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On $x>0$ with a homogeneous wall condition, positive diffusion $\gamma$ and growth $\mu (x)={\mu }_0-ϵ\lambda x$, remove constant drift by $A=e^{Ux/(2\gamma )-i\omega t}f$. The scale $\xi =(ϵ\lambda /\gamma {)}^{1/3}x$ reduces the eigenvalue problem to the Airy ordinary differential equation. Decay selects $f=\mathrm{Ai}(\xi +z_n)$, where $\mathrm{Ai}(z_n)=0$ enforces the wall. The least negative zero $z_1$ sets the global instability threshold ${\mu }_0>U^2/(4\gamma )+|z_1|({ϵ}^2\gamma {\lambda }^2{)}^{1/3}$. A locally absolutely unstable pocket can therefore exist without an unstable global mode.

Analytical ultracentrifugation observes concentration profiles during rotation, rather than only recovering a separated pellet. Sector-shaped sample cells rotate past optical detectors. Absorbance scans measure attenuation at selected radii. A Rayleigh interferometer compares a sample sector with a solvent reference, converting concentration-dependent refractive-index changes into fringe shifts. Repeating scans follows a sedimentation boundary or the approach to a stationary distribution.^[ref]

For a dilute solute of buoyant mass $m_b$ and translational friction $f_h$, radial force balance gives $s=m_b/f_h$, where $s$ is the sedimentation coefficient. Diffusion supplies $D=k_{B}T/f_h$. A velocity experiment resolves boundary motion and broadening; a sedimentation equilibrium experiment instead measures $d\ln c/d(r^2)=m_b{\omega }^2/(2k_{B}T)$. This follows from the Boltzmann distribution in centrifugal potential $-m_b{\omega }^2{r}^2/2$. The two approaches separate frictional size information from buoyant mass information; shape, hydration and solvent properties must be specified to infer a unique geometric dimension.

Anderson localization is the confinement of electron wave functions, or other wave modes, by interference between repeated wave scattering paths in static disorder. A localized eigenstate occupies a limited region even in a much larger sample. Disorder can therefore suppress diffusion without absorbing the wave. In 1958, Philip Warren Anderson studied random site energies and intersite coupling, finding a regime without transport. His paper develops a theoretical model; it does not report an Anderson laboratory measurement.^[ref]

The distinction from ordinary multiple scattering is the retention of wave phases: alternative paths contribute probability amplitudes before their squared modulus is taken. Adding path probabilities alone loses the cross terms responsible for interference. This makes Anderson localization useful as a warning about transport models: repeated deflections need not produce unlimited diffusion. It also connects electron transport with wave propagation in a random medium, although the detailed equations and experimental diagnostics differ between electrons, light and acoustic waves.

Philip Warren Anderson's 1977 lecture distinguishes localized magnetic moments from disorder-localized states. It credits Nevill Francis Mott with recognizing the significance of the mobility edge. A Mott insulator instead requires electron interactions and can exist without disorder.^[ref]

John Hasbrouck Van Vleck developed quantum mechanics explanations of magnetic susceptibility, including its temperature dependence; this is a distinct contribution from Anderson localization.^[ref]

A sinusoidal surface temperature applied to a homogeneous semi-infinite conductor produces a periodic depth profile with penetration length $\delta =\sqrt{2\kappa /(\omega C)}$. It follows by substituting a complex exponential into $C{\partial }_{t}T=\kappa {\partial }_z^{2}T$ and taking the root that decays at depth. A depth $z$ attenuates the amplitude by $e^{-z/\delta }$ and delays the phase by $z/\delta$. This is diffusion with phase delay, rather than a nondispersive travelling wave.

