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Diffusion is net transport associated with random microscopic motion. A localized patch of a dilute solute can spread through a stationary solvent: individual molecules move in both directions, but more cross from the crowded side of an imaginary surface than from the less crowded side. This imbalance gives a net diffusive flux down the concentration gradient; the molecules do not need to sense which direction has lower concentration. At uniform concentration, microscopic exchanges continue even though the net diffusive flux vanishes. ^[ref]

The distinction from advection is easiest to see in a dye patch. Uniform bulk motion carries its centre downstream; diffusion broadens it relative to that centre. Both can happen together. A moving or spreading patch by itself does not identify the mechanism: a nonuniform velocity field can also stretch a patch. ^[ref]

For a dilute nonreacting solute in a uniform medium at fixed temperature, Fick's first law models the relative diffusive flux as $\mathbf{j}=-D\nabla C$, with scalar diffusion coefficient $D>0$ of units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$. Here $C$ is the amount of substance per unit volume, so $\mathbf{j}$ has units $\mathrm{mol}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. With carrier velocity field $\mathbf{u}$, the total flux is $C\mathbf{u}+\mathbf{j}$. The continuity equation, with no sources or chemical reactions, givesFor constant $D$ and incompressible flow, this reduces to $C_t+\mathbf{u}\cdot \nabla C=D{\nabla }^{2}C$, where ${\nabla }^2$ is the Laplacian. In a stationary carrier it is the diffusion equation. These assumptions matter: variable $D$ must remain inside the divergence, and changing sources require extra terms. ^[ref]

The useful surprise is the common mathematics: thermal conduction smooths temperature through the analogous heat equation, while molecular diffusion smooths composition. This analogy already motivated the original formulation of Fick's laws. ^[ref] However, the thermal diffusivity and the solutal diffusivity need not be equal. A diffusive time scale $t_D\sim L^2/D$ means that crossing ten times the distance requires roughly one hundred times the time. This is why molecular diffusion is especially effective across thin layers, and why chaotic advection can assist mixing by making those layers thinner. Physical diffusion is a transport mechanism, distinct from the broader probabilistic class of Markov diffusion.

A diffusive flux transports a density $u$ down its gradient according to Fick's first law. For scalar diffusion coefficient $D>0$, $\mathbf{j}=-D\nabla u$ and the conservation law $u_t+\mathrm{div}\mathbf{j}=0$ gives a diffusion equation. If $D$ depends on $u$, this becomes a nonlinear diffusion equation.

Fick's laws describe transport down a concentration gradient and the resulting change of concentration with time. Combining Fick's first law with a conservation law gives the diffusion equation.

Treat the target as a stationary perfectly absorbing sphere of reaction radius $r_0$, surrounded by an effectively uniform reservoir of protein number density $C_A$. At steady state the radial diffusion equation givesThe inward capture rate per target is the integral of the Fick's first law:For dilute independent targets of number density $C_B$, the volume association rate is $J{C}_B$, soThis is the Smoluchowski association rate, in volume per time because the concentrations are number densities. If the target also diffuses, use the relative diffusion coefficient $D_A+D_B$. Orientation constraints, incomplete absorption and chromosome geometry can reduce the actual association rate relative to this ideal collision limit.

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.

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.

The diffusive flux is $\mathbf{J}=-D\nabla C$ by Fick's first law. Conservation in any fixed control volume $V$ gives $\frac{d}{dt}{\int }_{V}C=-{\int }_{\partial V}\mathbf{J}\cdot \mathbf{n}+{\int }_{V}F$. The divergence theorem and arbitrariness of $V$ yieldAfter the initial point injection, $F=0$ and $D=kC$ with $k>0$, so radial symmetry gives $C_t=k{r}^{-2}(r^{2}C{C}_r{)}_r$. Total mass is $4\pi {\int }_0^{\infty }{r}^{2}Cdr=4\pi M$.

