Read · Brownian motion · Explanation preview

§5 · From displacement to molecular scale

Read the displacement argument, ask for its missing steps, and investigate the same relationships in three working laboratories.

This is newly authored explanatory text in modern notation, with editorial review pending. It is not the German source, an English translation, or a complete edition of the paper. The reviewed source faces and pinned facsimile remain in preparation. Section headings identify the argument being discussed, not a completed source inventory.

New explanatory text authored with AI assistance. Mathematical and editorial review remains pending; none of these passages is presented as Einstein’s wording.

Read the whole available argument →

Detail

Show modern qualifications

JavaScript is off. The full explanation is shown. Each passage’s “Show every step here” disclosure works without scripts, and foundation links open complete reading pages. The embedded laboratory remains a static worked example.

§5 · From displacement to molecular scale

derivation · Model approximation

Why viscosity changes the spread

What fixes D for a small spherical tracer in a liquid?

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/(6πηa).

Let c be number density. At isothermal balance the force density cF balances the osmotic-pressure gradient. With ideal osmotic pressure Π = 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.

D=\frac{RT}{6\pi\eta aN}=\frac{k_BT}{6\pi\eta a}

Stokes–Einstein diffusivity is R T over six pi viscosity radius N, or k B T over six pi viscosity radius.

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: Counting what crosses a boundary

Name the quantities: T is absolute temperature, η dynamic viscosity, a tracer radius, R molar gas constant, and N molecular number per mole.
Mobility b is drift speed divided by force. Stokes drag supplies b = 1/(6πηa).
The isothermal ideal osmotic-pressure relation is Π = cRT/N, where c counts tracers per volume.
Differentiate with respect to position: the gradient of Π is (RT/N) times the gradient of c.
Mechanical balance gives cF equal to that pressure gradient. Multiply by b to obtain the drift flux.
Fick’s law gives the opposing diffusive flux as −D times the gradient of c.
Zero total flux for this balance gives D = bRT/N. Substitute the Stokes mobility.
Holding T, radius and constants fixed, doubling η halves D. The RMS displacement at a fixed time then falls by √2, not by two.

A tool for this step

Counting what crosses a boundary

Why does a difference of flows change the density?

Take a short interval. If more tracers enter through its boundaries than leave, its count increases. Flux counts signed crossings per area per time; number density counts particles per volume. They are not the same quantity.

\frac{\partial c}{\partial t}=-\frac{\partial J}{\partial x}

Density changes at the negative spatial derivative of the flux.

In the one-dimensional description, the same balance applies per unit transverse area. The minus sign says an increasing rightward flux removes more particles at the right boundary than arrive at the left.

One worked example

If seven particles enter and five leave during a time interval, the count rises by two. If both flows are five, the count is unchanged even though particles keep crossing.

A stopping point: Conservation counts net crossings; it does not require each individual particle to stop.

Explore the equation · Modern model notation

Resistance to motion controls spreading

D = \frac{k_B\,T}{6\,\pi\,\eta\,a}

The diffusion coefficient is thermal energy divided by the viscous drag coefficient.

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Input · Model result · Constant

Symbolic equation

Symbolic here. Open the linked laboratory for a worked example and live values.

Diffusion coefficient: No accepted value is available here.
Boltzmann constant: No accepted value is available here.
Absolute temperature: No accepted value is available here.
Dynamic viscosity: No accepted value is available here.
Particle radius: No accepted value is available here.

Read the equation aloud in words

The diffusion coefficient equals the Boltzmann constant times the absolute temperature, divided by six times pi times the dynamic viscosity times the particle radius.

Thermal energy competes with viscous drag. Doubling viscosity halves the diffusion coefficient while reducing RMS displacement only by the square root of two. Radius means radius, not diameter.

Model assumptions and every term’s meaning

Dilute, approximately spherical tracers with no-slip Stokes drag in a homogeneous Newtonian liquid.
Wall corrections, interactions, inertia, and observation noise are not included.
The modern SI 2019 constant set is used; this is not a historical inversion exercise.

An ideal model: This equation assumes the dilute spherical tracer and Stokes-drag model; dimensional consistency alone does not establish those assumptions. Read the prerequisite
Diffusion coefficient: A squared-distance-per-time coefficient. This value comes from the accepted model calculation. Read the prerequisite
Why divide by drag?: Increasing the denominator reduces diffusivity at fixed thermal energy. This is a model dependence, not a rule that every larger quantity must reduce motion. Read the prerequisite
Thermal energy scale: Boltzmann constant times absolute temperature supplies energy per particle. Read the prerequisite
A known modern constant: The SI 2019 constant set supplies this value. It must not be treated as independent historical evidence when trying to infer molecular number. Read the prerequisite
Absolute temperature: Use kelvin. The laboratory holds viscosity as an independently supplied parameter. Read the prerequisite
The drag coefficient: Six pi times viscosity times radius is Stokes drag per speed for an isolated sphere in the assumed low-Reynolds regime. Read the prerequisite
Dynamic viscosity: Viscosity is in the denominator. Doubling it at fixed temperature and radius halves D, not the RMS displacement. Read the prerequisite
Particle radius: The input is a radius, not a diameter. Confusing them changes the predicted coefficient by a factor of two. Read the prerequisite

Exact rational SI dimensions were checked at build time. This is a unit check, not a proof of the model. This teaching record is a draft, not a transcription of a printed equation.

Show every step here: Why viscosity changes the spread

Name the quantities: T is absolute temperature, η dynamic viscosity, a tracer radius, R molar gas constant, and N molecular number per mole.
Mobility b is drift speed divided by force. Stokes drag supplies b = 1/(6πηa).
The isothermal ideal osmotic-pressure relation is Π = cRT/N, where c counts tracers per volume.
Differentiate with respect to position: the gradient of Π is (RT/N) times the gradient of c.
Mechanical balance gives cF equal to that pressure gradient. Multiply by b to obtain the drift flux.
Fick’s law gives the opposing diffusive flux as −D times the gradient of c.
Zero total flux for this balance gives D = bRT/N. Substitute the Stokes mobility.
Holding T, radius and constants fixed, doubling η halves D. The RMS displacement at a fixed time then falls by √2, not by two.

