{ "arguments": [ { "citations": [ "ap-17-549" ], "experiments": [ "bm-01" ], "help": { "example": "mean-variance-rms", "missingStep": "bridge-squaring-square-roots", "why": "mean-variance-rms" }, "id": "arg-bm-observable", "kind": "argument", "limitations": [ "The model expectation need not equal the mean of one small sample.", "Net displacement is not total path length." ], "meaning": { "executionStatus": "static-illustration", "historicalStatus": "pedagogical-reconstruction", "logicalRole": "definition", "modelStatus": "exact-within-model" }, "paper": "brownian-motion", "premises": [ "All displacements use the same origin, axis, units and observation interval." ], "prerequisites": [], "question": "What can we measure when left and right cancel?", "readings": { "full": [ { "kind": "paragraph", "text": "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." }, { "kind": "formula", "latex": "\\langle x\\rangle=0,\\qquad\\langle x^2\\rangle>0", "spoken": "The model’s mean displacement can be zero while its mean-square displacement is positive." }, { "kind": "paragraph", "text": "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." }, { "id": "mean-variance-rms", "kind": "foundation", "returnCaption": "Return to the observable that keeps movement when signed displacements cancel." } ], "margin": [ { "kind": "paragraph", "text": "This is an authored mathematical example, not a transcription, translation or historical observation. The source pointer is the Brownian paper’s displacement argument; no claim is made that these four numbers appear there." } ], "overview": [ { "kind": "paragraph", "text": "Left and right can cancel in the average while every tracer moves. Squaring before averaging keeps track of the spread." } ], "steps": [ { "id": "bridge-negative-numbers-direction", "kind": "foundation", "returnCaption": "Return to why leftward and rightward displacements can cancel." }, { "items": [ "Choose the common starting point as zero and right as positive.", "Use four displacements, in micrometres: −3, −1, +1, +3.", "Their sum is zero. Divide by four: the signed mean is zero.", "Square each first: 9, 1, 1, 9 square micrometres.", "The squares add to 20. Divide by four: the mean square is 5 square micrometres.", "Take the square root: the RMS is approximately 2.236 micrometres.", "The mean distance is instead (3 + 1 + 1 + 3) / 4 = 2 micrometres. These are different questions, not inconsistent answers." ], "kind": "steps" }, { "id": "mean-variance-rms", "kind": "foundation", "returnCaption": "Return to the distinction between signed mean, mean square and RMS." } ] }, "recap": "Signed displacements can cancel. Their squares still record movement.", "review": "draft", "schemaVersion": 1, "section": "s4", "title": "Zero average is not no movement" }, { "citations": [ "ap-17-549" ], "experiments": [ "bm-05", "bm-01" ], "help": { "example": "random-walks", "missingStep": "random-walks", "why": "probability-independence" }, "id": "arg-bm-independent-steps", "kind": "argument", "limitations": [ "Correlated steps, bias or an infinite second moment change the argument.", "This pedagogical walk does not describe fixed physical jumps in a liquid." ], "meaning": { "executionStatus": "static-illustration", "historicalStatus": "pedagogical-reconstruction", "logicalRole": "derivation", "modelStatus": "exact-within-model" }, "paper": "brownian-motion", "premises": [ "Increments are independent, have zero mean, and share a finite mean square ℓ².", "One step corresponds to a declared interval τ." ], "prerequisites": [ { "edge": "premise", "id": "arg-bm-observable" } ], "question": "What permits us to add the contributions of many random steps?", "readings": { "full": [ { "kind": "paragraph", "text": "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." }, { "kind": "formula", "latex": "\\left\\langle\\left(\\sum_{i=1}^n\\Delta_i\\right)^2\\right\\rangle=\\sum_{i=1}^n\\langle\\Delta_i^2\\rangle=nl^2", "spoken": "The mean square of the sum of independent centred increments is the sum of their mean squares." }, { "kind": "formula", "latex": "t=n\\tau,\\qquad D=\\frac{l^2}{2\\tau},\\qquad\\langle x^2\\rangle=2Dt", "spoken": "Elapsed time is n tau; defining D as ell squared over two tau gives mean-square displacement two D t." }, { "id": "probability-independence", "kind": "foundation", "returnCaption": "Return to why the expected cross terms vanish." } ], "margin": [ { "kind": "paragraph", "text": "The coin walk is a teaching route. Independence is a coarse-grained model premise, not a statement about arbitrarily short physical times. An agreeing simulated sample checks the implementation within that model; it is not independent evidence for molecules." } ], "overview": [ { "kind": "paragraph", "text": "Under the stated independent, unbiased, finite-variance step model, mean square grows in proportion to time. Four times the time gives twice the RMS displacement, not four times." } ], "steps": [ { "id": "probability-independence", "kind": "foundation", "returnCaption": "Return to the independence and centring assumptions." }, { "kind": "formula", "latex": "(A+B)^2=A^2+2AB+B^2", "spoken": "The square of A plus B contains two squares and twice the cross product." }, { "items": [ "Let A and B be two signed increments. Both have mean zero.", "Independence gives average AB = average A times average B = 0.", "Averaging the expansion leaves average A² plus average B².", "For n increments, each squared increment contributes ℓ²; all pairwise products vanish in the expectation.", "Thus the total mean square is nℓ². Since t = nτ, replace n with t/τ.", "Name the coefficient ℓ²/(2τ) as D. The result becomes 2Dt.", "Four times t gives four times the mean square. Taking the square root gives twice the RMS." ], "kind": "steps" }, { "id": "random-walks", "kind": "foundation", "returnCaption": "Return to the step-count and elapsed-time comparison." } ] }, "recap": "Independent centred steps add mean squares; the root therefore grows as the square root of time.", "review": "draft", "schemaVersion": 1, "section": "s4", "title": "Why the square grows with time" }, { "citations": [ "ap-17-549" ], "experiments": [ "bm-05", "bm-06" ], "help": { "example": "taylor-expansion", "missingStep": "taylor-expansion", "why": "distributions" }, "id": "arg-bm-diffusion-equation", "kind": "argument", "limitations": [ "A