A spreading radius $R$ and typical concentration obey $C{R}^3\sim M$ and $R^2\sim kCt$. Thus $R^5\sim Mkt$. In the proposed ansatz this givesDirect substitution, including the time derivative of $\xi$, givesMultiplying by ${\xi }^2$ makes the left side $-({\xi }^{3}f{)}^{\prime }/5$. Regularity and zero radial flux at the origin set the integration constant to zero, so $f{f}^{\prime }=-\xi f/5$. On the positive region $f^{\prime }=-\xi /5$. Nonnegativity, compact support and continuity at its edge giveIts normalization is $1={\int }_0^{{\xi }_0}{\xi }^{2}fd\xi ={\xi }_0^5/75$. Therefore the spherical nonlinear-diffusion source profile isThe concentration and flux vanish at the free boundary, so there is no distributional flux jump despite the derivative kink. The solution satisfies the diffusion equation weakly across that boundary and classically inside its positive region. Its support shrinks to the origin as $t\downarrow 0$, and testing against a continuous compactly supported function gives the limit $4\pi M$ times its value at zero. This verifies the prescribed point injection as well as the mass normalization.

With no chemotaxis, the stationary oxygen equation is $D_c{c}^{\prime \prime }=k\bar{n}$. Symmetry and the two prescribed surface values giveBy Fick's first law, the vertical oxygen flux is $j_c=-D_c{c}^{\prime }=-k\bar{n}z$. Thus each face supplies an inward flux $k\bar{n}d/2$ per unit area, and the total influx is $k\bar{n}d$, equal to the integrated consumption across the layer. The profile is physically nonnegative only if $c_0\ge k\bar{n}{d}^2/(8D_c)$; otherwise the constant-consumption approximation must be modified.

For bacteria retained in the fluid, impose zero bacterial flux at the faces. The stationary bacterial equation says that $j_n=-D_n{n}^{\prime }+\mu n{c}^{\prime }$ is constant, hence zero. Integrating $n^{\prime }/n=(\mu /D_n)c^{\prime }$ gives $n(z)=n_0\exp (\mu c(z)/D_n)$, with $n_0$ fixed by the total bacterial population. Substitution into the oxygen equation provesZero bacterial through-flux is the boundary assumption behind this equilibrium relation.

Fick's first law gives the diffusive fluxLocal conservation says $C_t+\nabla \cdot \mathbf{J}=0$, and therefore

For $D(C)=kC$, conservation of the deposited amount requiresThe dimensions satisfy $[M]=[C]L^2$ and $[k]=L^2/([C]T)$. Thus $Mkt$ has dimension $L^4$, so the similarity length is $(Mkt{)}^{1/4}$. Requiring the concentration scale to have dimension $[C]$ givesHence

In polar coordinates, substitution into
$C_t=k{r}^{-1}(rC{C}_r{)}_r$ givesMultiplying by $\xi$ and integrating once, with regularity and zero radial flux at the origin, yieldsWhere $F>0$, this reduces to $F^{\prime }=-\xi /4$, soNonnegativity and zero flux at the moving front give the compactly supported profileThe normalization givesand hence $A=1/\sqrt{2}$ and ${\xi }_0^4=32$. This is the Two-dimensional Barenblatt solution with diffusivity proportional to concentration, and its support radius isso $N=32$.

Now add the linear reaction term and writeSubstitution cancels the terms $aC$ and leavesChoosingproduces $G_{\tau }=k\nabla \cdot (G\nabla G)$, as in the linear-reaction time change for quadratic nonlinear diffusion. Therefore

For $a=0$, the front grows like $t^{1/4}$ while the central concentration decays like $t^{-1/2}$. For $a>0$, $\tau \sim e^{at}/a$, so the front grows like $e^{at/4}$ and the central concentration like $e^{at/2}$. For $a<0$, $\tau \to 1/|a|$: the front approaches a finite limiting radius while the concentration decays exponentially to zero.

At times long compared with the encounter time at depth $H$, the fixed depth $H$ is negligible relative to the growing diffusion lengths. Salt obeys the diffusion equation with diffusivity $D$. Withthe self-similar concentration profile in the liquid isIt has $C(h,t)=C_i$ and tends to $C_0$ in the far field.

Because the solid contains no salt, conservation of solute at the moving boundary requires the rejected solute flux from Fick's first law to equal the rate at which the interface sweeps up salt:Substituting the similarity profile givesThis relation determines $C_i$ if $\lambda$ is already known. More generally it must be solved together with the thermal Stefan condition and the interfacial phase-equilibrium relation $T_i=-m{C}_i$ from the liquidus. For $ϵ\ll 1$, salt occupies a much thinner boundary layer than heat and $C_i$ can be much larger than $C_0$.