A concrete comparison used a Biacore T100 with a CM5 chip bearing the target receptor; unused reactive surface groups were blocked. At $25{}^{\circ }\mathrm{C}$ in a pH 7.4 buffer solution, intact IgG and its Fab fragment were injected for $180\mathrm{s}$ at $5\mu \mathrm{L}/\min$; dissociation was monitored for $3600\mathrm{s}$. The optical response tracks changes in refractive index near the surface, rather than counting occupied paratopes directly. A reference surface and blank injections corrected background; association and dissociation responses were fitted with a 1:1 model. The reported intact-IgG detachment was slower than that of the Fab fragment. ^[ref]

This comparison supports format-dependent attachment, not an intrinsic single-site antibody affinity for intact IgG. The reference and blanks address background, not two-arm avidity, surface accessibility or delivery by diffusion. The paper's fitted 1:1 constants alone cannot distinguish those mechanisms. Its Fab fragment comparison therefore motivates specifying the binding format before interpreting a dissociation constant; a good fit does not establish a unique molecular model.

Historical background on the replacement of gaseous diffusion by gas centrifuges and diffusion-plant closures. The later webpage is not used to assign present-day plant counts to 2019.

A trajectory with a persistent random velocity $V$ has $X(t)-X(0)=Vt$. An ensemble with finite velocity variance therefore has displacement variance $t^2\mathrm{Var}V$, contrasting with the linear-in-time variance of diffusion. In an impulsively kicked sinusoidal shear dispersion model with zero transverse diffusivity, the shear phase never changes and repeated identical kicks have this ballistic scaling. Transverse diffusion instead decorrelates the velocity and permits a long-time diffusive limit.

For $A_t=A+(1+ib)A_{xx}-(1+ic)|A{|}^{2}A$, the uniform oscillation $A=e^{-ict}$ has phase diffusion coefficient $D=1+bc$. Its amplitude-phase dispersion relation is ${\lambda }^2+(2+2k^2)\lambda +2D{k}^2+(1+b^2)k^4=0$. Thus $D<0$ gives a long-wavelength modulational instability, whereas $D>0$ gives linear stability of all nonzero Fourier modes. On a fixed periodic domain, instability requires an allowed wavenumber with $k^2<-2D/(1+b^2)$.

Patrick Maynard Stuart Blackett and Giuseppe Occhialini began their cosmic-ray collaboration in 1931. They used the coincidence methods of Walther Bothe and Bruno Rossi to make a particle initiate its own recording. Blackett later reported tracks on about $80\%$ of their counter-controlled photographs, compared with roughly $2$–$5\%$ for random expansions in small chambers. Those figures describe the reported apparatus and conditions, not a universal detection efficiency.^[ref]

In their 1933 account, Geiger counters above and below the chamber fed a coincidence circuit. Its pulse operated a thyratron, magnet and spring-release valve, letting a light piston expand the gas. A mercury-vapor lamp then illuminated the drops. A water-cooled solenoid supplied the magnetic field; two oblique cameras allowed track reconstruction. Moving the lower counter changed which trajectories could be recorded.^[ref]

Two timing problems must be separated. The coincidence circuit's window distinguishes nearly simultaneous counter signals. The mechanical expansion latency determines how much the existing ionization trail spreads before droplets form. For a chosen one-coordinate diffusion model with initial variance ${\sigma }_0^2$, diffusion coefficient $D$ and expansion delay $t_e$,Taking ${\sigma }_0=0$, $D=2.0\times 10^{-6}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $t_e=20\mathrm{ms}$ gives ${\sigma }_{\mathrm{track}}\simeq 0.283\mathrm{mm}$. This is an original illustrative calculation, not the historical track width. It shows why a fast electronic pulse does not by itself guarantee a sharp photograph: the subsequent mechanical response also matters.

The photographs revealed groups containing positive and negative electron-like tracks, complementing the Anderson positron track experiment. Their interpretation depended on track curvature, direction, ionization and particle range, not simply on the presence of a coincidence.^[ref] Blackett explicitly noted that the counter control favored larger particle showers.^[ref] The multiplicity bias from a detector trigger gives a small model of that effect: a recording method can improve the study of rare interactions while making its photographs unrepresentative of the incident population.

