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Why does an isopycnic centrifugation band return after displacement? Choose a dilute mathematical model with outward coordinate $z$, constant local centrifugal acceleration $a>0$, fixed displaced volume $V>0$, linear friction coefficient $f>0$, and an imposed liquid density ${\rho }_f(z)={\rho }_0+Gz$ with $G>0$. Treat the species' buoyant density ${\rho }_b$ as constant in this medium. The sedimentation coefficient force balance gives the ordinary differential equationIts exponential function solution is $z(t)=z_{\ast }+(z(0)-z_{\ast })e^{-kt}$. The sign of the velocity reverses across $z_{\ast }$, proving stable equilibrium when $z_{\ast }$ lies inside the liquid. Changing $V/f$ changes the approach time $1/k$, while $z_{\ast }$ depends on buoyant density and the imposed density gradient. A real centrifuge rotor has radially varying centrifugal acceleration; the constant-$a$ approximation describes a short radial interval. Here the density gradient is held fixed, rather than modeling its formation.

For an original dimensionless example, let $u=z/L$ and $\tau =t/T$, where $L$ is column length and $T$ a chosen time scale. Write $q=z_{\ast }/L$ and $\lambda =kT$, so $u(\tau )=q+(u(0)-q)e^{-\lambda \tau }$. Species A and B have the same $q=0.45$ but different friction-dependent rates: $({\lambda }_A,u_A(0))=(0.4,0.1)$ and $({\lambda }_B,u_B(0))=(1.6,0.95)$. A moves outward and B inward, both toward $u=0.45$. Species C has $(q_C,{\lambda }_C,u_C(0))=(0.8,0.8,0.1)$ and approaches a separate band at $u=0.8$. At $\tau =4$, their positions are approximately $0.3793$, $0.4508$ and $0.7715$: a finite-time photograph would still show A short of its limiting position.

An equilibrium band has finite width because diffusion opposes focusing. In the same local model, let $c(z)$ be concentration and let the constant diffusion coefficient be $D>0$. Combining the restoring drift with Fick's laws, zero net flux requiresDividing by $c$ and integrating proves the Gaussian function profile, whose standard deviation is ${\sigma }_z=\sqrt{D/k}$. This local result assumes that the band is narrow relative to its distance from the boundaries and that $D$, $k$ and the density gradient vary negligibly across it. Equal buoyant densities therefore imply coincident centers in this model, not identical widths. A band outside the available density range has no such interior restoring point.

For a positive density $u$, logarithmic diffusion is the nonlinear diffusion equation $u_t=\kappa \Delta \log u$, equivalently $u_t=\kappa \mathrm{div}(u^{-1}\nabla u)$. Its diffusion coefficient is inversely proportional to the density. In one dimension, a mass-$M$ self-similar solution isIts integral is $M$, and substitution verifies the equation. This is $M$ times a Cauchy distribution density of scale $2\pi \kappa t/M$. It tends weakly to $M$ times the Dirac delta distribution as $t\downarrow 0$; the slow algebraic tails illustrate the enhanced diffusion where density is small.

Molecular diffusion is the component of material transport caused by the random thermal motion of molecules, measured relative to bulk motion of their surroundings. A diffusion coefficient is a property of a particular substance in a particular medium under specified conditions, not a universal speed of spreading. Its dependence on the solute, solvent and temperature is precisely what makes a spreading experiment informative. A random walk idealizes this motion; the Brownian motion description provides a continuous approximation over sufficiently long times. ^[ref]

A motivating experiment: watch dye enter a gel. Samprovalaki, Robbins and Fryer studied rhodamine 6G and methylene blue entering agar gel from two side reservoirs. A covered slab, $10\times 5\times 1\mathrm{mm}$, sat on a controlled thermal stage; the dye was brought to the same temperature. They photographed the gel every ten minutes, averaged image intensity across a strip, and fitted the changing concentration to a diffusion equation model. The cover limited evaporation and stopped dye travelling over the gel surface. Runs at $30$, $50$ and $70{}^{\circ }\mathrm{C}$ were repeated in triplicate. The measured spatial colour change therefore supplied a quantitative transport test, rather than merely showing that dye eventually spreads. Changing lighting also changes image intensity, so the optical response had to be checked. Importantly for interpretation, their constant-diffusion coefficient fit was imperfect and the gel coefficient is an effective transport coefficient, not automatically the value in free water. ^[ref]