Taylor truncation is an approximation unless justified by a suitable limiting procedure.", "Physical independence breaks down at sufficiently short times; the limit is not a literal collision movie." ], "meaning": { "executionStatus": "static-illustration", "historicalStatus": "pedagogical-reconstruction", "logicalRole": "derivation", "modelStatus": "approximation" }, "paper": "brownian-motion", "premises": [ "The transition density is normalized, symmetric and has finite second moment.", "The density is smooth on the step scale; the retained expansion has a controlled coarse-grained range." ], "prerequisites": [ { "edge": "premise", "id": "arg-bm-independent-steps" } ], "question": "How can random individual steps produce a deterministic equation?", "readings": { "full": [ { "kind": "paragraph", "text": "Let φ(Δ) be the probability density for a displacement Δ during τ. To end at x, a tracer must start at x − Δ and then make that displacement. Adding over all possible increments gives the transition relation, written here in modern notation." }, { "kind": "formula", "latex": "p(x,t+\\tau)=\\int_{-\\infty}^{\\infty}p(x-\\Delta,t)\\varphi(\\Delta)\\,d\\Delta", "spoken": "The next density is the integral of the previous shifted density times the step density." }, { "kind": "paragraph", "text": "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." }, { "kind": "formula", "latex": "D=\\frac{1}{2\\tau}\\int_{-\\infty}^{\\infty}\\Delta^2\\varphi(\\Delta)\\,d\\Delta", "spoken": "D is half the mean-square step divided by the step interval." }, { "kind": "formula", "latex": "\\frac{\\partial p}{\\partial t}=D\\frac{\\partial^2p}{\\partial x^2}", "spoken": "The retained equation is the diffusion equation." }, { "id": "taylor-expansion", "kind": "foundation", "returnCaption": "Return to the expansion at fixed position or fixed time." } ], "margin": [ { "kind": "paragraph", "text": "The equations here are explanatory modern forms, not facsimile-checked printed formulas. A formal τ → 0 operation is not sufficient by itself: step sizes must scale with τ so that diffusivity stays finite and discarded terms vanish. The reader should distinguish this limit from the physical range where independent increments are a useful approximation." } ], "overview": [ { "kind": "paragraph", "text": "The exact step rule averages over possible displacements. With symmetry, finite variance and a justified smooth-scale approximation, its leading evolution is the diffusion equation." } ], "steps": [ { "id": "taylor-expansion", "kind": "foundation", "returnCaption": "Return to the two expansions and their remainders." }, { "kind": "formula", "latex": "p(x,t+\\tau)\\approx p(x,t)+\\tau\\frac{\\partial p}{\\partial t}", "spoken": "Keep the density and its first time change on the left." }, { "kind": "formula", "latex": "p(x-\\Delta,t)\\approx p-\\Delta\\frac{\\partial p}{\\partial x}+\\frac{\\Delta^2}{2}\\frac{\\partial^2p}{\\partial x^2}", "spoken": "Keep value, slope and curvature of the spatial density on the right." }, { "items": [ "The time derivative holds position fixed; the spatial derivatives hold time fixed.", "Insert the spatial expansion into the transition integral. The derivatives do not depend on the integration variable Δ.", "The integral of φ is one, so the first term integrates to p.", "For a symmetric φ, positive and negative Δ contributions cancel in the integral of Δφ.", "The remaining second-order contribution is half the second spatial derivative times the mean-square increment.", "Subtract p from both sides and divide by τ.", "Identify the mean-square increment divided by 2τ as D. This is the retained diffusion law, with the stated approximation conditions." ], "kind": "steps" }, { "id": "integration", "kind": "foundation", "returnCaption": "Return to normalization and cancellation inside the transition integral." }, { "id": "diffusion-equation", "kind": "foundation", "returnCaption": "Return to what the resulting density equation predicts." } ] }, "recap": "Normalization preserves probability; symmetry removes drift; the second moment supplies diffusion.", "review": "draft", "schemaVersion": 1, "section": "s4", "title": "From a step law to a density law" }, { "citations": [ "ap-17-549" ], "experiments": [ "bm-06", "bm-01" ], "help": { "example": "distributions", "missingStep": "gaussian-distributions", "why": "distributions" }, "id": "arg-bm-gaussian", "kind": "argument", "limitations": [ "A finite closed box has a different long-time distribution.", "At t = 0 the distribution is a point mass, not an ordinary density.", "Coordinate RMS is not the three-dimensional RMS distance." ], "meaning": { "executionStatus": "static-illustration", "historicalStatus": "pedagogical-reconstruction", "logicalRole": "derivation", "modelStatus": "exact-within-model" }, "paper": "brownian-motion", "premises": [ "Constant positive D, an unbounded line, no drift, and a localized initial ensemble." ], "prerequisites": [ { "edge": "premise", "id": "arg-bm-diffusion-equation" } ], "question": "How does the density law become a measurable displacement?", "readings": { "full": [ { "kind": "formula", "latex": "p(x,t)=\\frac{e^{-x^2/(4Dt)}}{\\sqrt{4\\pi Dt}}\\quad(t>0)", "spoken": "The point-source solution is the normalized Gaussian density for positive time." }, { "kind": "paragraph", "text": "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." }, { "kind": "formula", "latex": "\\lambda_x=\\sqrt{\\langle x^2\\rangle}=\\sqrt{2Dt}", "spoken": "Coordinate RMS displacement is the square root of two D t." }, { "kind": "paragraph", "text": "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." }, { "id": "gaussian-distributions", "kind": "foundation", "returnCaption": "Return to normalization and the Gaussian second moment." } ], "margin": [ { "kind": "paragraph", "text": "The displayed derivation verifies a modern normalized form. It does not replace the diplomatic transcription or establish which notation appears on a printed page. Three independent coordinates give total mean square 6Dt, whereas the paper’s one-coordinate displacement question uses 2Dt." } ], "overview": [ { "kind": "paragraph", "text": "For the unbounded, drift-free point-source model, spread is Gaussian at positive times and coordinate RMS displacement is √(2Dt). The starting distribution is a point mass." } ], "steps": [ { "id": "gaussian-distributions", "kind": "foundation", "returnCaption": "Return to the normalized point-source solution." }, { "items": [ "For positive t, change variable to u = x/√(4Dt). Then x = √(4Dt)u and dx = √(4Dt)du.", "The normalization factor cancels the factor in dx, leaving exp(−u²)/√π. Its integral is one.", "The first moment vanishes because x times the density is odd: the negative side cancels the positive side.", "For the second moment substitute x² = 4Dt u².", "The normalized integral of u² exp(−u²) is one half, by integration by parts.", "Multiplying 4Dt by one half yields 2Dt. Taking the nonnegative square root gives √(2Dt)." ], "kind": "steps" }, { "kind": "formula", "latex": "\\langle x^2\\rangle=\\frac{4Dt}{\\sqrt\\pi}\\int_{-\\infty}^{\\infty}u^2e^{-u^2}\\,du=2Dt", "spoken": "The second moment is four D t times the normalized Gaussian second-moment integral, giving two D t." }, { "kind": "paragraph", "text": "To check the density equation directly: the time derivative is p times (−1/(2t) + x²/(4Dt²)). The second spatial derivative is p times (−1/(2Dt) + x²/(4D²t²)). Multiplying the latter by D gives the former." }, { "id": "distributions", "kind": "foundation", "returnCaption": "Return to probabilities of intervals rather than heights." } ] }, "recap": "For an unbounded point source, the Gaussian has mean square 2Dt and coordinate RMS √(2Dt).", "review": "draft", "schemaVersion": 1, "section": "s4", "title": "What the spreading curve predicts" }, { "citations": [ "ap-17-549" ], "experiments": [ "bm-01", "bm-06" ], "help": { "example": "diffusion-equation", "missingStep": "diffusion-equation", "why": "flux-continuity" }, "id": "arg-bm-diffusivity", "kind": "argument", "limitations": [ "Slip, inertia, interactions, non-Newtonian response and gas corrections are not included.", "The argument imports constitutive laws; conservation alone does not derive them." ], "meaning": { "executionStatus": "static-illustration", "historicalStatus": "pedagogical-reconstruction", "logicalRole": "derivation", "modelStatus": "approximation" }, "paper": "brownian-motion", "premises": [ "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." ], "prerequisites": [ { "edge": "premise", "id": "arg-bm-gaussian" } ], "question": "What fixes D for a small spherical tracer in a liquid?", "readings": { "full": [ { "kind": "paragraph", "text": "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)." }, { "kind": "paragraph", "text": "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." }, { "kind": "formula", "latex": "D=\\frac{RT}{6\\pi\\eta aN}=\\frac{k_BT}{6\\pi\\eta a}", "spoken": "Stokes–Einstein diffusivity is R T over six pi viscosity radius N, or k B T over six pi viscosity radius." }, { "kind": "paragraph", "text": "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." }, { "id": "flux-continuity", "kind": "foundation", "returnCaption": "Return to balancing drift and diffusive fluxes." } ], "margin": [ { "kind": "paragraph", "text": "This is an explanatory summary of the physical relation developed before the displacement calculation, not a complete edition of §§1–3. The live laboratories use the explicitly labeled modern SI constant set, not claimed facsimile-verified 1905 numerical inputs." } ], "overview": [ { "kind": "paragraph", "text": "For the stated dilute-sphere liquid model, higher viscosity lowers D. Doubling viscosity halves the predicted mean square at fixed time and divides RMS displacement by √2." } ], "steps": [ { "items": [ "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." ], "kind": "steps" }, { "id": "flux-continuity", "kind": "foundation", "returnCaption": "Return to the distinction between density and flux." }, { "id": "bridge-squaring-square-roots", "kind": "foundation", "returnCaption": "Return to the square-root response when diffusivity changes." } ] }, "recap": "Combining ideal osmotic pressure with mobility and diffusive balance relates diffusivity to temperature, viscosity and radius.", "review": "draft", "schemaVersion": 1, "section": "s5", "title": "Why viscosity changes the spread" }, { "citations": [ "ap-17-549" ], "experiments": [ "bm-07", "bm-01" ], "help": { "example": "error-and-inference", "missingStep": "diffusion-equation", "why": "error-and-inference" }, "id": "arg-bm-inference", "kind": "argument", "limitations": [ "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." ], "meaning": { "executionStatus": "static-illustration", "historicalStatus": "pedagogical-reconstruction", "logicalRole": "derivation", "modelStatus": "approximation" }, "paper": "brownian-motion", "premises": [ "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." ], "prerequisites": [ { "edge": "premise", "id": "arg-bm-diffusivity" } ], "question": "Which additional measurements turn displacement into an estimate of N?", "readings": { "full": [ { "kind": "formula", "latex": "D=\\frac{\\langle x^2\\rangle}{2t},\\qquad N=\\frac{RTt}{3\\pi\\eta a\\langle x^2\\rangle}", "spoken": "Diffusivity is mean-square displacement over twice time; molecular number is R T t over three pi viscosity radius mean-square displacement." }, { "kind": "paragraph", "text": "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." }, { "kind": "paragraph", "text": "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." }, { "id": "mean-variance-rms", "kind": "foundation", "returnCaption": "Return to the observable that must be estimated from displacement data." }, { "id": "error-and-inference", "kind": "foundation", "returnCaption": "Return to what a finite sample and independent inputs can identify." } ], "margin": [ { "kind": "paragraph", "text": "No historical dataset or numerical estimate is supplied here as evidence. The bibliography points to the original argument; source transcription, translation, measurement-error instruments and historical-data inference remain in preparation. This keeps an algebraic inversion separate from a successful empirical determination." } ], "overview": [ { "kind": "paragraph", "text": "A displacement measurement can constrain molecular number only with additional independently known physical quantities and a valid model. A simulation using an assumed answer is not a new measurement of that answer." } ], "steps": [ { "items": [ "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." ], "kind": "steps" }, { "id": "mean-variance-rms", "kind": "foundation", "returnCaption": "Return