For ordinary long-time diffusion, the signed mean displacement vanishes while one-coordinate mean-square displacement is $2Dt$ and the three-dimensional mean-square radial displacement is $6Dt$. An rms distance is neither mean signed displacement nor cumulative path length.

Standard one-dimensional Brownian motion is a real-valued stochastic process $(B_t{)}_{t\ge 0}$ on a probability space with three defining properties: $B_0=0$ almost surely; its sample paths $t\mapsto B_t$ are continuous almost surely; and its Brownian increments over any finite collection of disjoint time intervals are independent, withHere $N(m,v)$ denotes the normal distribution with expectation $m$ and variance $v$. The alternative symbol $W_t$ denotes the same kind of stochastic process. Continuity is a simultaneous statement about an entire sample path, rather than a separate assertion at each fixed time. Relative to its natural Brownian filtration, a future Brownian increment is independent of all observations up to its starting time.

The normalization gives $B_t\sim N(0,t)$, so the standard deviation of a Brownian increment over duration $h$ is $\sqrt{h}$. For $h=1$, $0.01$, and $0.0001$, the corresponding standard deviations are $1$, $0.1$, and $0.01$. Time becoming one hundred times shorter makes typical displacement ten times smaller. Brownian scaling expresses this rule for the whole stochastic process, while the Brownian covariance kernel describes how positions at different times remain related despite independent increments.

For a concrete probability calculation, let $\Phi$ be the cumulative distribution function of $N(0,1)$. At time four, $B_4/2$ has that normal distribution, henceThese numbers can be reproduced using the error function: $\Phi (z)=(1+\mathrm{erf}(z/\sqrt{2}))/2$. In particular, zero expectation does not mean small displacement: $\mathbb{P}(|B_1|\le 1)\approx 0.6826894921$, but the same fixed interval contains only about $38.3\%$ of the probability at time four.

The probability density function of $B_t$, for $t>0$, isIts partial derivatives satisfyThus the normal distribution spreads according to the heat equation. Starting at a fixed position $x_0$ replaces $x$ by $x-x_0$. This is a useful bridge between individual random trajectories and the deterministic evolution of their probability density function: the same Brownian motion provides both descriptions.

Physical diffusion uses its own scale. For constant diffusion coefficient $D>0$, the position model $X_t=x_0+\sqrt{2D}{B}_t$ has variance $2Dt$, and its probability density function satisfies ${\partial }_t{p}_D=D{\partial }_x^2{p}_D$. For example, $D=0.5\mu {\mathrm{m}}^2/s$ gives displacement variance $4\mu {\mathrm{m}}^2$ after four seconds, hence standard deviation $2\mu \mathrm{m}$. Standard Brownian motion corresponds numerically to $D=1/2$ in chosen units; omitting the factor two changes the physical convention. Einstein's 1905 paper relates diffusion to the mean squared displacement in precisely this convention. ^[ref]

Continuity does not supply a smooth physical velocity. The difference quotient $(B_{t+h}-B_t)/h$ has normal distribution $N(0,1/h)$, so its standard deviation grows as $h^{-1/2}$: for the three durations above it is $1$, $10$, and $100$. This calculation motivates the nowhere differentiability of Brownian motion, but it alone is not a proof of the simultaneous assertion at every time. The theorem says that, with probability one, no finite derivative exists anywhere along the sample path. A drawn polygon only displays a finite grid; its straight segments cannot establish this theorem. See Theorem 7.1.6 in Probability: Theory and Examples (fifth edition) by Rick Durrett.

