Experiment: Brownian Motion and Avogadro's Number
At the start of the twentieth century the existence of atoms was still a live question: the energetic school held that thermodynamics needed no molecules, and no measurement had yet counted them. Brownian motion settled it. The theory of Statistical Mechanics predicts that a particle large enough to be watched under a microscope, but small enough for the bombardment it receives to be unbalanced, must wander, and that the wandering must obey a quantitative law containing Avogadro's constant and nothing else that is unknown. Perrin measured that law four independent ways — from sedimentation equilibrium, from translational displacements, from rotational displacements, and from diffusion — and each gave the same number to within his stated errors. That agreement, not any single value, is what ended the argument.
This chapter reports the measurement as an experiment: the emulsions
and optics Perrin used, the counting procedure, the observed numbers
with their uncertainties in SI units, the interpretation, and the
modern repetitions that reach the regime Einstein himself declared
unobservable — the instantaneous velocity of a Brownian particle.
Written out it will use the structured experiment environment
of the treatise, in the order of the sections below. Its theoretical
neighbours are Statistical Mechanics for the fluctuation
formulae, Kinetic Theory of Gases for the molecular picture,
Nonequilibrium Thermodynamics and Transport for the general
fluctuation–dissipation relation, and
Probability and Statistics for the stochastic processes the
episode created. Primary sources are Brown [Brown:1828], Einstein
[Einstein:1905d], Smoluchowski [Smoluchowski:1906], Langevin
[Langevin:1908] and Perrin [Perrin:1909] [Perrin:1913].
Experiment: Brownian Motion and Avogadro's Number: all derivations of this chapter are pending.
Historical context and the prediction under test
Brown's observation
[Reserved: the ceaseless irregular motion of pollen-derived particles suspended in water, and Brown's decisive control — the same motion in particles ground from window glass, from rock and from a fragment of the Sphinx, which excluded any vital cause [Brown:1828]; the seventy years of failed explanations (convection, evaporation, capillarity, electricity) and the observations that excluded each; the qualitative facts any theory had to reproduce — the motion never stops, is faster for smaller particles, and is faster in less viscous liquids.]
Particles about a micrometre across, suspended in a liquid and watched under a microscope, execute a perpetual irregular motion that never ceases, is more vigorous the smaller the particle and the less viscous the liquid, and does not depend on the chemical nature or the origin of the particle: Brown found the same agitation in pollen-derived grains, in particles ground from window glass and from rock, and in a fragment of the Sphinx, which excludes any vital cause [Brown:1828]. No external agency — convection, evaporation, illumination or capillarity — has been found that switches it off, and the motion persists indefinitely in a sealed, thermally equilibrated cell.
Molecular origin of the agitation: the unbalanced momentum delivered by molecular impacts on a grain of radius \(a\), and the reason the relative size of the resulting fluctuation grows as the grain shrinks
The atomic hypothesis under attack
[Reserved: the positivist objection that atoms are unobservable and therefore not physics, argued from the economy of thought [Mach:1883]; the energetic programme's claim that thermodynamics is complete without them; what would count as an answer — not a plausibility argument but a measurement of how many molecules there are; Perrin's own framing of the question [Perrin:1913], and the connection to the epistemology of Epistemology and the Scientific Method.]
Einstein's prediction
[Reserved: the 1905 paper deriving, from the osmotic pressure of the suspended particles and Stokes drag, the diffusion coefficient \(D = RT/(6\pi\eta a N_{A})\) and the mean square displacement \(\langle x^{2}\rangle = 2Dt\) [Einstein:1905d]; the crucial point that the displacement grows as \(\sqrt{t}\), not as \(t\), so that a velocity measured over any accessible interval is meaningless — the statement Einstein made explicitly and that Section 119.7.3 eventually overturned; the second paper extending the treatment and adding rotational diffusion \(\langle \theta^{2}\rangle = RT t/(4\pi\eta a^{3}N_{A})\) [Einstein:1906].]
