ranging over the volume $V^*$, gives

and so the free energy is

The pressure on the wall is minus the rate at which F changes as $V^*$ grows:

Open the foundation: Partial derivatives and held-fixed quantities

This shows, Einstein concludes, that osmotic pressure is a consequence of the molecular-kinetic theory of heat, and that on this theory equal numbers of dissolved molecules and suspended bodies behave exactly alike as regards osmotic pressure at great dilution. The question of how the theory avoids solving every molecular motion has a plain answer: the hard part of B, the factor J, is never computed. Only its independence of $V^*$ is needed, and that follows from equal boxes being equally probable. The volume enters only through $V^{*n}$, and its logarithm, $n\lg V^*$, gives the pressure.

Left and right can cancel in the average while every tracer moves. Squaring before averaging keeps track of the spread.

A signed mean answers where the ensemble’s centre has moved. It does not answer how far its members have wandered. For a centred distribution, rightward and leftward contributions balance even while the distribution broadens.

To retain the movement, square each displacement before averaging. Taking the square root of that mean square returns a length: the root-mean-square displacement, or RMS. A mean absolute displacement is a different observable, and neither is the length of a wandering trajectory.

Open the foundation: Mean, variance and RMS

If each step is independent of the last and as likely to go left as right, the mean square grows in proportion to time. Four times the time gives twice the RMS displacement, not four times.

Write the displacement after n steps as the sum of their increments. Expanding its square exposes both the squared increments and their cross terms. Independence factors each expected cross term into the product of two means; centring makes that product zero.

Open the foundation: Probability and independence

Each tracer’s next position is its present one plus a random jump. Average that rule over every possible jump, take left and right jumps to be equally likely, and on scales larger than one jump the density obeys the diffusion equation.

Let $\varphi(\Delta)$ be the probability density for a displacement $\Delta$ during $\tau$. To end at $x$, a tracer must start at $x - \Delta$ and then make that displacement. Adding over all possible increments gives the transition relation, written here in modern notation.

Expand to first order in time and second order in displacement. Normalization cancels the zeroth-order term. Symmetry removes the first moment. The second moment remains.

Open the foundation: Approximating a curve near a point

Tracers that all start at one point spread into a bell curve whose width grows with time: along one axis the RMS displacement is $\sqrt{2Dt}$.

Its symmetry gives zero mean. Its second moment is 2Dt. These are ensemble statements: the curve assigns probabilities to intervals, not destinations to individual particles.

The probability of finding a displacement between a and b is the integral of this density over that interval. At time zero, an interval containing the starting point has probability one; no finite bell represents that state.

Open the foundation: The Gaussian and its width

For small spheres in a liquid, a more viscous liquid gives a smaller $D$: doubling the viscosity halves the mean square at a fixed time and divides the RMS displacement by $\sqrt{2}$, not by 2.

The displacement law tells us what a given $D$ predicts. A separate model connects $D$ to a tracer’s physical surroundings. Let $b$ be mobility, so a small force $F$ produces mean drift $bF$. Stokes drag for a sphere gives $b = 1/(6\pi\eta a)$.

Open the foundation: Viscosity and Stokes drag

Let $c$ be number density. At isothermal balance the force density $cF$ balances the osmotic-pressure gradient. With ideal osmotic pressure $cRT/N$, the drift flux $cbF$ becomes $b\,(RT/N)$ times the density gradient. Equating it with the opposite diffusive flux gives $D = bRT/N$.

Open the foundation: Osmotic pressure and free energy

The equality kB = R/N relates the two forms. For a prediction using modern constants it is convenient. For an inference of N, using a value of kB derived from that same N would defeat the point.

Open the foundation: Temperature and thermal energy

Open the foundation: Counting what crosses a boundary

Measured displacements give $D$. Turning $D$ into a count of molecules also needs the temperature, the viscosity and, above all, an independently measured radius. A simulation that assumes the answer cannot measure it.

The first expression refers to the model mean square, or to an estimate obtained from an appropriate sample. The second is an inversion under the stated physical assumptions. An estimate needs uncertainty and checks of those assumptions; rearranging symbols does not remove experimental error.

Without an independent radius, the same D can result from many pairs of a and N. The data then select a compatible family rather than a unique molecular number. The existing synthetic laboratories explore this relationship but do not supply a historical measurement.

Open the foundation: Two measurements, two unknowns

Open the foundation: Mean, variance and RMS

Open the foundation: Uncertainty, evidence and inference

Open the foundation: The units of 1905