For physical particles, Brownian motion is a model of position on suitable observation scales. It does not claim that microscopic velocity is a derivative of this idealized path. Einstein explicitly restricts his assumption of independent increments to intervals that are not too short. ^[ref]

Another way to measure the roughness is quadratic variation. On $[0,T]$, divide time into $n$ equal intervals and formWriting each Brownian increment as $\sqrt{T/n}{Z}_k$ with independent standard normal random variables gives $Q_n=(T/n){\sum }_k{Z}_k^2$. Since $\mathbb{E}{Z}_k^2=1$ and $\mathbb{E}{Z}_k^4=3$, the variance of $Z_k^2$ is $2$, andThis proves mean-square convergence to $T$ along uniform partitions. For $T=2$ and $n=100$, the expectation is $2$, the variance is $0.08$, and the root mean square of the error is $\sqrt{0.08}\approx 0.2828427125$. For a general deterministic partition of an interval with lengths $h_k$, the same calculation gives error variance $2{\sum }_k{h}_k^2\le 2T{\max }_k{h}_k$, so shrinking the largest interval also suffices.

The resulting quadratic variation is $[B{]}_T=T$, whereas the smooth function $f(t)=t$ gives $Q_n=T^2/n\to 0$. This difference explains why the ordinary rules of calculus need the second-order correction in the Itô formula. In the figure, coarser sums reuse the same finest-grid Brownian increments; their individual values can oscillate with refinement even though the error tends to zero in mean-square convergence. Refining a fixed interpolating polygon forever would instead give zero quadratic variation, so the finite-grid illustration must not be confused with the limiting Brownian motion.

Historically, Brown's 1828 report describes microscopic observations made in 1827, starting with particles released from pollen and extending to inorganic materials; this challenged an explanation based solely on living matter. His observations and the later mathematical model are distinct contributions. ^[ref] Einstein's 1905 paper derived a quantitative connection between kinetic theory, diffusion, and displacement. ^[ref] Wiener's 1923 Differential-Space developed integration on spaces of paths, a foundation for Wiener measure; it also acknowledges earlier work and his own previous papers. These dates mark documented contributions rather than an assertion that each subject had a single discoverer. ^[ref]

Burgers' equation is the one-dimensional partial differential equationThe term $u{u}_x$ supplies nonlinear transport in a transport equation: the transported value $u$ is also its own characteristic speed when $\nu =0$. Larger values catch up with smaller values ahead of them, steepening a decreasing profile. The diffusion term $\nu u_{xx}$ opposes this steepening. Positive $\nu$ gives the viscous Burgers equation; setting $\nu =0$ gives the Inviscid Burgers equation.

This small equation makes three different behaviors accessible in one model: Burgers characteristic breaking from continuous decreasing data, an entropy shock for a downward step, and a rarefaction wave for an upward step. The Burgers Riemann problem with positive flux and the Viscous Burgers travelling shock below give exact examples. These are mathematical calculations with chosen data; the figures contain no experimental measurements.

Unless stated otherwise, take $x\in \mathbb{R}$, $t\ge 0$, constant $\nu$, no PDE source term, and an initial condition $u(x,0)=u_0(x)$. There are no finite endpoints requiring boundary conditions. Where a profile has different limiting states at $x=\pm \infty$, those states are specified explicitly. On a finite interval, boundary conditions become part of the problem: positive diffusion normally requires a condition at each endpoint, whereas the Inviscid Burgers equation requires appropriate inflow transport boundary conditions. With dimensional variables, $u$ has units of velocity and $\nu$ of kinematic viscosity; all numerical examples below use dimensionless quantities.

Can two readout values locate a hit between wires? Only with a specified response model. Place two readout wires at $0$ and $p$, put $u=x/p\in [0,1]$, and stipulate calibrated, noiseless electric chargesThese are assumptions for a toy sensor response, not the natural response law of adjacent anode wires. Under these assumptions the centroidrecovers position exactly. The apparent gain in precision comes from adding calibrated analogue information. Taking the mean of two fired wire addresses without their electric charges would instead give the fixed midpoint $p/2$.