Smoluchowski and Sutherland
[Reserved: Smoluchowski's independent kinetic derivation by direct collision counting, differing from Einstein's coefficient by a numerical factor and reaching the same scaling [Smoluchowski:1906]; Sutherland's independent statement of the same diffusion relation, obtained while estimating molecular masses from solution data [Sutherland:1905]; the resulting name Stokes–Einstein–Sutherland relation, and the honest attribution.]
The Langevin equation
[Reserved: the phenomenological equation of motion \(m\ddot{x} = -6\pi\eta a\dot{x} + F(t)\) with a random force of zero mean and vanishing correlation time, and Langevin's derivation of \(\langle x^{2}\rangle = 2Dt\) from it in a page — what he called an infinitely simpler demonstration [Langevin:1908]; the momentum relaxation time \(\tau_{p}=m/(6\pi\eta a)\) that the overdamped limit discards, and which reappears as the observable of Section 119.7.3; the equation as the first stochastic differential equation.]
Apparatus
The gamboge and mastic emulsions
[Reserved: gamboge resin and mastic emulsified in water and purified by repeated fractional centrifuging to obtain grains uniform in radius to a few percent; typical radii from \(0.2\,\mu\mathrm{m}\) to \(0.5\,\mu\mathrm{m}\) and densities near \(1.2\times 10^{3}\,\mathrm{kg}/\mathrm{m}^{3}\); the months of preparation required for a usable fraction, and why monodispersity is the controlling systematic of the whole measurement [Perrin:1909] [Perrin:1913].]
Microscope, cell and recording
[Reserved: the shallow observation cell of depth about \(100\,\mu\mathrm{m}\) sealed against evaporation and convection; the short depth of focus used to select a horizontal layer; the calibrated eyepiece grid and the camera lucida for recording successive positions at fixed time intervals of \(30\,\mathrm{s}\); temperature control and its uncertainty; the counting of grains per layer by direct enumeration [Perrin:1909].]
Calibration of the grain radius
[Reserved: three independent determinations of the grain radius — from Stokes-law sedimentation velocity, from direct microscopic measurement of a chain of grains, and from weighing a counted number of grains — and their mutual agreement as the internal check on the most sensitive input; the viscosity of water at the working temperature and its uncertainty; the propagation of these into the final result [Perrin:1909], in the manner of Measurement, SI Units, and the Theory of Errors and [Taylor:1997].]
Procedure
Sedimentation equilibrium
[Reserved: the vertical distribution of grains in equilibrium as a barometric law with the molecular mass replaced by the buoyant mass of a grain, so that the scale height is measurable under a microscope; the counting of grains at four levels separated by tens of micrometres; the exponential fit and its slope giving \(N_{A}\) directly; the demonstration that a visible suspension obeys the same statistics as a gas [Perrin:1909].]
A dilute suspension of uniform grains, left undisturbed, settles into a stationary vertical distribution in which the number density falls off exponentially with height,
where \(m^{*}\) is the buoyant mass of a grain of radius \(a\) and density \(\rho\) suspended in a fluid of density \(\rho_{f}\). Perrin measured the scale height by counting grains at four levels separated by tens of micrometres and found the law obeyed over a fourfold change of concentration [Perrin:1909]. A suspension one can see therefore obeys the same barometric statistics as a gas one cannot.
Derivation. Treat the grains as a dilute solute, so rare that they do not interact with one another; their osmotic pressure is then that of an ideal gas, \(p=n k_{B}T\). In mechanical equilibrium the gradient of that pressure must support the net downward force per unit volume, which is the weight of the grains less the buoyancy of the fluid they displace,
Substituting \(p=nk_{B}T\) at uniform temperature gives \(k_{B}T\,\dd n/\dd z=-n m^{*}g\), a first-order linear equation whose solution is Equation (119.1). Equivalently, the grains are distributed over the potential energy \(U=m^{*}gz\) with the Boltzmann weight \(\ee^{-U/k_{B}T}\); the two readings agree, which is precisely the point at issue, since the second treats a visible object as a statistical-mechanical degree of freedom. Because \(m^{*}\) is known from the measured radius and densities, the slope of \(\ln n\) against \(z\) delivers \(k_{B}=R/N_{A}\), and hence \(N_{A}\), from a photograph.