A tool for this step

Counting what crosses a boundary

Why does a difference of flows change the density?

\frac{\partial c}{\partial t}=-\frac{\partial J}{\partial x}

Density changes at the negative spatial derivative of the flux.

One worked example

If seven particles enter and five leave during a time interval, the count rises by two. If both flows are five, the count is unchanged even though particles keep crossing.

A stopping point: Conservation counts net crossings; it does not require each individual particle to stop.

Assumptions and limits: Why viscosity changes the spread

Assumed here

Dilute spherical tracers in a homogeneous Newtonian liquid, Stokes mobility and ideal osmotic pressure.
R is a separately known molar gas constant; N denotes molecular number per mole.

What this does not establish

Slip, inertia, interactions, non-Newtonian response and gas corrections are not included.
The argument imports constitutive laws; conservation alone does not derive them.

Earlier step: What the spreading curve predicts

Source context: Annalen der Physik (4), 17, 549–560 (1905), §§1–5. Bibliographic pointer; this preview is not a source transcription or translation.

derivation · Model approximation

What would let us count molecules?

Which additional measurements turn displacement into an estimate of N?

D=\frac{\langle x^2\rangle}{2t},\qquad N=\frac{RTt}{3\pi\eta a\langle x^2\rangle}

Diffusivity is mean-square displacement over twice time; molecular number is R T t over three pi viscosity radius mean-square displacement.

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: Mean, variance and RMS

Open the foundation: Uncertainty, evidence and inference

Show every step here: What would let us count molecules?

Begin with mean square = 2Dt. Divide both sides by 2t, for positive t.
The result is D = mean square/(2t).
The separate physical relation is D = RT/(6πηaN). Multiply both sides by 6πηaN.
Divide by 6πηaD to obtain N = RT/(6πηaD).
Insert mean square/(2t) for D. Dividing by that fraction multiplies by 2t/mean square.
Cancel the factor two against six, giving N = RTt/(3πηa mean square).
Notice that a and N occur as a product in the original relation. Without one, D alone cannot determine the other.
A simulation using an assumed molecular number can test this inversion’s arithmetic, but cannot independently establish that number in nature.

A tool for this step

Mean, variance and RMS

Which average can reveal wandering that cancels in the signed mean?

\langle x\rangle=\frac{1}{M}\sum_{i=1}^{M}x_i

Add the M signed displacements and divide by M.

\langle x^2\rangle=\frac{1}{M}\sum_{i=1}^{M}x_i^2

Square each displacement before adding and dividing by M.

\operatorname{Var}(x)=\langle x^2\rangle-\langle x\rangle^2

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

RMS means root mean square: take the square root after averaging the squares. RMS and standard deviation coincide only when the mean is zero. The average absolute displacement is yet another statistic.

One worked example

For −3, −1, +1, +3, the signed sum is 0, so the mean is 0.
The absolute values are 3, 1, 1, 3; their mean is 2.
The squares are 9, 1, 1, 9; their mean is 5.
The RMS is the square root of 5, approximately 2.236.
A small random sample need not have a signed mean of exactly zero even when the model mean is zero.

A stopping point: Specify whether you square before averaging, average before squaring, or take absolute values.

Assumptions and limits: What would let us count molecules?

Assumed here

A suitable estimate of the model mean-square coordinate displacement or its time slope.
Independent values of R, T, η and tracer radius a, under the stated dilute-liquid model.

What this does not establish

A synthetic run generated from an assumed N is not independent evidence for N.
Diffusivity alone constrains the product aN when radius is unknown.
Measurement noise, drift, exposure and finite sampling require separate treatment.

Earlier step: Why viscosity changes the spread

Source context: Annalen der Physik (4), 17, 549–560 (1905), §§1–5. Bibliographic pointer; this preview is not a source transcription or translation.

Explanation

Einstein's displacement argument: the typical distance grows with the square root of time, not with time itself. Every explanation stays on the page when reading-only is on.

Static worked case

For radius 0.5 μm, viscosity 1.35×10⁻³ Pa·s, and T = 290.15 K, the RMS displacement is about 0.8 μm in one second. This static worked case stays in the markup; loading the live ensemble does not replace it.

Keep the experiment beside the argument

This laboratory stays mounted while you change detail or open a foundation. Applying its controls explicitly starts a host calculation; opening an explanation never starts or restarts a trial.

Open the tracer ensemble in this reading

BM-01 · A reproducible trial

Investigate the displacement argument

Static worked example

CurrentThese numbers match the current settings.