A finite-pulse version to test width growth. The following is a proposed teaching variant, not that paper's apparatus or a report of measurements. Make a long, narrow, transparent gel channel of uniform cross-section. Join a thin dyed gel band between two undyed lengths of the same gel, with good contact and no air gaps; the dye should initially fill the cross-section so the dominant concentration variation is along the channel coordinate $x$. After assembly, keep the covered channel horizontal at uniform temperature, away from vibration. The dye is a finite initial pulse: no reservoir supplies more dye. Choose a dye whose image response can be calibrated and whose interaction with the gel is small enough for an approximately constant diffusion coefficient to be plausible.

Photograph the whole pulse at recorded times with fixed lighting, focus and exposure, including a length scale. Use uniformly dyed standards of the same gel thickness and a blank gel to convert image intensity to concentration $C(x,t)$; do not assume raw brightness is proportional to concentration. Average across the channel width and examine several parallel strips to check the one-dimensional approximation. Measure the initial profile instead of treating the band as infinitely thin. For a finite pulse with negligible end effects, define its spatial integral $Q$, centre $\bar{x}$ and width $\sigma$ byHere ${\sigma }^2$ is the variance of the normalized profile $C/Q$, and $\sigma$ is its standard deviation. If $C$ is measured in $\mathrm{mol}{\mathrm{m}}^{-3}$, then $Q$ has units $\mathrm{mol}{\mathrm{m}}^{-2}$; multiplying by the channel cross-section gives the total amount of substance. For the unbounded one-dimensional diffusion equation with constant $D$, conserved $Q$, finite variance and sufficiently decaying tails,Thus plot width squared against time: its slope determines the diffusion coefficient, while its intercept includes the finite starting width. This moment law holds even for a non-Gaussian distribution initially; it follows by applying integration by parts twice to the diffusion equation. ^[ref]

For the particularly readable case of an initially Gaussian distribution centred at zero, uniform advection at speed $u$ and constant diffusion coefficient $D$ give the one-dimensional concentration profileIts centre moves as $\bar{x}=ut$ and its variance grows as ${\sigma }_0^2+2Dt$. Our illustrative choice ${\sigma }_0=0.50\mathrm{mm}$ and $D=0.20\mathrm{m}{\mathrm{m}}^2{\mathrm{h}}^{-1}$ gives widths $0.50$, $0.81$ and $1.36\mathrm{mm}$ at zero, one and four hours. These are calculated values, not a measured dye coefficient. Once $2Dt$ dominates the initial variance, width grows as $\sqrt{2Dt}$: four times the elapsed time gives approximately twice the width. A rectangular starting band will not immediately have a Gaussian distribution, although the same width-squared law still applies under the stated assumptions.

Controls and limits of the interpretation. In the proposed experiment, check that $Q$ stays constant and that the centre remains stationary while the width increases. Include a blank and a uniformly dyed gel photographed over the same period: an apparent signal change there exposes lighting drift, dye fading or evaporation. Repeat with a lower starting dye concentration and a second initial width; agreement of the slopes is a check on constant $D$ and on preparation effects. Keep the pulse far from the channel ends and record its faint tails: truncation and background errors can strongly bias a variance. Poor contact, dye binding, chemical reactions or transport in a liquid film can invalidate the simple model. A gel suppresses bulk liquid motion, but its inferred diffusivity need not equal that in the pure solvent. A stationary centre alone does not exclude symmetric flows, and linear width-squared growth alone does not prove a molecular mechanism: turbulence can also produce an effective diffusive description. These are reasons to combine the profile test with the controlled setup.

A morphogen gradient is a spatially varying concentration profile of a morphogen. The profile $c(x)$ specifies the local amount; its mathematical gradient, $c^{\prime }(x)$ in one dimension, specifies how rapidly that amount changes with position. In the simplest positional interpretation, a biological cell responds to its local concentration, without having to measure the gradient across itself. Production, transport and removal together determine a morphogen gradient.