∎Displacement statistics
[Reserved: the recording of several thousand successive horizontal displacements over fixed intervals; the histogram of displacements tested against the predicted Gaussian; the mean square displacement per unit time compared with \(\langle x^{2}\rangle = 2Dt\) and hence with Section 119.1.3 [Einstein:1905d] [Perrin:1909]; the explicit test that the increments are independent, by checking \(\langle x^{2}\rangle\) against interval length.]
Measured along any fixed horizontal axis, the displacement of a suspended grain over an interval \(t\) has zero mean and a variance proportional to the interval itself, not to its square,
for a sphere of radius \(a\) in a fluid of viscosity \(\eta\), and the histogram of displacements is Gaussian. The root mean square displacement therefore grows as \(\sqrt{t}\), so the ratio of an observed displacement to the interval over which it was recorded diverges as that interval is shortened: a Brownian “velocity” read off a finite interval is a property of the stopwatch and not of the particle. Equation (119.3) was predicted by Einstein [Einstein:1905d], obtained independently by Sutherland [Sutherland:1905] and by Smoluchowski [Smoluchowski:1906], and confirmed on emulsions of gamboge and mastic by Perrin [Perrin:1909].
Derivation. Follow Langevin [Langevin:1908]. Resolve the force on the grain into the systematic Stokes drag it would feel if the fluid were a continuum and a residual random force \(F(t)\) of zero mean, uncorrelated with the position of the grain,
Multiply by \(x\) and use \(x\ddot{x}=\dd(x\dot{x})/\dd t-\dot{x}^{2}\):
The grain is one mechanical degree of freedom in thermal equilibrium with the liquid, so equipartition fixes \(m\avg{\dot{x}^{2}}=k_{B}T\). Writing \(z=\avg{x\dot{x}}=\tfrac{1}{2}\dd\avg{x^{2}}/\dd t\) turns Equation (119.5) into the linear equation
whose solution relaxes to the constant \(z=k_{B}T/(6\pi\eta a)\) over the momentum relaxation time \(\tau_{p}=m/(6\pi\eta a)\), which for Perrin's sub-micrometre resin grains in water is a small fraction of a microsecond and so utterly unresolvable at his \(30\,\mathrm{s}\) sampling interval. For \(t\gg\tau_{p}\), then, \(\dd\avg{x^{2}}/\dd t=2k_{B}T/(6\pi\eta a)\), and integrating from \(x(0)=0\) gives Equation (119.3) with \(D=k_{B}T/(6\pi\eta a)\). Substituting \(k_{B}=R/N_{A}\) puts Avogadro's constant into a quantity measurable with a microscope and a clock; every other symbol in Equation (119.3) is independently known. The Gaussian shape follows because the displacement over an interval long compared with \(\tau_{p}\) is a sum of many independent increments, which is the central limit theorem of Probability and Statistics.
∎Rotational Brownian motion
[Reserved: larger grains of about \(13\,\mu\mathrm{m}\) carrying visible internal inclusions used as angular markers; the measurement of the mean square angular displacement against Einstein's rotational prediction [Einstein:1906]; the very different power of the radius in the two laws, which makes their agreement a strong test rather than a repetition [Perrin:1909].]
Observations
Measured displacements and distributions
[Reserved: tables of the observed mean square displacements with their standard errors, the fitted diffusion coefficients in \(\mathrm{m}^{2}/\mathrm{s}\), and the observed Gaussian form of the displacement histogram; the sedimentation scale heights; the rotational data; all quantities in SI with uncertainties, as required by the treatise convention [Perrin:1909].]