to estimating a mean square rather than a squared mean." } ] }, "recap": "Displacement constrains diffusivity; independent radius, viscosity, temperature and gas-constant information are needed to infer molecular number.", "review": "draft", "schemaVersion": 1, "section": "s5", "title": "What would let us count molecules?" } ], "citations": [ { "id": "ap-17-549", "kind": "citation", "locator": "Annalen der Physik (4), 17, 549–560 (1905), §§1–5. Bibliographic pointer; this preview is not a source transcription or translation.", "schemaVersion": 1, "title": "A. Einstein, On the motion of particles suspended in liquids at rest required by the molecular-kinetic theory of heat", "url": "https://doi.org/10.1002/andp.19053220806" }, { "id": "bipm-si-definitions", "kind": "citation", "locator": "Modern SI definitions; not historical evidence", "schemaVersion": 1, "title": "BIPM: SI measurement units and defining constants", "url": "https://www.bipm.org/en/measurement-units" }, { "id": "nist-normal-variance", "kind": "citation", "locator": "Modern statistical reference: chi-square confidence limits", "schemaVersion": 1, "title": "NIST/SEMATECH e-Handbook: confidence limits for a standard deviation", "url": "https://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm" } ], "equations": [ { "argument": "arg-bm-observable", "assumptions": [ "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." ], "bindings": [ { "experimentId": "bm-01", "instanceSlot": "primary", "outputId": "modelApparentSpeed", "quantityId": "modelApparentSpeed", "termId": "eq-model-bm-apparent-speed.t.apparentSpeed" }, { "experimentId": "bm-01", "instanceSlot": "primary", "outputId": "rmsDisplacement1d", "quantityId": "rmsDisplacement1d", "termId": "eq-model-bm-apparent-speed.t.rms" }, { "experimentId": "bm-01", "instanceSlot": "primary", "outputId": "observationInterval", "quantityId": "observationInterval", "termId": "eq-model-bm-apparent-speed.t.time" } ], "explanation": "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.", "id": "eq-model-bm-apparent-speed", "kind": "equation", "notation": "modern-pedagogical", "notes": [ { "explanation": "This defines an interval-dependent comparison. It does not introduce a physical instantaneous Brownian velocity.", "foundation": "mean-variance-rms", "nodeId": "eq-model-bm-apparent-speed.op.definition", "title": "Define the observable" }, { "explanation": "A distance-per-interval statistic. It is not a molecular collision speed.", "foundation": "random-walks", "nodeId": "eq-model-bm-apparent-speed.t.apparentSpeed", "title": "Apparent coordinate speed" }, { "explanation": "The distance grows as the square root of time, while the denominator grows linearly. The quotient therefore decreases as the interval grows.", "foundation": "bridge-squaring-square-roots", "nodeId": "eq-model-bm-apparent-speed.op.division", "title": "Divide by the same interval" }, { "explanation": "The same accepted one-coordinate model displacement used by the neighboring RMS equation.", "foundation": "mean-variance-rms", "nodeId": "eq-model-bm-apparent-speed.t.rms", "title": "Model coordinate RMS" }, { "explanation": "A zero interval yields no apparent-speed value. The interface keeps the explanation instead of fabricating zero or infinity.", "foundation": "random-walks", "nodeId": "eq-model-bm-apparent-speed.t.time", "title": "Positive observation interval" } ], "paper": "brownian-motion", "review": "draft", "schemaVersion": 1, "sentence": [ { "nodeId": "eq-model-bm-apparent-speed.t.apparentSpeed", "text": "The apparent coordinate speed" }, { "nodeId": "eq-model-bm-apparent-speed.op.division", "text": " divides" }, { "nodeId": "eq-model-bm-apparent-speed.t.rms", "text": " the typical coordinate distance" }, { "nodeId": "eq-model-bm-apparent-speed.t.time", "text": " by the chosen positive interval." } ], "spoken": "Apparent coordinate speed equals the model coordinate root mean square displacement divided by the observation interval.", "title": "Why the apparent speed depends on how you watch", "tree": { "kind": "relation", "left": { "kind": "symbol", "quantityId": "modelApparentSpeed", "termId": "eq-model-bm-apparent-speed.t.apparentSpeed" }, "opId": "eq-model-bm-apparent-speed.op.definition", "operator": "define", "right": { "denominator": { "kind": "symbol", "quantityId": "observationInterval", "termId": "eq-model-bm-apparent-speed.t.time" }, "kind": "quotient", "numerator": { "kind": "symbol", "quantityId": "rmsDisplacement1d", "termId": "eq-model-bm-apparent-speed.t.rms" }, "opId": "eq-model-bm-apparent-speed.op.division" } }, "unitSystem": "si" }, { "argument": "arg-bm-diffusivity", "assumptions": [ "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." ], "bindings": [ { "experimentId": "bm-01", "instanceSlot": "primary", "outputId": "diffusionCoefficient", "quantityId": "diffusionCoefficient", "termId": "eq-model-bm-diffusivity.t.diffusion" }, { "experimentId": "bm-01", "instanceSlot": "primary", "outputId": "boltzmannConstant", "quantityId": "boltzmannConstant", "termId": "eq-model-bm-diffusivity.t.boltzmann" }, { "experimentId": "bm-01", "instanceSlot": "primary", "outputId": "temperature", "quantityId": "temperature", "termId": "eq-model-bm-diffusivity.t.temperature" }, { "experimentId": "bm-01", "instanceSlot": "primary", "outputId": "viscosity", "quantityId": "viscosity", "termId": "eq-model-bm-diffusivity.t.viscosity" }, { "experimentId": "bm-01", "instanceSlot": "primary", "outputId": "particleRadius", "quantityId": "particleRadius", "termId": "eq-model-bm-diffusivity.t.radius" } ], "explanation": "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.", "id": "eq-model-bm-diffusivity", "kind": "equation", "notation": "modern-pedagogical", "notes": [ { "explanation": "This equation assumes the dilute spherical tracer and Stokes-drag model; dimensional consistency alone does not establish those assumptions.", "foundation": "diffusion-equation", "nodeId": "eq-model-bm-diffusivity.op.equality", "title": "An ideal model" }, { "explanation": "A squared-distance-per-time coefficient. This value comes from the accepted model calculation.", "foundation": "diffusion-equation", "nodeId": "eq-model-bm-diffusivity.t.diffusion", "title": "Diffusion coefficient" }, { "explanation": "Increasing the denominator reduces diffusivity at fixed thermal energy. This