For a worked example, take a stationary one-dimensional tissue occupying $x\ge 0$, constant diffusion coefficient $D>0$, and uniform first-order reaction removal at rate $k>0$. At steady state, this specialization of the morphogen reaction-diffusion equation gives the ordinary differential equation and boundary conditionsHere $D$ has units of length squared per time, $k$ has units of inverse time, and $c_0$ is a prescribed source concentration. The Dirichlet boundary condition fixes the concentration at $x=0$. It does not prescribe a source flux; any required flux follows from the solution. These assumptions define a toy morphogen gradient, not a universal account of transport during biological development.

Define the diffusion length $\ell =\sqrt{D/k}$. Substitution into the ordinary differential equation shows that its general solution is $A{e}^{x/\ell }+B{e}^{-x/\ell }$. The vanishing boundary condition at infinity forces $A=0$, and the source Dirichlet boundary condition forces $B=c_0$. HenceOver every additional diffusion length, the concentration falls by the same factor $e^{-1}$. Faster diffusion increases $\ell$; faster removal decreases $\ell$. This makes the diffusion length a useful bridge between microscopic transport and the spatial reach of a morphogen gradient.

Choose $c_0=100$ arbitrary concentration units and $\ell =100\mu \mathrm{m}$, where $1\mu \mathrm{m}=10^{-6}\mathrm{m}$. Then $c(x)=100e^{-x/(100\mu \mathrm{m})}$: at $x=0$, $100\mu \mathrm{m}$ and $200\mu \mathrm{m}$, the concentrations are $100$, approximately $36.79$ and approximately $13.53$. The following diagram combines this morphogen gradient with the idealized responses calculated under morphogen response threshold.

For dilute, independently migrating carriers of number density $n$, charge $q$ and diffusion coefficient $D$, balancing drift with diffusion gives mobility $D/(k_{B}T)$ per applied force. Multiplying drift velocity by charge density yields $\sigma =n{q}^{2}D/(k_{B}T)$. State which species defines $n$ and $D$: a vacancy diffusivity must be combined with vacancy density. Correlated motion requires a correction rather than automatic use of a tracer diffusivity.

The coefficient in $\mathbf{J}=-D\nabla \varphi$ for the scalar neutron flux convention. It has dimensions of length, unlike a particle-density diffusion coefficient with dimensions length squared per time. For a single fixed speed $v$, the corresponding density diffusion coefficient is $vD$.

Lipid diversity allows a membrane to remain a selective, mechanically coherent barrier while retaining the molecular mobility required for function. Stability and fluidity are different properties: a lipid bilayer can be well ordered yet permit rapid lateral diffusion, or be mobile yet leaky and vulnerable to rupture. The effects depend on composition, temperature, leaflet organization and interactions among components rather than on one universal ranking of “rigid” and “fluid” lipids.

The basic stability comes from amphiphilic organization. Hydrophobic chains avoid exposure to water, while polar headgroups remain hydrated. A pore exposing chains has an energetic cost, encouraging local defects to close, but stress, chain disorder or incompatible molecular shapes can make defects easier to form. Useful measurements therefore include membrane permeability, leakage and mechanical failure as well as rotational mobility, lateral diffusion and acyl-chain order. Calling all of these “fluidity” conceals distinctions relevant to cell survival.

Acyl-chain length changes the contact area between neighboring chains. Other things equal, longer chains generally strengthen attractive contacts, increase bilayer thickness and raise the tendency to form an ordered phase. Shorter chains tend to lower the chain-melting transition and can increase permeability. A double bond between carbon atoms in the cis configuration restricts the chain geometry and disrupts close packing, generally lowering the chain-melting temperature relative to a comparable saturated chain. A double bond in the trans configuration has a different geometry and commonly permits closer packing than the corresponding cis configuration. Branching also changes packing. These comparisons hold the headgroup and other variables fixed; they are not predictions that all unsaturated lipids behave identically in a complex mixture.