Model note

Primary output temperature: Host calculation (bm01.acceptedInputs). Owner bm01.acceptedInputs.
Primary output viscosity: Host calculation (bm01.acceptedInputs). Owner bm01.acceptedInputs.
Primary output particleRadius: Host calculation (bm01.acceptedInputs). Owner bm01.acceptedInputs.
Primary output observationInterval: Host calculation (bm01.acceptedInputs). Owner bm01.acceptedInputs.
Primary output boltzmannConstant: Host calculation (constants.modernSI2019). Owner constants.modernSI2019.
Primary output diffusionCoefficient: Host calculation (diffusion.stokesEinsteinD). Owner diffusion.stokesEinsteinD.
Primary output rmsDisplacement1d: Host calculation (diffusion.rmsDisplacement). Owner diffusion.rmsDisplacement.
Primary output modelSecondMoment: Host calculation (diffusion.moments). Owner diffusion.moments.
Primary output modelMeanNorm: Host calculation (diffusion.moments). Owner diffusion.moments.
Primary output modelRmsNorm: Host calculation (diffusion.moments). Owner diffusion.moments.
Primary output tracerPositions: Host reference calculation (recordTracers). Owner diffusion.recordTracers.
Primary output modelApparentSpeed: Host calculation (diffusion.apparentSpeed). Owner diffusion.apparentSpeed.
Primary output sampledApparentSpeed: Host calculation (diffusion.ensembleMoments). Owner diffusion.ensembleMoments.
Primary output traceCoordinates: Host calculation (diffusion.recordTracers). Owner diffusion.recordTracers.
Primary output traceTimes: Host calculation (bm01.measure). Owner bm01.measure.
Primary output histogramEdges: Host calculation (diffusion.displacementHistogram). Owner diffusion.displacementHistogram.
Primary output histogramCounts: Host calculation (diffusion.displacementHistogram). Owner diffusion.displacementHistogram.
Primary output histogramFrequencies: Host calculation (diffusion.displacementHistogram). Owner diffusion.displacementHistogram.
Primary output histogramModel: Host calculation (diffusion.intervalProbability). Owner diffusion.intervalProbability.
Primary output underflow: Host calculation (diffusion.displacementHistogram). Owner diffusion.displacementHistogram.
Primary output overflow: Host calculation (diffusion.displacementHistogram). Owner diffusion.displacementHistogram.
Primary output plotTimes: Host calculation (bm01.measure). Owner bm01.measure.
Primary output plotSampleMean: Host calculation (bm01.measure). Owner bm01.measure.
Primary output plotSampleMsd: Host calculation (bm01.measure). Owner bm01.measure.
Primary output plotSampleRms: Host calculation (bm01.measure). Owner bm01.measure.
Primary output plotSampleApparent: Host calculation (bm01.measure). Owner bm01.measure.
Primary output plotModelMean: Host calculation (bm01.measure). Owner bm01.measure.
Primary output plotModelMsd: Host calculation (bm01.measure). Owner bm01.measure.
Primary output plotModelRms: Host calculation (bm01.measure). Owner bm01.measure.
Primary output plotModelApparent: Host calculation (bm01.measure). Owner bm01.measure.
Primary output meanBand: Host calculation (diffusion.ensembleMomentBands). Owner diffusion.ensembleMomentBands.
Primary output secondMomentBand: Host calculation (diffusion.ensembleMomentBands). Owner diffusion.ensembleMomentBands.
Primary output recordingDraws: Host calculation (diffusion.recordTracers). Owner diffusion.recordTracers.
Primary output reusedRecording: Host calculation (bm01.measure). Owner bm01.measure.
Primary output ensembleSize: Host calculation (bm01.acceptedInputs). Owner bm01.acceptedInputs.
Primary output signedMean: Host calculation (diffusion.ensembleMoments). Owner diffusion.ensembleMoments.
Primary output meanSquare: Host calculation (diffusion.ensembleMoments). Owner diffusion.ensembleMoments.
Primary output lambdaX1s: Host calculation (diffusion.rmsDisplacement). Owner diffusion.rmsDisplacement.
Primary output lambdaX60s: Host calculation (diffusion.rmsDisplacement). Owner diffusion.rmsDisplacement.
Primary output signedMeanLowerBand: Host calculation (diffusion.ensembleMomentBands). Owner diffusion.ensembleMomentBands.
Primary output signedMeanUpperBand: Host calculation (diffusion.ensembleMomentBands). Owner diffusion.ensembleMomentBands.
Primary output meanSquareLowerBand: Host calculation (diffusion.ensembleMomentBands). Owner diffusion.ensembleMomentBands.
Primary output meanSquareUpperBand: Host calculation (diffusion.ensembleMomentBands). Owner diffusion.ensembleMomentBands.
Primary output kolmogorovDistance: Host calculation (diffusion.displacementHistogram). Owner diffusion.displacementHistogram.
Secondary output sampleMean: Host reduction (ensembleMoments). Owner diffusion.ensembleMoments.
Secondary output sampleMeanAbsolute: Host reduction (ensembleMoments). Owner diffusion.ensembleMoments.
Secondary output sampleMeanSquare: Host reduction (ensembleMoments). Owner diffusion.ensembleMoments.
Secondary output sampleRms: Host reduction (ensembleMoments). Owner diffusion.ensembleMoments.
Secondary output sampleMeanNorm: Host reduction (ensembleMoments). Owner diffusion.ensembleMoments.
Secondary output sampleMeanSquareNorm: Host reduction (ensembleMoments). Owner diffusion.ensembleMoments.
Secondary output sampleRmsNorm: Host reduction (ensembleMoments). Owner diffusion.ensembleMoments.
Seed 1905.
Accepted input revision 1.
Snapshot version 1.
Not modeled: Molecular collisions (no collision bath owns the displacement).
Show the code

JavaScript is off. The traces, histogram, tables and explanations below are a seeded worked example computed when this edition was built. Changing the settings requires JavaScript.

Compare the movement of individual tracers with the statistics of the whole ensemble. Changing when or how you observe the trial does not generate different paths.

Same-seed viscosity comparisons use common random numbers, not independent trials.

Predict before comparing observation times

Watching four times as long: how does the typical distance change?

The independent-step model predicts four times the mean square, hence twice its square root. Your prediction never locks the explanation or controls.

These buttons use the accepted trial, not unsaved draft edits. The requested time must lie on its recording grid.

Accepted synthetic trial: 400 tracers, 1 seconds, coordinate mean 0.0029237 micrometres and coordinate RMS 0.98602 micrometres.

Accepted trial: seed 1905; 400 tracers; 293.15 K; viscosity 1 mPa·s; radius 0.5 μm. Observe at 1 s in a 10-second recording. Constants: modern SI 2019.