is a model dependence, not a rule that every larger quantity must reduce motion.", "foundation": "flux-continuity", "nodeId": "eq-model-bm-diffusivity.op.division", "title": "Why divide by drag?" }, { "explanation": "Boltzmann constant times absolute temperature supplies energy per particle.", "foundation": "mean-variance-rms", "nodeId": "eq-model-bm-diffusivity.op.thermalEnergy", "title": "Thermal energy scale" }, { "explanation": "The SI 2019 constant set supplies this value. It must not be treated as independent historical evidence when trying to infer molecular number.", "foundation": "mean-variance-rms", "nodeId": "eq-model-bm-diffusivity.t.boltzmann", "title": "A known modern constant" }, { "explanation": "Use kelvin. The laboratory holds viscosity as an independently supplied parameter.", "foundation": "diffusion-equation", "nodeId": "eq-model-bm-diffusivity.t.temperature", "title": "Absolute temperature" }, { "explanation": "Six pi times viscosity times radius is Stokes drag per speed for an isolated sphere in the assumed low-Reynolds regime.", "foundation": "flux-continuity", "nodeId": "eq-model-bm-diffusivity.op.drag", "title": "The drag coefficient" }, { "explanation": "Viscosity is in the denominator. Doubling it at fixed temperature and radius halves D, not the RMS displacement.", "foundation": "flux-continuity", "nodeId": "eq-model-bm-diffusivity.t.viscosity", "title": "Dynamic viscosity" }, { "explanation": "The input is a radius, not a diameter. Confusing them changes the predicted coefficient by a factor of two.", "foundation": "bridge-squaring-square-roots", "nodeId": "eq-model-bm-diffusivity.t.radius", "title": "Particle radius" } ], "paper": "brownian-motion", "review": "draft", "schemaVersion": 1, "sentence": [ { "nodeId": "eq-model-bm-diffusivity.t.diffusion", "text": "The diffusion coefficient" }, { "nodeId": "eq-model-bm-diffusivity.op.thermalEnergy", "text": " is thermal energy" }, { "nodeId": "eq-model-bm-diffusivity.op.division", "text": " divided by" }, { "nodeId": "eq-model-bm-diffusivity.op.drag", "text": " the viscous drag coefficient." } ], "spoken": "The diffusion coefficient equals the Boltzmann constant times the absolute temperature, divided by six times pi times the dynamic viscosity times the particle radius.", "title": "Resistance to motion controls spreading", "tree": { "kind": "relation", "left": { "kind": "symbol", "quantityId": "diffusionCoefficient", "termId": "eq-model-bm-diffusivity.t.diffusion" }, "opId": "eq-model-bm-diffusivity.op.equality", "operator": "=", "right": { "denominator": { "args": [ { "kind": "number", "value": "6" }, { "kind": "constant", "name": "pi" }, { "kind": "symbol", "quantityId": "viscosity", "termId": "eq-model-bm-diffusivity.t.viscosity" }, { "kind": "symbol", "quantityId": "particleRadius", "termId": "eq-model-bm-diffusivity.t.radius" } ], "kind": "product", "opId": "eq-model-bm-diffusivity.op.drag" }, "kind": "quotient", "numerator": { "args": [ { "kind": "symbol", "quantityId": "boltzmannConstant", "termId": "eq-model-bm-diffusivity.t.boltzmann" }, { "kind": "symbol", "quantityId": "temperature", "termId": "eq-model-bm-diffusivity.t.temperature" } ], "kind": "product", "opId": "eq-model-bm-diffusivity.op.thermalEnergy" }, "opId": "eq-model-bm-diffusivity.op.division" } }, "unitSystem": "si" }, { "argument": "arg-bm-observable", "assumptions": [ "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." ], "bindings": [ { "experimentId": "bm-01", "instanceSlot": "primary", "outputId": "rmsDisplacement1d", "quantityId": "rmsDisplacement1d", "termId": "eq-model-bm-rms.t.rms" }, { "experimentId": "bm-01", "instanceSlot": "primary", "outputId": "diffusionCoefficient", "quantityId": "diffusionCoefficient", "termId": "eq-model-bm-rms.t.diffusion" }, { "experimentId": "bm-01", "instanceSlot": "primary", "outputId": "observationInterval", "quantityId": "observationInterval", "termId": "eq-model-bm-rms.t.time" } ], "explanation": "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.", "id": "eq-model-bm-rms", "kind": "equation", "notation": "modern-pedagogical", "notes": [ { "explanation": "This equality concerns the ideal model. A finite synthetic ensemble fluctuates around it; its sample RMS is displayed separately.", "foundation": "mean-variance-rms", "nodeId": "eq-model-bm-rms.op.equality", "title": "A model relation" }, { "explanation": "One coordinate, not the total three-dimensional distance. Model RMS and sample RMS have different meanings.", "foundation": "mean-variance-rms", "nodeId": "eq-model-bm-rms.t.rms", "title": "Coordinate RMS" }, { "explanation": "Two D t has units of squared length. Its positive square root has units of length and defines the typical displacement.", "foundation": "bridge-squaring-square-roots", "nodeId": "eq-model-bm-rms.op.squareRoot", "title": "Why a square root?" }, { "explanation": "Independent zero-mean increments add their variances. The definition of D makes the coordinate mean square equal to 2 D t.", "foundation": "probability-independence", "nodeId": "eq-model-bm-rms.op.meanSquare", "title": "Build the mean square" }, { "explanation": "This is the accepted model diffusivity, not a rate inferred from the synthetic data.", "foundation": "diffusion-equation", "nodeId": "eq-model-bm-rms.t.diffusion", "title": "Diffusion coefficient" }, { "explanation": "This is the interval used for the displayed displacement, not the simulation frame rate. Re-observing the trial preserves its paths.", "foundation": "random-walks", "nodeId": "eq-model-bm-rms.t.time", "title": "Observation interval" } ], "paper": "brownian-motion", "review": "draft", "schemaVersion": 1, "sentence": [ { "nodeId": "eq-model-bm-rms.t.rms", "text": "The typical coordinate distance" }, { "nodeId": "eq-model-bm-rms.op.squareRoot", "text": " is the positive square root" }, { "nodeId": "eq-model-bm-rms.t.diffusion", "text": " of twice the diffusion coefficient" }, { "nodeId": "eq-model-bm-rms.t.time", "text": " times the observation interval." } ], "spoken": "The model coordinate root mean square displacement equals the square root of two times the diffusion coefficient times the observation interval.", "title": "From spreading to a measurable distance", "tree": { "kind": "relation", "left": { "kind": "symbol", "quantityId": "rmsDisplacement1d", "termId": "eq-model-bm-rms.t.rms" }, "opId": "eq-model-bm-rms.op.equality", "operator": "=", "right": { "degree": 