An immediate consequence is that composition has to be interpreted relative to the growth temperature. A membrane made mostly of long saturated chains can become too ordered on cooling, slowing membrane protein conformational changes and molecular redistribution. Greater unsaturation can preserve mobility. On heating, the converse adjustments can limit excessive disorder and leakage. The original homeoviscous adaptation experiments found temperature-dependent changes in the saturated and long-chain fatty acid content of Escherichia coli, using spin-label electron paramagnetic resonance to compare membrane-lipid viscosity at the cells' growth temperatures.^[ref] This establishes an adaptive relationship rather than a claim that every organism maintains a single invariant value by the same biosynthetic route.

Headgroup diversity changes both local interactions and preferred geometry. Phosphatidylcholine is approximately cylindrical in many common bilayer compositions, whereas phosphatidylethanolamine has a smaller headgroup relative to its chains and can favor negative monolayer spontaneous curvature. A single-chain lysophospholipid often favors curvature of the opposite sign. Mixing these shapes can support bending and membrane-fusion intermediates, while too much of a nonbilayer-forming component can destabilize a planar lipid bilayer. The amphiphile packing parameter is a useful qualitative guide, but hydrated headgroup size and chain volume depend on conditions; a molecular cartoon does not supply a quantitative phase diagram.

Headgroup charge adds another control. phosphatidylserine and phosphatidylinositol 4,5-bisphosphate contribute negative inner-leaflet charge, recruiting basic membrane-binding regions of proteins and changing local electrostatics. Ionic strength, pH and divalent ions can screen, alter or bridge those charges. A lipid-induced change in domain organization could therefore arise from ion-mediated interactions rather than acyl-chain order alone. Sphingomyelin can pack tightly through its chain structure and headgroup/backbone interactions, influencing thickness and local order. A comparison of headgroups must specify the attached chains, because changing a headgroup while simultaneously changing chain saturation confounds the interpretation.

Cholesterol is particularly effective because its small polar head and rigid ring system fit between phospholipid chains. In a relatively disordered bilayer, it often increases chain order, condenses packing and reduces permeability to small solutes. In a tendency toward a tightly packed gel, it can disrupt regular packing and inhibit the cooperative freezing transition. Thus “cholesterol decreases fluidity” is incomplete: it can reduce some local motions while preventing an immobile gel and retaining lateral mobility. Its effects depend on concentration, temperature and the surrounding lipids; excessive amounts can exceed the mixture's capacity to accommodate it.

Mixtures of saturated lipids, unsaturated lipids and cholesterol can form coexisting liquid-ordered phase and liquid-disordered phase. The former combines greater acyl-chain order with lateral mobility; it is distinct from a solid gel. fluorescence microscopy of defined giant lipid vesicles directly demonstrated domains and mapped composition- and temperature-dependent coexistence.^[ref] The result proves that lipid interactions can organize a membrane without a protein scaffold. It does not establish that living plasma membranes contain large equilibrium domains of the same sizes. Cellular crowding, cytoskeleton coupling, asymmetry, trafficking and active turnover can change or suppress those patterns.

Diversity also works across the membrane. The outer and inner membrane leaflets differ in headgroups and chain distributions; equal total composition does not imply equal surface properties. Lipidomic and packing measurements found greater unsaturation and looser packing in the cytoplasmic leaflet of the examined mammalian plasma membranes.^[ref] Such asymmetry can couple to membrane protein shape and binding preferences. flippases and scramblases change transverse lipid distribution; because spontaneous translocation of many polar phospholipids is slow, this organization can be maintained and regulated. Leaflet imbalance creates curvature stress, but the result depends on lipid exchange and mechanical constraints rather than on headgroup charge alone.

Membrane composition also sets the energetic cost of deforming the surface around a membrane protein. Longer ordered chains may increase thickness and stiffness; mismatch with a transmembrane protein's hydrophobic span can favor local deformation, changed association or altered activity. Curvature-preferring lipids can lower the cost of particular conformational or membrane-shape transitions. Greater mechanical stiffness can resist bending but need not mean a larger rupture threshold. Oxidation of polyunsaturated chains can change molecular geometry and produce defects or reactive breakdown products, making chemical stability another tradeoff alongside mobility. Appropriate experiments should measure oxidation state and leakage rather than infer stability from saturation alone.