Displacements from a common origin, not a literal microscope image. Lines connect recorded positions and do not supply an instantaneous velocity. 0 / 400 endpoints lie outside this view; none are removed from the ensemble.

t = 1.00 s (true rate (1 s/s))

Natural rate: ~0.8 μm per second Brownian walk (scale bar: 1 μm)Simulation view: accelerated snapshot across 10 s

View only: no calculation or new draws.

Solid bars: the synthetic sample. Dashed line: probabilities of the same bins under the unbounded model, not a density curve. Counts beyond the plotted range: 0 left, 0 right.

Whole-ensemble statistics: signed coordinate x, total over 1 coordinate
Diffusion coefficient (model)	0.42944 μm²/s
Signed mean (sample)	0.0029237 μm (modern-si-2019)
Mean absolute coordinate displacement	0.77808 μm (modern-si-2019)
Mean-square coordinate displacement	0.97223 μm²
Coordinate RMS (sample / model)	0.98602 / 0.92676 μm (modern-si-2019)
Mean distance (sample / model)	0.77808 / 0.73945 μm (modern-si-2019)
Total mean square (sample / model)	0.97223 / 0.85888 μm²
Apparent coordinate speed (sample / model)	0.98602 / 0.92676 μm/s

Sampling bands under the model

99.9% per comparison, not a simultaneous guarantee across the table or repeated observations. These ranges are calculated from the model variance, not estimated from the observed sample, and are not uncertainties of the model itself.

Signed coordinate mean (μm): -0.15248 to 0.15248

Total mean square (μm²): 0.67299 to 1.0729

Inspect the model behind this trial

These equations describe the same accepted trial as the plots and table. Select a symbol or an operation to see what it means; model predictions are not sample estimates.

Explore the equation · Modern model notation

Why the apparent speed depends on how you watch

v_{\mathrm{app}} := \frac{\lambda_x}{t}

The apparent coordinate speed divides the typical coordinate distance by the chosen positive interval.

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Input · Model result · Constant

Static worked example

Values describe this accepted snapshot, not unsaved input edits.

Apparent coordinate speed: 0.92676 μm/s
Coordinate RMS displacement: 0.92676 μm
Observation interval: 1 s

Read the equation aloud in words

Apparent coordinate speed equals the model coordinate root mean square displacement divided by the observation interval.

This quotient depends on the observation interval. It is not instantaneous physical velocity. At zero interval the quotient is undefined, even though the displacement is zero.

Model assumptions and every term’s meaning

The same one-coordinate ideal Brownian model as the RMS relation.
A positive observation interval is required for the quotient.
Lines joining recorded points are a rendering convention, not a velocity measurement.

Define the observable: This defines an interval-dependent comparison. It does not introduce a physical instantaneous Brownian velocity. Read the prerequisite
Apparent coordinate speed: A distance-per-interval statistic. It is not a molecular collision speed. Read the prerequisite
Divide by the same interval: The distance grows as the square root of time, while the denominator grows linearly. The quotient therefore decreases as the interval grows. Read the prerequisite
Model coordinate RMS: The same accepted one-coordinate model displacement used by the neighboring RMS equation. Read the prerequisite
Positive observation interval: A zero interval yields no apparent-speed value. The interface keeps the explanation instead of fabricating zero or infinity. Read the prerequisite

Exact rational SI dimensions were checked at build time. This is a unit check, not a proof of the model. This teaching record is a draft, not a transcription of a printed equation.

Explore the equation · Modern model notation

Resistance to motion controls spreading

D = \frac{k_B\,T}{6\,\pi\,\eta\,a}

The diffusion coefficient is thermal energy divided by the viscous drag coefficient.

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Input · Model result · Constant

Static worked example

Values describe this accepted snapshot, not unsaved input edits.

Diffusion coefficient: 0.42944 μm²/s
Boltzmann constant: 1.3806e-23 J/K
Absolute temperature: 293.15 K
Dynamic viscosity: 1 mPa·s
Particle radius: 0.5 μm

Read the equation aloud in words

The diffusion coefficient equals the Boltzmann constant times the absolute temperature, divided by six times pi times the dynamic viscosity times the particle radius.

Thermal energy competes with viscous drag. Doubling viscosity halves the diffusion coefficient while reducing RMS displacement only by the square root of two. Radius means radius, not diameter.

Model assumptions and every term’s meaning

Dilute, approximately spherical tracers with no-slip Stokes drag in a homogeneous Newtonian liquid.
Wall corrections, interactions, inertia, and observation noise are not included.
The modern SI 2019 constant set is used; this is not a historical inversion exercise.

An ideal model: This equation assumes the dilute spherical tracer and Stokes-drag model; dimensional consistency alone does not establish those assumptions. Read the prerequisite
Diffusion coefficient: A squared-distance-per-time coefficient. This value comes from the accepted model calculation. Read the prerequisite
Why divide by drag?: Increasing the denominator reduces diffusivity at fixed thermal energy. This is a model dependence, not a rule that every larger quantity must reduce motion. Read the prerequisite
Thermal energy scale: Boltzmann constant times absolute temperature supplies energy per particle. Read the prerequisite
A known modern constant: The SI 2019 constant set supplies this value. It must not be treated as independent historical evidence when trying to infer molecular number. Read the prerequisite
Absolute temperature: Use kelvin. The laboratory holds viscosity as an independently supplied parameter. Read the prerequisite
The drag coefficient: Six pi times viscosity times radius is Stokes drag per speed for an isolated sphere in the assumed low-Reynolds regime. Read the prerequisite
Dynamic viscosity: Viscosity is in the denominator. Doubling it at fixed temperature and radius halves D, not the RMS displacement. Read the prerequisite
Particle radius: The input is a radius, not a diameter. Confusing them changes the predicted coefficient by a factor of two. Read the prerequisite

Exact rational SI dimensions were checked at build time. This is a unit check, not a proof of the model. This teaching record is a draft, not a transcription of a printed equation.