Molecular transport across gradients. A gas can transport momentum, thermal energy and species abundance even when its average bulk velocity is zero. Molecules arriving at a point carry properties acquired near their preceding collisions. A small gradient makes the populations arriving from opposite sides slightly different, producing transport from larger to smaller values of the carried property. A continuum transport law requires a mean free path $\lambda$ small compared with the gradient length, and relaxation times short compared with the observation time.

For a dilute hard-sphere gas,where $d$ is molecular collision diameter. Both speed and collision distance matter: high speed alone does not determine a transport coefficient.

A simple kinetic estimate makes the common structure explicit. For a quantity $a(y)$ carried per particle, a molecule moving at angle $\theta$ to $+y$ brings approximately $a(y-\lambda \cos \theta )$. Expanding to first order and averaging an isotropic velocity distribution givessince $⟨\cos \theta ⟩=0$ and $⟨{\cos }^2\theta ⟩=1/3$. This mean-free-path argument estimates the coefficients; a detailed collision calculation changes their numerical prefactors and distinguishes different relaxation rates.

Momentum transport and viscosity. For a shear flow with mean velocity $U_x(y)$, the transported directed momentum per molecule is $m{U}_x$. The net flux of $x$ momentum in the $+y$ direction isHere $\eta$ is dynamic viscosity in $\mathrm{Pa}\mathrm{s}$. The associated shear stress on a material surface has the corresponding traction sign convention; the flux carries momentum from the faster-moving layer towards the slower-moving one. The kinematic viscosity is $\nu =\eta /(nm)$, with units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$. Viscosity smooths velocity gradients, and the lost ordered mechanical energy becomes heat.

Heat transport. Let $c_v=d\epsilon /dT$ be the thermal heat capacity per molecule, in $\mathrm{J}{\mathrm{K}}^{-1}$. Substituting $a=\epsilon (T)$ gives Fourier's law,Thermal conductivity has units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$. For a monatomic gas, translational equipartition gives $c_v=3k_B/2$. Rotational or vibrational excitations in other gases can increase the energy carried per molecule, while collisions govern how those modes exchange energy. A temperature gradient can therefore be transported differently from directed momentum even when the dimensional estimates look similar.

Species transport. For a dilute tracer or a mixture with prescribed total density and negligible bulk flow, molecules diffuse down concentration gradients. The first-gradient number flux is Fick's first law,The diffusivity $D$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$; multiplying number flux by the species mass gives mass flux. More generally one uses diffusion flux relative to the mixture's bulk velocity and specifies mass or mole fractions. The approximation is not a claim that an arbitrary pure-gas pressure gradient can remain at rest while only a Fick flux acts. In a mixture, collision cross-sections and relative molecular speeds, including the reduced mass, determine the species diffusivities.

Dependence on density and temperature. The kinetic estimates of gas transport coefficients predict, for fixed collision diameter and molecular heat capacity,Thus dynamic viscosity and thermal conductivity are approximately density independent in the dilute continuum regime. Reducing density reduces the number of carriers but increases their mean free path in the same proportion. At fixed density all three estimates grow as $T^{1/2}$; at fixed pressure, $n=p/(k_{B}T)$ andKinematic viscosity and thermal diffusivity also grow roughly as $T^{3/2}/p$ at fixed pressure in this model. Real interaction potentials make effective collision cross-sections temperature dependent, and internal excitations change $c_v$. Dense-gas correlations and boundaries invalidate the simple cancellation. In particular, when the mean free path approaches the apparatus scale, transport becomes free molecular; it cannot be described by extending the same local, pressure-independent Fourier and viscosity coefficients indefinitely.

Heat diffusion equation. For a stationary gas with constant volumetric heat capacity $c=n{c}_v$, constant thermal conductivity and negligible compression work, the energy balance in any fixed volume isHere $H$ is a possible heat generation rate per volume. The divergence theorem and Fourier's law yieldThermal diffusivity $\alpha$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$ and is of order $\overline{v}\lambda /3$. In a moving or compressing gas, the material derivative, pressure-volume work and viscous heating must also enter the internal-energy equation. The displayed pure diffusion equation describes the approximation in which those effects are absent or negligible.