A good causal experiment uses compositionally defined lipid vesicles, independently varying chain saturation, chain length, headgroup and cholesterol. Keep vesicle size, temperature, ionic strength and osmotic conditions controlled. differential scanning calorimetry identifies cooperative transitions; spin-label electron paramagnetic resonance or calibrated fluorescence anisotropy probes local order and rotational freedom; fluorescence recovery after photobleaching measures lateral redistribution. Recovery also depends on geometry, immobile fractions and probe partitioning, so distinguish these from the diffusion coefficient. Test barrier stability by entrapped-dye leakage with dye-quenching and complete-release controls, and mechanical behavior with defined membrane-tension measurements. Repeat key comparisons with different probes because a probe can partition preferentially into one phase. In living biological cells, combine lipidomics with targeted biosynthetic perturbation, rescue and viability measurements to separate compositional effects from secondary stress responses.

Lipid diversity stabilizes useful combinations of properties: a sealed barrier, adaptable curvature, controlled local order and sufficient mobility. It also allows different compartments and leaflets to satisfy different physical demands; no single lipid composition maximizes all of those properties simultaneously.

All three marked points are at strongly oxidizing potentials, where X is depleted at the electrode: $c_X(0)\simeq 0$. Assuming only X initially in bulk, $c_{X^{+}}(\infty )=0$ and $c_X(\infty )=c_b$. Oxidation produces X${}^{+}$ at the interface.

For equal diffusion coefficients, the two concentration profiles are complementary, $c_X+c_{X^{+}}=c_b$. More generally equality of their interfacial molar fluxes requires $D_X{c}_X^{\prime }(0)=-D_{X^{+}}{c}_{X^{+}}^{\prime }(0)$, so the product surface concentration need not equal $c_b$. A and B are approximately steady hydrodynamic profiles; C is a time-dependent diffusion profile.

The diffusion coefficient relates particle flux to concentration gradient through Fick's first law, $\mathbf{j}=-D\nabla c$. For unbiased isotropic motion in $d$ dimensions it can equivalently be defined by the long-time mean squared displacement:An uncorrelated random walk with jump length $a$ and total jump rate $\Gamma$ has $⟨r^2⟩=\Gamma a^{2}t$, hence $D=\Gamma a^2/(2d)$; correlations introduce a multiplicative correlation factor. The factor $d$ counts the directions sharing the total displacement. It does not mean that the same material's single-coordinate diffusivity changes merely by choosing to observe fewer coordinates: $⟨x^2⟩=2Dt$ in every Cartesian direction of the same isotropic system.

At $500$ K, silver iodide is in its fast-ion-conducting $\alpha$ phase. For an order-of-magnitude estimate, take a jump length $a\sim 3$ Å and an effective successful jump rate $\Gamma \sim 10^{11}{\mathrm{s}}^{-1}$. ThenThus $D$ of order $10^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ is a reasonable estimate for this superionic conductor. The length and rate are explicitly assumed material scales; the question supplies neither a measured conductivity nor jump data, so an exact numerical coefficient cannot be deduced from temperature alone. The estimate is not presented as an experimentally measured value.

NMR probes the chosen isotope's local environments and motion in the bulk without electrical contacts. Line narrowing, relaxation and exchange measurements can expose hopping across a broad range of timescales, separate sites with resolved shifts, and provide microscopic activation energies. Pulsed-field-gradient measurements can sometimes measure longer-range displacement directly; local relaxation measurements usually infer it through a model.

Limitations include low sensitivity for dilute or low-abundance nuclei, overlap of inequivalent sites, quadrupolar broadening for spins above one-half, and contributions from paramagnetic impurities or other relaxation mechanisms. Heating can change defect populations or phase composition. Most importantly, local hopping need not produce long-range ionic conductivity: back-and-forth motion, correlated jumps and trapped ions can relax spins without carrying charge over macroscopic distances. Connecting $T_1$ to a diffusion coefficient requires an explicit jump model and comparison with a bulk transport measurement.