Explore the equation · Modern model notation

From spreading to a measurable distance

\lambda_x = \sqrt{2\,D\,t}

The typical coordinate distance is the positive square root of twice the diffusion coefficient times the observation interval.

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Input · Model result · Constant

Static worked example

Values describe this accepted snapshot, not unsaved input edits.

Coordinate RMS displacement: 0.92676 μm
Diffusion coefficient: 0.42944 μm²/s
Observation interval: 1 s

Read the equation aloud in words

The model coordinate root mean square displacement equals the square root of two times the diffusion coefficient times the observation interval.

Squaring measures spread without cancellation between directions. Taking the positive square root turns squared distance back into a distance. Four times the observation interval gives twice the model RMS, not four times.

Model assumptions and every term’s meaning

Independent, zero-mean Gaussian displacement increments in a homogeneous liquid.
One-coordinate model statistic, not a measured speed or a sample estimate.
The overdamped regime is assumed rather than established from additional particle and fluid measurements.

A model relation: This equality concerns the ideal model. A finite synthetic ensemble fluctuates around it; its sample RMS is displayed separately. Read the prerequisite
Coordinate RMS: One coordinate, not the total three-dimensional distance. Model RMS and sample RMS have different meanings. Read the prerequisite
Why a square root?: Two D t has units of squared length. Its positive square root has units of length and defines the typical displacement. Read the prerequisite
Build the mean square: Independent zero-mean increments add their variances. The definition of D makes the coordinate mean square equal to 2 D t. Read the prerequisite
Diffusion coefficient: This is the accepted model diffusivity, not a rate inferred from the synthetic data. Read the prerequisite
Observation interval: This is the interval used for the displayed displacement, not the simulation frame rate. Re-observing the trial preserves its paths. Read the prerequisite

Exact rational SI dimensions were checked at build time. This is a unit check, not a proof of the model. This teaching record is a draft, not a transcription of a printed equation.

RMS distance over recorded time

Solid: this sample. Dashed: the model. The time axis is logarithmic. Zero sample values, if any, remain in the table but cannot appear on a log scale. All times refer to the same recorded paths.

Read the comparison as a table

RMS distance (μm)
Time (s)	Sample	Model
0.02	0.13038	0.13106
0.04	0.18556	0.18535
0.1	0.30368	0.29307
0.2	0.40416	0.41446
0.4	0.61388	0.58613
1	0.98602	0.92676
2	1.3495	1.3106
4	1.9035	1.8535
10	2.8723	2.9307

What is, and is not, being simulated

These synthetic paths are Gaussian independent increments for dilute spherical tracers in a homogeneous Newtonian liquid. They do not simulate individual molecular collisions, inertia, interactions, sedimentation, walls, localization error or motion blur.

Low Reynolds number and observation times long compared with momentum relaxation are assumed, not verified from fluid and particle data. Lines between samples are a drawing convention; they do not define an instantaneous Brownian speed.

The three latent coordinates are recorded together. Axis, dimension, observation and statistic changes reuse those coordinates. The histogram and statistics use every tracer, including those outside the view.

No FrankenSim WASM artifact is used. Integer random draws follow the pinned Philox mapping; Gaussian conversion uses this host’s math functions and is not claimed bitwise identical across browser engines.

Logical recording draws: 1200000. This result was assembled from a newly generated deterministic recording.

Evaluator source digest: source:sha256:06f140ad57b373415f0056e72c0a61cee543139245eb0f242333675672ccd7b2

Show the code

Audited TypeScript reference evaluator: the owner on this device, or the host fallback for a FrankenSim capability.

stokesEinsteinD · src/physics/reference/diffusion/distributions.ts · revision workspace · sha256:a41fa1ff39026dbed9a701b368028012995d1835d9acebb2e32d5010f47404ec

This is the function that produced the current snapshot.

In words

Take the temperature and the gas constant, divide by the number of molecules in a mole to get the energy scale of one molecule, then divide by the drag on a sphere of this radius in a liquid of this viscosity.

Mathematics

Implementation

/** SI: temperature K, viscosity Pa s, radius m, output m2/s. No ambient constants. */
export function stokesEinsteinD(
  {
    T,
    eta,
    a,
    medium = "liquid",
  }: { T: number; eta: number; a: number; medium?: "liquid" | "gas" },
  set: ConstantSet,
): Evaluation {
  if (![T, eta, a].every((v) => Number.isFinite(v) && v > 0))
    return outside(
      "diffusionCoefficient",
      "stokesEinsteinD",
      "T > 0, eta > 0, a > 0",
      "Enter positive finite temperature, viscosity, and particle radius.",
      "input",
      set,
    );
  if (medium !== "liquid")
    return outside(
      "diffusionCoefficient",
      "stokesEinsteinD",
      "stokes-gas-medium",
      "The liquid Stokes-drag model does not include the slip correction needed in a gas.",
      "model",
      set,
    );
  const k = thermalConstant(set);
  return number(
    "diffusionCoefficient",
    "stokesEinsteinD",
    (k.value * T) / (6 * Math.PI * eta * a),
    set,
    undefined,
    true,
  );
}

One worked example of this calculation. Constant set: Declared 1905-plan inputs (source review pending).
Step	Expression	Value	Unit
The gas constant in SI	R = 8.31e7 erg mol^{-1} K^{-1}	8.31	J mol^{-1} K^{-1}
Thermal energy per mole	RT	2411.1465	J mol^{-1}
Drag on one sphere	6 π η a	1.2723450247038662e-8	kg s^{-1}
Drag on a mole of spheres	N 6 π η a	7634070148223197	kg s^{-1} mol^{-1}
Diffusion coefficient	RT / (N 6 π η a)	3.158402337396894e-13	m^2 s^{-1}
Coordinate RMS at 1 s	λ_x = √(2 D t) at t = 1 s	7.947832833416785e-7	m
Coordinate RMS at 60 s	λ_x = √(2 D t) at t = 60 s	0.000006156364840452743	m

Audited TypeScript reference evaluator: the owner on this device, or the host fallback for a FrankenSim capability.

rmsDisplacement · src/physics/reference/diffusion/distributions.ts · revision workspace · sha256:6d2d381222179a43397683f83d95eecf9dab253844f72dfcaa1eaab1302222eb

This function computes the listed outputs when it runs.