2, "kind": "root", "opId": "eq-model-bm-rms.op.squareRoot", "radicand": { "args": [ { "kind": "number", "value": "2" }, { "kind": "symbol", "quantityId": "diffusionCoefficient", "termId": "eq-model-bm-rms.t.diffusion" }, { "kind": "symbol", "quantityId": "observationInterval", "termId": "eq-model-bm-rms.t.time" } ], "kind": "product", "opId": "eq-model-bm-rms.op.meanSquare" } } }, "unitSystem": "si" } ], "foundations": [ { "citations": [], "example": [ { "kind": "paragraph", "text": "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." } ], "explanation": [ { "kind": "paragraph", "text": "Put the starting point at zero on a ruler. A final mark three units right has displacement +3; one three units left has displacement −3. Adding the signed displacements gives zero. Adding the distances from the start gives six." }, { "kind": "paragraph", "text": "Displacement compares the final and initial positions. It is not the length of the path travelled between them." } ], "id": "bridge-negative-numbers-direction", "kind": "foundation", "prerequisites": [ "bridge-sum-average" ], "question": "How can movement add up to zero?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "The sign names a direction relative to a chosen axis; it does not mean a negative distance.", "summary": "Choose right as positive and left as negative. Opposite displacements can cancel without either particle staying still.", "title": "A sign records direction" }, { "citations": [], "example": [ { "kind": "paragraph", "text": "The mean square of −3, −1, +1, +3 is 5. Doubling each displacement gives −6, −2, +2, +6, whose mean square is 20. The RMS changes from approximately 2.236 to 4.472: twice as large, not four times as large." } ], "explanation": [ { "kind": "paragraph", "text": "Both +3 and −3 have square 9. Squaring therefore removes the sign while giving greater weight to a larger magnitude. A square root returns the result to the original kind of unit: the square root of a square micrometre is a micrometre." }, { "kind": "formula", "latex": "\\sqrt{4q}=2\\sqrt q\\quad(q\\ge 0)", "spoken": "The square root of four q is twice the square root of q, for nonnegative q." } ], "id": "bridge-squaring-square-roots", "kind": "foundation", "prerequisites": [ "bridge-sum-average" ], "question": "Why does four times a mean square mean twice the RMS?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "Use the nonnegative square root when reporting a magnitude.", "summary": "Squaring multiplies a number by itself. The nonnegative square root reverses that operation.", "title": "Squares and square roots" }, { "citations": [], "example": [ { "items": [ "Add 3 + 1 + 1 + 3 to obtain 8.", "Count four values.", "Divide 8 by 4 to obtain 2.", "Repeating the list gives 16 divided by 8, still 2." ], "kind": "steps" } ], "explanation": [ { "kind": "paragraph", "text": "Imagine four equally sized containers holding 3, 1, 1 and 3 units. Combining them gives 8 units. Sharing the total equally between the four containers gives 2 in each. Two is the average; the original containers were not all equal." }, { "kind": "paragraph", "text": "The number of observations and the sum of their values are different quantities. Doubling the number of repeated observations doubles the sum but leaves the average unchanged." } ], "id": "bridge-sum-average", "kind": "foundation", "prerequisites": [], "question": "What does an average keep, and what does it lose?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "An average is a total shared equally among the number of observations.", "summary": "Add the values, then divide by how many there are. An average does not preserve the individual values.", "title": "Adding and averaging" }, { "citations": [], "example": [ { "kind": "paragraph", "text": "Holding everything else fixed, doubling D doubles the mean square at a given time and multiplies RMS displacement by √2. Four times the elapsed time doubles RMS displacement. Neither comparison specifies a single particle’s path." } ], "explanation": [ { "kind": "formula", "latex": "\\frac{\\partial p}{\\partial t}=D\\frac{\\partial^2p}{\\partial x^2}", "spoken": "The time rate of change of density is D times its spatial curvature." }, { "kind": "paragraph", "text": "At a local dip with positive curvature, the diffusion term increases density. At a local peak with negative curvature it decreases density. Boundary conditions matter: a closed box conserves its contents, but its long-time distribution is not the unbounded Gaussian." }, { "kind": "formula", "latex": "J=-D\\frac{\\partial c}{\\partial x}", "spoken": "Fick’s constitutive law sends flux down the density gradient." }, { "kind": "paragraph", "text": "Combining the flux law with conservation gives the diffusion equation for constant D. The flux law itself is a model premise in this route, not something conservation alone proves." } ], "id": "diffusion-equation", "kind": "foundation", "prerequisites": [ "flux-continuity", "bridge-squaring-square-roots" ], "question": "What changes when D changes?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "D has units of length squared per time; boundary conditions and model premises must be stated.", "summary": "D measures the rate of growth of a coordinate’s mean square: on an unbounded line that growth is 2D per unit time.", "title": "What the diffusion coefficient means" }, { "citations": [], "example": [ { "kind": "paragraph", "text": "A uniform density of 0.25 per micrometre on a four-micrometre interval has total probability one. A one-micrometre subinterval has probability 0.25; a two-micrometre subinterval has probability 0.5." } ], "explanation": [ { "kind": "paragraph", "text": "Divide a ruler into bins. The fraction of observations in a bin estimates its probability. Dividing that fraction by the bin width estimates a density. Changing the ruler from metres to micrometres changes the numerical density but not the probability of the same physical interval." }, { "kind": "formula", "latex": "P(a\\le X\\le b)=\\int_a^b p(x)\\,dx", "spoken": "The probability of the interval from a to b is the integral of the density over that interval." }, { "kind": "paragraph", "text": "A continuous distribution gives probability zero to any single exact position. At the starting time the ideal point source is instead an atom of probability one, not a finite curve with infinite height." } ], "id": "distributions", "kind": "foundation", "prerequisites": [ "bridge-sum-average" ], "question": "What does the height of a probability curve mean?