Values of Avogadro's constant
[Reserved: the four independent routes giving values clustered near \(6.8\times 10^{23}\,/\mathrm{mol}\), within roughly \(15\,\mathrm{\%}\) of the modern figure; Perrin's tabulation of the same constant obtained from thirteen unrelated phenomena — viscosity of gases, blackbody radiation, radioactivity, opalescence, blue of the sky, electric charge — agreeing to within their errors [Perrin:1913]; why the concordance of independent routes carries the evidential weight, an argument revisited in Epistemology and the Scientific Method.]
Four measurements made on the same emulsions but resting on different physics — sedimentation equilibrium (Equation (119.1)), translational displacements (Equation (119.3)), rotational displacements, and diffusion of the grains themselves — return one and the same value of \(N_{A}\) to within their stated errors, and that value agrees to within roughly \(15\,\mathrm{\%}\) with values obtained from a dozen entirely unrelated phenomena, among them the viscosity of gases, blackbody radiation, the counting of radioactive decays, critical opalescence and the blue of the sky [Perrin:1909] [Perrin:1913]. The constant is today fixed by definition at \(N_{A}=6.02214076\times 10^{23}\,/\mathrm{mol}\) [BIPM:2019]. It is the concordance, and not the accuracy of any single route, that closed the question of whether molecules are real.
The rotational displacement law for a sphere, in which the mean square angular displacement grows linearly in time with a coefficient carrying the inverse cube of the grain radius rather than its first power — the different power being what makes agreement between the two routes an independent check and not a repetition
Uncertainty budget
[Reserved: the dominant contributions — grain radius, polydispersity, viscosity, temperature drift, finite observation interval, and the two-dimensional projection of a three-dimensional walk; the systematic from tracking a grain that leaves the focal layer; comparison with the modern uncertainty conventions of Measurement, SI Units, and the Theory of Errors and [Taylor:1997].]
Interpretation
The fluctuation–dissipation content
[Reserved: the relation \(D = kT/(6\pi\eta a)\) read as a link between a fluctuation (\(D\)) and a dissipation (\(\eta\)) at one temperature [Einstein:1905d] [Sutherland:1905]; the same structure in the thermal noise of Statistical Mechanics and in the general theorem of Nonequilibrium Thermodynamics and Transport; the physical content — the same molecular collisions that damp a motion also drive it, so the two coefficients cannot be independent.]
Why this settled the atomic hypothesis
[Reserved: the difference between explaining a phenomenon by atoms and counting the atoms; Perrin's argument that a definite \(N_{A}\) obtained concordantly from unrelated phenomena leaves no economical alternative to their reality [Perrin:1913]; Ostwald's and the energeticists' public change of position; the modern reading of this as the standard example of the argument from consilience.]
The second law as a statistical statement
[Reserved: the visible violation of the second law in a suspended grain, which rises against gravity for part of every second; Smoluchowski's analysis of the fluctuation-driven limits on any perpetual-motion device built from them [Smoluchowski:1906], and its relation to the Landauer analysis in Statistical Mechanics; the entropy statement of Classical Thermodynamics restated as a statement about overwhelming probability, not impossibility.]
Recognition
[Reserved: the 1926 Nobel Prize in Physics awarded to Perrin for his work on the discontinuous structure of matter, and especially for his discovery of sedimentation equilibrium; Perrin's Nobel lecture as his own summary of the evidence [Perrin:1926]; the delayed and partial recognition of Smoluchowski, who died in 1917.]
Mathematical aftermath
The Wiener process
[Reserved: Wiener's construction of a measure on continuous paths realising Einstein's law, and the proof that almost every path is continuous and nowhere differentiable [Wiener:1923]; why nowhere-differentiability is the mathematical form of Einstein's warning about velocity in Section 119.1.3; the process as the basic object of Probability and Statistics.]