In words

The typical one-dimensional displacement is the square root of twice the diffusion coefficient times the observation interval.

Mathematics

Implementation

export function rmsDisplacement(D: number, t: number): Evaluation {
  if (!validDt(D, t))
    return outside(
      "rmsDisplacement1d",
      "rmsDisplacement",
      "D >= 0 and t >= 0",
      "Diffusivity and elapsed time must be finite and nonnegative.",
    );
  return number(
    "rmsDisplacement1d",
    "rmsDisplacement",
    scale(D, t),
    undefined,
    undefined,
    D > 0 && t > 0,
  );
}

Audited TypeScript reference evaluator: the owner on this device, or the host fallback for a FrankenSim capability.

apparentSpeed · src/physics/reference/diffusion/distributions.ts · revision workspace · sha256:1f2137435fdc15b0a9b5c91cc4dafba680e8d52b0f5143fdbefdfa9945f6baec

This function computes the listed outputs when it runs.

In words

Divide the model RMS displacement by the observation interval. The quotient depends on how long you watch.

Mathematics

Implementation

export function apparentSpeed(D: number, tau: number): Evaluation {
  if (!validDt(D, tau) || tau === 0)
    return outside(
      "apparentSpeed",
      "apparentSpeed",
      "D >= 0 and tau > 0",
      "An apparent speed needs a positive observation interval.",
    );
  return number(
    "apparentSpeed",
    "apparentSpeed",
    (Math.SQRT2 * Math.sqrt(D)) / Math.sqrt(tau),
    undefined,
    undefined,
    D > 0,
  );
}

Audited TypeScript reference evaluator: the owner on this device, or the host fallback for a FrankenSim capability.

ensembleMoments · src/physics/reference/diffusion/tracers.ts · revision workspace · sha256:1d87c8e69590b8871bcc7a6390089003f59d0b80bfc7eb1919a57dec661743e6

This function computes the listed outputs when it runs.

In words

Average the recorded displacements of every tracer, not a viewport subset.

Mathematics

Implementation

/** Compensated sums over ALL members, never a viewport-selected subset. Row-major M by d. */
export function ensembleMoments({
  displacements,
  d,
  dt,
  viewport,
}: {
  displacements: Float64Array;
  d: number;
  dt?: number;
  viewport?: ViewportBounds;
}): Computation<EnsembleMoments> {
  if (
    !(displacements instanceof Float64Array) ||
    ![1, 2, 3].includes(d) ||
    displacements.length === 0 ||
    displacements.length % d !== 0 ||
    displacements.length > 30000 ||
    !displacements.every(Number.isFinite)
  )
    return invalid(
      ["displacements", "d"],
      "Provide finite, row-major displacements with one, two or three coordinates per tracer.",
    );
  const M = displacements.length / d,
    sums = new Float64Array(d * 3 + 1),
    corrections = new Float64Array(d * 3 + 1);
  function add(i: number, v: number) {
    const corr = corrections[i] ?? 0,
      sum = sums[i] ?? 0,
      y = v - corr,
      t = sum + y;
    corrections[i] = t - sum - y;
    sums[i] = t;
  }
  for (let i = 0; i < M; i++) {
    let norm = 0;
    for (let j = 0; j < d; j++) {
      const v = displacements[i * d + j];
      if (v === undefined) return failure("A required displacement coordinate was undefined.");
      if (v !== 0 && v * v === 0)
        return failure("A nonzero squared displacement is below the representable range.");
      add(j * 3, v);
      add(j * 3 + 1, Math.abs(v));
      add(j * 3 + 2, v * v);
      norm = Math.hypot(norm, v);
    }
    add(d * 3, norm);
  }
  const axes = Array.from({ length: d }, (_, j) => {
    const mean = (sums[j * 3] ?? 0) / M;
    const meanAbsolute = (sums[j * 3 + 1] ?? 0) / M;
    const meanSquare = (sums[j * 3 + 2] ?? 0) / M;
    return {
      mean,
      meanAbsolute,
      meanSquare,
      rms: Math.sqrt(meanSquare),
    };
  });
  const meanSquareNorm = axes.reduce((s, a) => s + a.meanSquare, 0);
  if (![...sums, meanSquareNorm].every(Number.isFinite))
    return failure("The moment reduction exceeded the numerical range.");
  const rmsNorm = Math.sqrt(meanSquareNorm);
  let insideCount: number | undefined;
  let outsideCount: number | undefined;
  if (viewport) {
    let inc = 0;
    let outc = 0;
    for (let i = 0; i < M; i++) {
      let isInside = true;
      for (let j = 0; j < d; j++) {
        const v = displacements[i * d + j] ?? 0;
        const minVal = viewport.min[j] ?? -Infinity;
        const maxVal = viewport.max[j] ?? Infinity;
        if (v < minVal || v > maxVal) {
          isInside = false;
          break;
        }
      }
      if (isInside) inc++;
      else outc++;
    }
    insideCount = inc;
    outsideCount = outc;
  }
  const apparentSpeed =
    dt !== undefined && Number.isFinite(dt) && dt > 0
      ? (d === 1 ? (axes[0]?.rms ?? 0) : rmsNorm) / dt
      : undefined;
  return {
    kind: "accepted",
    data: {
      M,
      axes,
      meanNorm: (sums[d * 3] ?? 0) / M,
      meanSquareNorm,
      rmsNorm,
      ...(apparentSpeed !== undefined ? { apparentSpeed } : {}),
      ...(insideCount !== undefined && outsideCount !== undefined ? { insideCount, outsideCount } : {}),
    },
  };
}

Audited TypeScript reference evaluator: the owner on this device, or the host fallback for a FrankenSim capability.

displacementHistogram · src/physics/reference/diffusion/tracers.ts · revision workspace · sha256:746f24cee7b8573b8a986cd8e072b4786549eba47dffca106dbd38582d670c7b

This function computes the listed outputs when it runs.