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "Always distinguish curve height, interval area and an individual observation.", "summary": "A density measures probability per unit of position. Probability belongs to an interval and is its area under the curve.", "title": "Density is not probability" }, { "citations": [ "nist-normal-variance", "bipm-si-definitions" ], "example": [ { "items": [ "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." ], "kind": "steps" } ], "explanation": [ { "kind": "paragraph", "text": "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." }, { "kind": "paragraph", "text": "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." }, { "kind": "paragraph", "text": "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." }, { "kind": "paragraph", "text": "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." }, { "kind": "paragraph", "text": "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." } ], "id": "error-and-inference", "kind": "foundation", "prerequisites": [ "mean-variance-rms", "probability-independence" ], "question": "What makes an estimate informative, rather than just a number returned by a formula?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "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.", "summary": "Keep identifiability, estimator bias, repeated-sample uncertainty and independently measured inputs separate.", "title": "Uncertainty, evidence and inference" }, { "citations": [], "example": [ { "kind": "paragraph", "text": "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." } ], "explanation": [ { "kind": "paragraph", "text": "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." }, { "kind": "formula", "latex": "\\frac{\\partial c}{\\partial t}=-\\frac{\\partial J}{\\partial x}", "spoken": "Density changes at the negative spatial derivative of the flux." }, { "kind": "paragraph", "text": "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." } ], "id": "flux-continuity", "kind": "foundation", "prerequisites": [ "bridge-sum-average" ], "question": "Why does a difference of flows change the density?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "Conservation counts net crossings; it does not require each individual particle to stop.", "summary": "What accumulates in a region equals what enters minus what leaves, provided nothing is created or destroyed inside.", "title": "Counting what crosses a boundary" }, { "citations": [], "example": [ { "kind": "formula", "latex": "\\frac{1}{\\sqrt\\pi}\\int_{-\\infty}^{\\infty}u^2e^{-u^2}\\,du=\\frac12", "spoken": "The normalized integral of u squared times exp of minus u squared is one half." }, { "kind": "paragraph", "text": "Substituting back gives mean square 4Dt times one half, or 2Dt. About 68.27% of the probability lies within one RMS width of the centre, not all of it." } ], "explanation": [ { "kind": "formula", "latex": "p(x,t)=\\frac{1}{\\sqrt{4\\pi Dt}}\\exp\\left(-\\frac{x^2}{4Dt}\\right)", "spoken": "For positive D and t, the density is an exponential bell with normalization one over the square root of four pi D t." }, { "kind": "paragraph", "text": "Here x is displacement, t is elapsed time, and D is a constant diffusivity. Put u = x divided by the square root of 4Dt. Then dx is the square root of 4Dt times du. The normalization reduces to the integral of exp(−u²), which is the square root of π." }, { "kind": "paragraph", "text": "One way to evaluate that integral is to square it: I² is the integral of exp(−u²−v²) over the plane. In polar coordinates, area is r dr dθ; angles cover 0 to 2π. The radial integral of r exp(−r²) is one half, so I² = π. Since the integral is positive, I = √π. This is a modern verification calculation, not an attributed reconstruction of Einstein’s reasoning." }, { "kind": "paragraph", "text": "For the second moment use integration by parts: choose u as the factor and integrate u exp(−u²), whose antiderivative is −exp(−u²)/2. The endpoint term vanishes at both infinities. What remains is one half the normalization integral." } ], "id": "gaussian-distributions", "kind": "foundation", "prerequisites": [ "integration", "mean-variance-rms" ], "question": "How do we normalize the curve and find its mean square?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "A normalized density and its moments describe an ensemble, not a required endpoint for one tracer.", "summary": "A Gaussian is a family of densities whose spread is set by its variance; being bell-shaped is not a proof of its origin.", "title": "The Gaussian and its width" }, { "citations": [], "example": [ { "kind": "formula", "latex": "\\int_a^b u\\frac{dv}{dx}\\,dx=[uv]_a^b-\\int_a^b v\\frac{du}{dx}\\,dx", "spoken": "Integration by parts moves a derivative from one factor to another and retains the endpoint term." }, { "kind": "paragraph", "text": "A constant density of 0.25 per micrometre over 4 micrometres gives 0.25 times 4 = 1, regardless of how many equal rectangles we use." } ], "explanation": [ { "kind": "paragraph", "text": "Approximate the area under a curve by rectangles. Each contributes height times width. Add all contributions, then refine the widths. When those sums approach a limit, that limit is the integral." }, { "kind": "paragraph", "text": "For a probability density, a rectangle has units of inverse length times length, leaving a dimensionless probability. For total probability the whole area must be one." }, { "kind": "paragraph", "text": "Integration by parts comes from adding the product rule for differentiation over an interval: the integral of u times the derivative of v equals the endpoint product minus the integral of v times the derivative of u." } ], "id": "integration", "kind": "foundation", "prerequisites": [ "distributions" ], "question": "What operation does an integral describe here?