Use the supplied Arrhenius law as a diffusion-scale calculation. The stated temperature and duration become $T=573.15\mathrm{K}$ and $t=10800\mathrm{s}$. With $R=8.31446\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$,With the standard diffusion-distance convention, the characteristic scale isThe one-dimensional rms displacement $\sqrt{2Dt}$ is about30micrometres, and the three-dimensional rms displacement $\sqrt{6Dt}$ about53micrometres. These different prefactors follow from the diffusion-distance convention; the diffusion coefficient is unchanged.

The scale is much larger than atomic spacing and can permit local carbon redistribution and carbide formation over microstructural distances. It is not evidence that the entire macroscopic specimen becomes compositionally homogeneous. Trapping, concentration gradients, evolving carbides and the use of a ferrite diffusivity for an initially defect-rich transformation product limit a literal tracer-distance interpretation.

The numerical vacancy fraction depends on what the supplied diffusivity describes. In the usual effective mobile-vacancy calculation, the $0.9236$ carriers per cell are oxygen vacancies, each of effective charge $+2e$. There are eight oxygen sites, so that convention givesThere is a real qualification: the printed description calls the measured particles oxygen carrier ions. If $D$ is their tracer diffusivity and every occupied oxygen site is counted as an uncorrelated carrier, the inversion instead gives $N_O=0.9236$, formally $f_v=1-N_O/8=0.8845$. Such an enormous oxygen deficiency is inconsistent with the ordinary lightly doped fluorite picture; it is not a credible deduction of $88.5\%$ vacancies in stabilized zirconia.

More generally the conductivity involves a collective diffusion coefficient, whereas tracer diffusion includes jump correlations. With the explicitly defined convention $H_R=D_{\mathrm{tr}}/D_{\sigma }$,Without a carrier definition and the relevant correlation information, the literal tracer data do not uniquely establish the vacancy fraction. The consistent vacancy-carrier interpretation gives the boxed $11.5\%$ estimate. Interpreting the same measured $D$ as an oxygen-ion tracer diffusivity requires the collective/tracer conversion above; ordinary Nernst-Einstein relation arithmetic cannot identify these two carrier populations. This states the model assumption needed for the numerical result without treating the physically implausible uncorrelated oxygen-ion inversion as an established material composition.

Irradiation both creates dislocation loops and supplies mobile point defects that allow existing dislocations to climb, rearrange and annihilate. At low initial dislocation density, net production can dominate; at high initial density, sink absorption and recovery can dominate. A simple illustrative rate equation isThus both initial states approach ${\rho }_{\mathrm{ss}}=A/B$ in this fixed-condition model. Here $D$ is irradiation dose, not a diffusion coefficient. The asymptote reflects a dynamic balance, not the cessation of damage. It need not be the same for different alloys, temperatures, dose rates or applied stresses.

Write the Arrhenius equation as $D=D_0{e}^{-Q/(k_{B}T)}$. Eliminating the prefactor with two temperatures givesUse absolute temperatures $573.15$ and $873.15\mathrm{K}$. Because the input is a vacancy diffusion coefficient, this is a migration activation energy inferred from that diffusivity; an additional vacancy-formation energy should not automatically be added.

An oxygen vacancy has effective carrier charge $q_v=+2e$. Combining its density with its own diffusion coefficient in the Nernst-Einstein relation givesat $T=873.15\mathrm{K}$. The oxygen ions move oppositely to the vacancies and carry the same conventional current. This estimate assumes independently migrating vacancies, negligible correlation corrections, and negligible electronic contribution to the reported ionic conductivity.

At $227^{\circ }\mathrm{C}\simeq 500\mathrm{K}$, $1000/T\simeq 2$. Reading the carbon line gives $D_C\sim 2\times 10^{-15}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ to graphical accuracy. With $t=60\mathrm{s}$, the characteristic diffusion length isIf “distance” means a one-coordinate root-mean-square displacement, use $\sqrt{2Dt}\sim 0.49\mu \mathrm{m}$; the three-dimensional radial root-mean-square is $\sqrt{6Dt}\sim 0.85\mu \mathrm{m}$. The graph does not justify precision between these conventions. Equal diffusion distance in equal time requires equal diffusion coefficients. The copper line reaches this diffusivity around $1000/T\simeq 0.96$, giving $T_{\mathrm{Cu}}\sim 1.0\times 10^3\mathrm{K}$, about $770^{\circ }\mathrm{C}$. Carbon moves interstitially through ferrite, whereas copper self-diffusion is substitutional and vacancy-mediated, with a much higher effective activation barrier.