In words

Count how many recorded displacements fall in each interval of the histogram.

Mathematics

Implementation

export function displacementHistogram(
  values: Float64Array,
  edges: Float64Array,
): Computation<{ counts: Float64Array; underflow: number; overflow: number; total: number }> {
  if (
    values.length === 0 ||
    values.length > 10000 ||
    edges.length < 2 ||
    edges.length > 1001 ||
    !values.every(Number.isFinite) ||
    !edges.every((v, i) => {
      if (!Number.isFinite(v)) return false;
      if (i === 0) return true;
      const prev = edges[i - 1];
      return prev !== undefined && v > prev;
    })
  )
    return invalid(
      ["values", "edges"],
      "Use finite samples and strictly increasing histogram edges.",
    );
  const counts = new Float64Array(edges.length - 1);
  let underflow = 0,
    overflow = 0;
  const firstEdge = edges[0];
  const lastEdge = edges[edges.length - 1];
  if (firstEdge === undefined || lastEdge === undefined)
    return invalid(["edges"], "Histogram edges array must contain at least two finite bounds.");
  for (const value of values) {
    if (value < firstEdge) {
      underflow++;
      continue;
    }
    if (value > lastEdge) {
      overflow++;
      continue;
    }
    let lo = 0,
      hi = edges.length - 1;
    while (hi - lo > 1) {
      const mid = (lo + hi) >>> 1;
      const edgeMid = edges[mid];
      if (edgeMid !== undefined && value < edgeMid) hi = mid;
      else lo = mid;
    }
    const bin = Math.min(lo, counts.length - 1);
    const prevCount = counts[bin];
    if (prevCount !== undefined) {
      counts[bin] = prevCount + 1;
    }
  }
  return { kind: "accepted", data: { counts, underflow, overflow, total: values.length } };
}

Every histogram count, including tails

Coordinate intervals in μm; half-open bins, with the final endpoint included
Interval	Count
Below the plotted range	0
-4.6338 to -4.4021	0
-4.4021 to -4.1704	0
-4.1704 to -3.9387	0
-3.9387 to -3.707	0
-3.707 to -3.4753	0
-3.4753 to -3.2437	0
-3.2437 to -3.012	0
-3.012 to -2.7803	1
-2.7803 to -2.5486	1
-2.5486 to -2.3169	0
-2.3169 to -2.0852	7
-2.0852 to -1.8535	7
-1.8535 to -1.6218	6
-1.6218 to -1.3901	16
-1.3901 to -1.1584	12
-1.1584 to -0.92676	15
-0.92676 to -0.69507	28
-0.69507 to -0.46338	24
-0.46338 to -0.23169	39
-0.23169 to 0	39
0 to 0.23169	43
0.23169 to 0.46338	37
0.46338 to 0.69507	26
0.69507 to 0.92676	32
0.92676 to 1.1584	23
1.1584 to 1.3901	17
1.3901 to 1.6218	8
1.6218 to 1.8535	6
1.8535 to 2.0852	6
2.0852 to 2.3169	3
2.3169 to 2.5486	1
2.5486 to 2.7803	1
2.7803 to 3.012	1
3.012 to 3.2437	1
3.2437 to 3.4753	0
3.4753 to 3.707	0
3.707 to 3.9387	0
3.9387 to 4.1704	0
4.1704 to 4.4021	0
4.4021 to 4.6338	0
Above the plotted range	0
Total, including both tails	400

Notes and laboratory

Beside this passage

Margin notes, assumptions, and the laboratory stay here so the German argument keeps the main column.

Keep the tracer ensemble in view

Read in another form

Download the full explanation as Markdown · Structured reading records · Browse the foundation library

Uncertainty, evidence and inference

What makes an estimate informative, rather than just a number returned by a formula?

Inference asks what an observation permits you to learn under a specified model. Before calculating a narrow interval, ask whether the observations distinguish the parameters at all. Brownian diffusivity constrains the product of radius and molecular number when temperature and viscosity are given; it cannot identify both members of that product separately.

An estimator is a rule applied to a finite sample. Different samples produce different estimates. Bias concerns the estimator's mean across such repetitions, not whether one particular estimate happens to equal the answer. Fitting drift from the same displacements uses information and changes the degrees of freedom available to estimate spread.

A 95% confidence procedure covers a fixed true parameter in 95% of repetitions when its assumptions hold. That statement is not a 95% posterior probability that this one fixed parameter lies inside one realized interval. Keeping the missed intervals in a repeated-trial plot is essential to understanding coverage.

A conditional interval holds the input radius, viscosity and temperature exact. A combined interval can include their uncertainty only when their measurement procedures and coverage are declared. A plausible-looking plus-or-minus bound with no coverage is not a confidence statement. The union bound adds error probabilities without requiring statistical independence between the input intervals.

A synthetic path created from a chosen molecular number is useful for checking recovery and coverage, not as independent evidence for that number in nature. With modern SI definitions, the Avogadro constant is already fixed; the analogous inversion is a consistency check. An empirical determination requires noncircular physical inputs and an admitted observation model.