Representative solutions. With no source, a steady one-dimensional slab between temperatures $T_a,T_b$ at $x=0,L$ has $T=T_a+(T_b-T_a)x/L$ and uniform heat flux $-\kappa (T_b-T_a)/L$. Thus a constant spatial gradient transports heat without changing local temperature in time when the boundaries continually supply and remove it.

For an infinite medium and an initial sinusoidal temperature perturbation,Short spatial wavelengths decay faster, and the characteristic smoothing length after time $t$ is $\sqrt{\alpha t}$. A finite slab with both ends held at $T_b$ has a sine expansion with factors $e^{-\alpha (n\pi /L{)}^{2}t}$; its slowest relaxation time is $L^2/({\pi }^2\alpha )$.

An instantaneous heat input $Q_A$ per unit transverse area at $x=0$ in an otherwise uniform infinite medium gives the Gaussian solutionIts integral $\int c(T-T_b)dx=Q_A$ conserves the deposited heat. The peak falls as $t^{-1/2}$ while its width grows as $\sqrt{t}$; diffusion redistributes energy instead of transporting a sharp front at a single velocity.

Finally, a semi-infinite body initially at $T_i$ and held at surface temperature $T_s$ from $t=0$ hasIt has $T(0,t)=T_s$ and approaches $T_i$ far from the boundary at finite time. These solutions demonstrate steady conduction, mode decay, spreading of localized heat and boundary-driven diffusion. Their idealized behaviour at extremely short times or arbitrarily long distances must be interpreted within the continuum regime rather than as an assertion of instantaneous microscopic communication.

Let $C(x)$ be the particle number density, $U(x)$ the conservative potential energy and $F=-dU/dx$ the applied force. In the overdamped regime a viscous drag coefficient $\zeta$ gives drift velocity $v_d=F/\zeta$. Combining this drift with Fick's first law gives the number fluxAt thermal equilibrium, the Boltzmann distribution is $C_{\mathrm{eq}}\propto e^{-U/(k_{\mathrm{B}}T)}$, so $d{C}_{\mathrm{eq}}/dx=F{C}_{\mathrm{eq}}/(k_{\mathrm{B}}T)$. Setting its net flux to zero yieldsThis is the Einstein diffusion relation. The derivation assumes a dilute equilibrium suspension, uniform temperature, linear drift response and a well-defined long-time diffusion coefficient. With a nonzero conservative force, the full transport equation includes drift; the force-free equation in the next section describes spreading without that drift.

For constant $D$ and no drift, particle conservation and Fick's first law give the one-dimensional diffusion equation ${\partial }_{t}C=D{\partial }_x^{2}C$. For $C=A{t}^{-1/2}{e}^{-x^2/(4Dt)}$,so direct substitution verifies the equation for $t>0$. The Gaussian integral fixes the normalization:This is the heat kernel for a point release, with $C(x,0)=N\delta (x)$ in the distributional limit. It remains centered at zero while its width grows as $t^{1/2}$ and its peak falls as $t^{-1/2}$. To compute the second moment, set $y=x/\sqrt{4Dt}$:using $\int y^2{e}^{-y^2}dy=\sqrt{\pi }/2$ over the whole real line, obtained by integration by parts.

If the mean number of independent unbiased jumps in time $t$ is $⟨m⟩=ft$, conditioning on $m$ gives $⟨x^2(t)⟩=\Delta ⟨m⟩=f\Delta t$. Matching it to the continuum diffusion result yieldsFor fixed-magnitude steps $a$, $\Delta =a^2$. The equality of second moments also holds for Poisson jump times independent of the increments; a Gaussian continuum approximation additionally requires many steps and finite step variance.

Diffusion and mobility. For a dilute isolated sphere in a Newtonian fluid, Stokes's law gives the drag magnitude $F_{\mathrm{drag}}=6\pi \eta Rv=\xi v$, where $\eta$ is the dynamic viscosity and $R$ is the hydrodynamic radius. The mechanical mobility is therefore $1/\xi$; the translational hydrodynamic mobility matrix is ${\xi }^{-1}I$.