For unbiased one-dimensional diffusion, the mean squared displacement is $⟨x^2(t)⟩=2Dt$. With $t=n\tau$, compare this to the mutant expression:Therefore the mutant diffusion coefficient is

Take $x=0$ at chamber A and $x=L$ at chamber B. At steady state, the diffusion equation with constant positive diffusion coefficient givesThe two Dirichlet boundary conditions give $A=c_0$ and $B=-c_0/L$. ThusThe concentration gradient is constant, $dc/dx=-c_0/L$. By Fick's first law, the diffusive flux is $J=-Ddc/dx=D{c}_0/L$, directed from A towards B. The profile is linear independently of $D$, while the flux and the time needed to approach it depend on $D$.

This result assumes no significant bulk flow, chemoattractant loss to the walls, degradation or cellular uptake. Experimentally, a uniform-concentration control with matched reservoir concentrations tests for directional cell migration caused by flow or apparatus asymmetry; reversing the reservoir concentrations tests whether the directional response follows the chemical gradient. Keeping both reservoirs well mixed and measuring the concentration profile checks the assumed boundary concentrations and whether uptake has distorted it. These controls distinguish the assumed passive diffusion field from other directional cues; the derived profile alone does not establish the mechanism of chemotaxis.

For a cell of length $\ell$ aligned with the capillary, the steady linear gradient produces the front-to-back concentration differenceAt the given initial length, $L=5\times 10^{-3}\mathrm{m}$, this is $100(20\times 10^{-6})/(5\times 10^{-3})=0.4\mathrm{nM}$, above the stated threshold. Requiring $\Delta c\ge 0.1\mathrm{nM}$ givesThe diffusion coefficient does not enter this steady-gradient threshold. It does constrain experimental waiting time: for fixed reservoir concentrations, the slowest decaying mode of the diffusion equation has timescale $L^2/({\pi }^{2}D)$. With the supplied $D$, this is about $7.04$ hours at $0.5$ cm and $4.69$ days at $2$ cm; several such times may be required for an accurate steady state. Thus the maximum from the detection threshold is usable only if the reservoirs and cells remain suitable while the profile develops.

Let ${\sigma }_v^2=⟨v(0{)}^2⟩$, assumed finite. The displacement is $\Delta X(t)={\int }_0^{t}v(s)ds$. By stationarity, its two-time covariance is $⟨v(s)v(r)⟩={\sigma }_v^2\rho (s-r)$, with $\rho$ even for a real scalar process. Integrating over the two triangles of the square gives the Taylor turbulent dispersion identityA sufficient condition for ordinary long-time variance growth is ${\int }_0^{\infty }|\rho (s)|ds<\infty$ with ${\int }_0^{\infty }\rho (s)ds>0$. The dominated convergence theorem then givesMore generally the necessary asymptotic condition for a finite positive variance-based diffusion coefficient is convergence of ${\sigma }_v^2{\int }_0^t(1-s/t)\rho (s)ds$ to that coefficient. Absolute integrability is sufficient, not necessary. A zero integral does not give nondegenerate ordinary diffusion. Moreover a second-moment calculation alone does not establish a normal distribution or an entire diffusive scaling limit; those require additional probabilistic assumptions.

For the specified positive autocorrelation, direct integration gives, when $\alpha \ne 1,2$,Thus the power-law velocity-correlation dispersion has the following regimes. If $\alpha >1$, the Lagrangian integral time is $1/(\alpha -1)$ and ordinary variance growth holds, with $D={\sigma }_v^2/(\alpha -1)$. At $\alpha =2$ the exact expression is $2{\sigma }_v^2[t-\log (1+t)]$. If $\alpha =1$,For $0<\alpha <1$,The last two regimes spread faster than ordinary diffusion and have no finite long-time diffusion coefficient. Their slowly decaying velocity memory is responsible. At short times all these correlations instead give the ballistic law $⟨\Delta X^2⟩\sim {\sigma }_v^2{t}^2$.