Ornstein–Uhlenbeck and the velocity correlation
[Reserved: the Ornstein–Uhlenbeck solution of the Langevin equation, giving the exponentially decaying velocity autocorrelation \(\langle v(0)v(t)\rangle = (kT/m)\ee^{-t/\tau_{p}}\) and recovering \(\langle x^{2}\rangle = 2Dt\) only for \(t\gg\tau_{p}\) [Uhlenbeck:1930]; the ballistic regime \(\langle x^{2}\rangle = (kT/m)t^{2}\) at short times, and the equipartition value of the velocity variance — the prediction tested in Section 119.7.3.]
The Fokker–Planck description
[Reserved: the evolution equation for the probability density in position and velocity, its equivalence to the Langevin description, and the Maxwell–Boltzmann stationary solution; Chandrasekhar's review as the standard synthesis of the whole subject [Chandrasekhar:1943]; forward reference to Nonequilibrium Thermodynamics and Transport for transport coefficients and to Plasmas and Magnetohydrodynamics for the same equation in a plasma.]
Modern repetitions and precision
Kappler's torsion balance
[Reserved: a small mirror on a quartz fibre in a low-pressure gas, whose angular fluctuations obey \(\langle \theta^{2}\rangle = kT/\kappa\) by equipartition alone, independent of pressure over four decades; Kappler's determination of \(N_{A}\) to about \(1\,\mathrm{\%}\) from these fluctuations [Kappler:1931]; the experiment as the cleanest instance of the fluctuation formulae of Statistical Mechanics for a single degree of freedom.]
Optical tweezers
[Reserved: the single-beam gradient-force trap [Ashkin:1986] as the modern apparatus — a micrometre bead held in a harmonic potential of known stiffness, with position read out interferometrically at nanometre and microsecond resolution; the power spectrum of the trapped bead as a direct measurement of both \(D\) and \(\tau_{p}\); the same instrument used for the single-molecule work cited in Statistical Mechanics.]
The instantaneous velocity
[Reserved: Li, Kheifets, Medellin and Raizen's measurement of the instantaneous velocity of a silica bead in air, resolving the ballistic regime and recovering the equipartition value \(\langle v^{2}\rangle = kT/m\) [Li:2010]; the extension to liquids, where hydrodynamic memory modifies the correlation function and the simple Ornstein–Uhlenbeck form fails [Kheifets:2014]; the explicit overturning of Einstein's judgement that this velocity could never be measured, and what made it possible — trap stiffness, detection bandwidth and low pressure.]
Resolved on time scales short compared with the momentum relaxation time \(\tau_{p}\) of Equation (119.6), the motion of a Brownian particle is not diffusive but ballistic: the mean square displacement grows as \(t^{2}\) rather than as \(t\), the trajectory acquires a well-defined instantaneous velocity, and the variance of that velocity takes the equipartition value
This was measured for a silica bead of micrometre scale held in an optical trap in air, with the position read out at nanometre and sub-microsecond resolution [Li:2010]; in a liquid the same measurement finds the simple exponential velocity correlation replaced by a slower, hydrodynamic-memory form, so that Equation (119.7) is approached but the Ornstein–Uhlenbeck description of Section 119.6.2 fails in detail [Kheifets:2014]. The measurement overturns Einstein's explicit judgement [Einstein:1905d] that this velocity could never be observed.
The velocity autocorrelation of the Ornstein–Uhlenbeck process obtained from the Langevin equation, its short-time ballistic limit, and the crossover to the diffusive law at the momentum relaxation time; and separately the hydrodynamic-memory kernel that replaces it for a sphere in a dense fluid
Avogadro's constant today
[Reserved: the silicon-sphere X-ray crystal density route and the watt-balance route that superseded Brownian determinations; the 2019 redefinition fixing \(N_{A}\) exactly at \(6.02214076\times 10^{23}\,/\mathrm{mol}\) by convention, so that the quantity Perrin measured is now a defined constant [BIPM:2019]; the current values of the related constants [Tiesinga:2021]; what a modern Brownian measurement now tests instead — the Boltzmann constant, the viscosity, or the apparatus itself; the SI framework of Measurement, SI Units, and the Theory of Errors.]