To relate mobility to diffusion, put the particles in a weak external potential energy $U(x)$. Their drift speed is $-U^{\prime }(x)/\xi$. For number density $n(x)$, the sum of drift and Fick's first law fluxes isAt thermal equilibrium, the Boltzmann distribution gives $n\propto e^{-U/(k_{B}T)}$, and the net flux is zero. Substituting $n^{\prime }=-n{U}^{\prime }/(k_{B}T)$ into $J=0$ yields $D=k_{B}T/\xi$. Thus the Stokes–Einstein relation isHere $k_B$ is the Boltzmann constant and $T$ is absolute temperature. The assumptions include a no-slip approximately spherical particle, a continuum fluid, a small particle Reynolds number, and negligible hydrodynamic interactions with other particles or boundaries.

Gravitational sedimentation. The buoyant force must be included, since the mass of displaced water is comparable to the protein mass. Using $1\mathrm{Da}=1.66054\times 10^{-27}\mathrm{kg}$ givesso the buoyant mass in the stated spherical model isThe drag coefficient is $\xi =6\pi (10^{-3})(2.16\times 10^{-9})=4.0715\times 10^{-11}\mathrm{kg}{\mathrm{s}}^{-1}$. At terminal speed, $\xi v_g=m_{b}g$, soIgnoring buoyancy would give $1.47\times 10^{-11}\mathrm{m}{\mathrm{s}}^{-1}$ and would not use the supplied water density. This calculation treats the stated radius as both the hydrodynamic radius and the radius determining displaced volume; a real hydrated protein can require a separately measured partial specific volume.

Electrical drift. In the specified simple electrophoresis model, terminal force balance is $Ze\mathbf{E}=\xi \mathbf{v}$. Thereforewhere $\mu$ is the electrophoretic mobility and $e$ the elementary electric charge. For the given positive beta-lactoglobulin charge,in the direction of the electric field. This exceeds the gravitational sedimentation speed by about $8.6\times 10^6$.

Separation time. The two populations have equal drag coefficients because their radii are equal. Their difference in drift speed is therefore set by one elementary charge:Starting with coincident band centres, their separation is $\Delta vt$. HenceAt that time the original band has travelled $20\mathrm{mm}$ and the mutant $21\mathrm{mm}$. Their masses do not enter this ideal electrical force balance: equal-radius particles with different masses but the same charge would have equal electrophoretic mobility in this model.

Practical discrimination. A one-charge difference is enough for the stipulated separation, so the method is sensitive to mutations that change effective charge or friction. It cannot distinguish every mutation: substitutions that preserve charge and hydrodynamic size need not separate. Buffer pH changes protein ionization, and realistic electrophoresis also involves the ionic atmosphere, electro-osmosis, walls or gel interactions, and finite band widths.

For example, the Stokes–Einstein relation at $293.15\mathrm{K}$ gives $D=9.94\times 10^{-11}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. For an initially narrow band, diffusive broadening of electrophoretic bands givesat the calculated separation time. Thus the centres are about $3.15$ diffusion standard deviations apart, rather than forming infinitely sharp bands. The stated $1\mathrm{mm}$ centre-separation criterion is achievable in the model, but actual resolution also depends on the loading width and other dispersion mechanisms.

Equivalent centrifugal acceleration. For the original protein to attain its electrical drift speed by sedimentation, its buoyant centrifugal force must equal $20eE$:ConsequentlyThis is the acceleration at the protein's position in the rotating solution; no rotation rate can be assigned without a centrifuge radius. If the intended comparison is instead the relative separation speed of the two bands, use $\Delta m_b=600\mathrm{Da}$, since their displaced volumes are equal. Matching $|\Delta m_b|a_{\mathrm{eff}}=eE$ then gives $8.04\times 10^7\mathrm{m}{\mathrm{s}}^{-2}$. In that separation experiment the heavier original protein moves faster centrifugally, whereas the more highly charged mutant moves faster electrically.

Let $J$ be population flux. For any fixed control volume $V\subseteq \Omega$, mass conservation with local production givesAssume local, isotropic random dispersal, no imposed advection, and constant diffusion coefficient $D$. Fick's first law is $J=-D\nabla u$. The divergence theorem and arbitrariness of $V$ yieldThe coefficient has units length squared per time; $f$ has units population density per time. Smoothness sufficient to pass from the integral balance to the local equation is understood. This is a reaction-diffusion equation.