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "The width and the endpoint term are part of the calculation, not decorations on the integral sign.", "summary": "An integral adds contributions from narrow intervals; its units include the width of each interval.", "title": "Adding continuously" }, { "citations": [], "example": [ { "items": [ "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." ], "kind": "steps" } ], "explanation": [ { "kind": "formula", "latex": "\\langle x\\rangle=\\frac{1}{M}\\sum_{i=1}^{M}x_i", "spoken": "Add the M signed displacements and divide by M." }, { "kind": "formula", "latex": "\\langle x^2\\rangle=\\frac{1}{M}\\sum_{i=1}^{M}x_i^2", "spoken": "Square each displacement before adding and dividing by M." }, { "kind": "formula", "latex": "\\operatorname{Var}(x)=\\langle x^2\\rangle-\\langle x\\rangle^2", "spoken": "Variance is the mean square minus the square of the mean." }, { "kind": "paragraph", "text": "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." } ], "id": "mean-variance-rms", "kind": "foundation", "prerequisites": [ "bridge-negative-numbers-direction", "bridge-squaring-square-roots" ], "question": "Which average can reveal wandering that cancels in the signed mean?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "Specify whether you square before averaging, average before squaring, or take absolute values.", "summary": "The mean tracks the centre; the variance tracks spread around the centre; RMS measures distance from the chosen zero.", "title": "Mean, variance and RMS" }, { "citations": [], "example": [ { "items": [ "For two fair independent steps of size one, list (++), (+−), (−+), (−−). Each has probability one quarter.", "Their products are +1, −1, −1, +1. The average product is zero.", "Their sums are +2, 0, 0, −2. The squared sums are 4, 0, 0, 4, with average 2.", "If the second step always repeats the first, the squared sum is always 4. That is a different, correlated model." ], "kind": "steps" } ], "explanation": [ { "kind": "paragraph", "text": "For independent choices, the probability of one result together with another is the product of their probabilities. Independence says learning the first result does not change the probabilities of the second. It does not mean that the second result must undo the first." }, { "kind": "formula", "latex": "\\langle AB\\rangle=\\langle A\\rangle\\langle B\\rangle=0", "spoken": "The expected product of independent centred variables A and B is zero." }, { "kind": "paragraph", "text": "For nonzero means the product of the means survives. Correlations can also preserve a cross term. The diffusion argument must state which case it assumes." } ], "id": "probability-independence", "kind": "foundation", "prerequisites": [ "bridge-sum-average", "bridge-negative-numbers-direction" ], "question": "Why do the cross terms disappear?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "Ask whether learning one result changes the probabilities for the next.", "summary": "Independent centred steps have zero expected product. Independence alone is not enough: their means must also be zero.", "title": "Probability and independence" }, { "citations": [], "example": [ { "kind": "paragraph", "text": "One hundred steps of RMS size 1 have RMS net displacement 10. Four hundred such steps have RMS net displacement 20. This does not say that any individual walker ends exactly 10 or 20 units away, and it does not count the path length." } ], "explanation": [ { "kind": "formula", "latex": "X_n=\\Delta_1+\\ldots+\\Delta_n", "spoken": "Position after n steps is the sum of the n signed increments." }, { "kind": "paragraph", "text": "Assume each increment has zero mean and the same finite mean square ℓ². Expanding the square produces n squared increments plus pairwise products. Independence and zero means remove every expected cross term." }, { "kind": "formula", "latex": "\\langle X_n^2\\rangle=nl^2,\\qquad \\sqrt{\\langle X_n^2\\rangle}=l\\sqrt n", "spoken": "The mean square is n times ell squared, and RMS displacement is ell times the square root of n." } ], "id": "random-walks", "kind": "foundation", "prerequisites": [ "probability-independence", "mean-variance-rms" ], "question": "Why does typical displacement grow as the square root of the step count?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "This walk is a mathematical bridge, not a claim that liquid molecules force a tracer to make fixed jumps.", "summary": "Independent centred increments add their variances, not their distances.", "title": "From steps to spread" }, { "citations": [], "example": [ { "kind": "formula", "latex": "(x+\\Delta)^2=x^2+2x\\Delta+\\Delta^2", "spoken": "For the square function, the local expansion is exact through the quadratic term." }, { "kind": "paragraph", "text": "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." } ], "explanation": [ { "kind": "paragraph", "text": "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." }, { "kind": "formula", "latex": "f(x+\\Delta)=f(x)+\\Delta f'(x)+\\frac{\\Delta^2}{2}f''(x)+\\ldots", "spoken": "Nearby value equals the value plus displacement times slope plus half the squared displacement times curvature, followed by a remainder." }, { "kind": "paragraph", "text": "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." } ], "id": "taylor-expansion", "kind": "foundation", "prerequisites": [ "bridge-squaring-square-roots" ], "question": "Why can a smooth function be replaced by a few terms?", "review": "draft", "schemaVersion": 1, "stoppingPoint": "Name what is held fixed, which terms are retained, and why the remainder is small.", "summary": "Near a point, value, slope and curvature describe the leading change. Discarding the remaining terms requires a scale argument.", "title": "A local expansion" } ], "paper": { "citation": "ap-17-549", "description": "Read the displacement argument and ask for the steps it leaves out. The laboratories test the same relationships.", "germanTitle": "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen", "id": "brownian-motion", "kind": "paper", "schemaVersion": 1, "sections": [ { "arguments": [ "arg-bm-observable", "arg-bm-independent-steps", "arg-bm-diffusion-equation", "arg-bm-gaussian" ], "id": "s4", "title": "§4 · From random displacement to diffusion" }, { "arguments": [ "arg-bm-diffusivity", "arg-bm-inference" ], "id": "s5", "title": "§5 · From displacement to molecular scale" } ], "sourceNotice": "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.", "sourceStatus": "in-preparation", "status": "explanation-preview", "title": "Brownian motion: from wandering to a measurable law" }, "schemaVersion": 1 }