One worked example

Consider a purely numerical inverse exercise N = 12/D, with D estimated as 4. Its point estimate is N = 3.
Suppose an admitted procedure gives the D interval [3, 6]. Since division by a larger positive D produces a smaller N, the corresponding interval is [12/6, 12/3] = [2, 4]. The endpoints reverse.
Now suppose an extra unknown scale a enters as N = 12/(aD). The same D supports many a–N pairs. No interval for a unique N is justified until a is supplied independently.
Declaring a = 2 halves both the point estimate and the inverted bounds without changing any displacement. It changes an assumption, not the observations.
For a combined guarantee, three input procedures with at most 1% failure each and a diffusion procedure with at most 2% failure have at most 5% total failure by the union bound. Without their stated coverage, that arithmetic supplies no guarantee.

A stopping point: State what is identifiable, which estimator and observation model you used, what is held exact, and whether the result is synthetic recovery, an independent estimate or a consistency check.

Open this as a full reading page →

A local expansion

Why can a smooth function be replaced by a few terms?

The first derivative is local change per unit of the input. The second derivative describes how that first rate changes. A Taylor expansion writes the nearby value using those rates. The dots stand for a remainder; they are not automatically negligible.

f(x+\Delta)=f(x)+\Delta f'(x)+\frac{\Delta^2}{2}f''(x)+\ldots

Nearby value equals the value plus displacement times slope plus half the squared displacement times curvature, followed by a remainder.

For a diffusion limit, the mean square step shrinks in proportion to the time interval, while the retained variance per unit time stays finite. Higher moments and the variation scale of the density must justify neglecting higher terms. A real tracer does not have independent steps at arbitrarily short times.

One worked example

(x+\Delta)^2=x^2+2x\Delta+\Delta^2

For the square function, the local expansion is exact through the quadratic term.

At x = 2 and Δ = 0.1, the three terms are 4, 0.4 and 0.01, adding to 4.41. Keeping only the value and slope gives 4.4, with an error of 0.01.

A stopping point: Name what is held fixed, which terms are retained, and why the remainder is small.

Extension · Planned callers: Special Relativity §4, Mass–Energy

Binomial Series Expansion for the Relativistic Lorentz Factor γ

In relativistic mechanics, the Lorentz factor γ = (1 − v²/c²)^−1/2 governs time dilation, length contraction, and the relativistic kinetic energy E_k = (γ − 1)mc². Setting x = (v/c)², the function is expanded using Newton’s generalized binomial theorem:

(1 − x)−1/2 = 1 + ½ x + ⅜ x² + ⁵/₁₆ x³ + ³⁵/₁₂₈ x⁴ + …   (|x| < 1)

1. Moderate speeds: partial sums at v/c = 0.6 (x = 0.36)

At sixty percent the speed of light, x = 0.36. The exact value of γ − 1 is 1/√(1 − 0.36) − 1 = 1/0.8 − 1 = 0.25. The progressive partial sums illustrate how higher-order terms accumulate:

Partial sums of γ − 1 at x = 0.36 against exact 0.25
Order	Term added	Term value	Partial sum	Difference to exact (0.25)
1st (linear in x)	½ x	0.180000	0.180000	0.070000
2nd (quadratic)	⅜ x²	0.048600	0.228600	0.021400
3rd (cubic)	⁵/₁₆ x³	0.014580	0.243180	0.006820
4th (quartic)	³⁵/₁₂₈ x⁴	0.0045927	0.247773	0.002227

Notice that the leading quadratic term 0.18 corresponds to the 0.18L mass–energy fixture beside the exact 0.25L, making both the retained classical kinetic energy (½ mv²) and the neglected relativistic remainder visible.

2. Low speeds: catastrophic cancellation and the cancellation-free rule

At everyday velocities such as v/c = 10⁻⁴ (where x = 10⁻⁸), evaluating γ − 1 by naive floating-point subtraction suffers catastrophic loss of precision. Because γ ≈ 1.000000005, subtracting 1 throws away the leading digits and amplifies round-off error.

Comparison at v/c = 10⁻⁴ (x = 10⁻⁸)

First series term (½ x):: 5.0000000000e-9
Cancellation-free route (trustworthy):: 5.0000000375e-9 (computed: 5.0000000375e-9)
Relative diff to first term:: 7.500000e-9 (exact theoretical 0.75x = 7.500000e-9)
Planted naive route 1/√(1−x) − 1:: 5.0000001917e-9 (computed: 5.0000001917e-9)
Naive relative difference:: 3.833145e-8 (approx 3.833e-8; over five times too large)

Cancellation-free evaluation rule: The cancellation-free route usingexpm1(-0.5 * log1p(-x)) or x / (√(1−x) · (1 + √(1−x)))is required for trustworthy numerical evaluation. Naive subtraction in IEEE 754 double precision yields 5.0000001917e-9 instead of the true 5.0000000375e-9, corrupting relativistic corrections with false precision artifacts.

Open this as a full reading page →

Keep the explanation for offline reading

Each HTML file contains the available explanation at every detail level, linked foundations, source references, and available build-time scalar results. It includes no private notes or running simulations. This remains an explanation preview, not a reviewed source edition.

Save this chapter for offline reading: §5 · From displacement to molecular scale (455631 bytes · HTML)

Open the downloaded file in a browser. Online source links still need a connection. Interactive plots are not included. Browse saved-chapter downloads.

Original journal scan

§5 · From displacement to molecular scale

Why viscosity changes the spread

What this relies on

Assumed here

What this does not establish

What would let us count molecules?

What this relies on

Assumed here

What this does not establish

Explanation

Static worked case

Keep the experiment beside the argument

Inspect the model behind this trial

RMS distance over recorded time

What is, and is not, being simulated

In words

Mathematics

Implementation

In words

Mathematics

Implementation

In words

Mathematics

Implementation

In words

Mathematics

Implementation

In words

Mathematics

Implementation

Beside this passage

Read in another form

Keep the explanation for offline reading