For a chosen noise model, replace the two electric charges by $q_L+n_L$ and $q_R+n_R$, with independent random variables $n_L,n_R$ of zero expected value and common standard deviation $s$. Assume $s/Q\ll 1$ and no threshold censoring. First-order expansion of the ratio givesFor stipulated $p=2\mathrm{mm}$ and $s/Q=0.02$, this approximation gives $0.0283\mathrm{mm}$ at the midpoint and $0.0400\mathrm{mm}$ at either endpoint. The endpoint values describe uncensored signed channel readouts, including small negative values after baseline subtraction. They do not assume that both channels would pass a positive threshold. A ratio with a fluctuating denominator also need not have exactly zero bias; this is a local small-noise approximation.

Three original counterexamples show why calibration and selection matter:

Likewise, two equal hits at $u=1/4$ and $u=3/4$ have exactly the same total two-channel response as one hit at $u=1/2$ with twice the electric charge. A centroid can hide multiplicity even when each charge measurement is exact.

Diffusion enabled the lecture's Orsay interpolation, without establishing this model's linear response.^[ref]

Catalyst deactivation is loss of catalytic activity during use. Regeneration in an ideal catalytic cycle does not rule out a separate damaging chemical reaction, blocking of a surface, or physical loss of catalyst from the apparatus. A route can also be slowed by diffusion to the surface or by accumulation of a reaction intermediate. Such effects limit the usefulness of a low schematic reaction energy barrier: measured throughput depends on the whole reaction mechanism and on transport. Demonstrating regeneration requires more evidence than observing a faster first run. For a recovered solid catalyst, a useful further comparison is a second run with fresh reactants, matched recovered amount, and the same mixing and temperature, alongside a fresh-solid run and a blank. Loss during recovery or residual reactants can otherwise imitate catalyst deactivation or apparent reuse. This is a proposed control, not an observation reported for the catalysed peroxide comparison.

Consider an original geometric model of $N$ separate nonporous solid spheres, each with radius $r$. Hold their total volume $V_s$ fixed. The sphere formulas giveAt fixed mass density, fixed $V_s$ also means fixed total mass. Halving $r$ requires eight times as many particles and doubles their total external surface area $S$; this follows by eliminating $N$, without any assumption about chemical kinetics.

For example, take one sphere of radius $1.00\mathrm{mm}$, so $V_s=4\pi /3\mathrm{m}{\mathrm{m}}^3$. Eight spheres of radius $0.50\mathrm{mm}$ have the same $V_s$. Their total surface areas are respectively $4\pi \mathrm{m}{\mathrm{m}}^2$ and $8\pi \mathrm{m}{\mathrm{m}}^2$, approximately $12.57$ and $25.13\mathrm{m}{\mathrm{m}}^2$. Sixty-four particles of radius $0.25\mathrm{mm}$ would instead give $16\pi \mathrm{m}{\mathrm{m}}^2$, four times the original area.

To obtain a kinetic prediction, add assumptions explicitly. Let $f$ be the accessible fraction of external surface area, $\sigma$ the amount of substance of catalytic surface sites per unit accessible area, and $\nu$ the mean number of completed conversions per site per unit time under the chosen conditions. Then the model's reaction product formation rate isHere $\nu$ has units ${\mathrm{s}}^{-1}$ and $\sigma$ has units $\mathrm{mol}{\mathrm{m}}^{-2}$, giving $\mathrm{mol}{\mathrm{s}}^{-1}$ overall. Halving $r$ doubles this predicted rate only if $f$, $\sigma$ and $\nu$ remain unchanged. Particle contact can hide area; pores can add area; diffusion, altered surface structure and catalyst deactivation can change the other factors. Consequently, twice the calculated external area is not a universal promise of twice the measured reaction rate. The geometric argument changes contact, not the net Gibbs free energy or the equilibrium constant of the same overall chemical reaction.

If $C$ is the cell number density, $\mathbf{u}$ the fluid velocity, $V_s⟨\mathbf{p}⟩$ the mean swimming velocity, and $D$ an isotropic translational diffusion coefficient, conservation is ${\partial }_{t}C+\nabla \cdot [C(\mathbf{u}+V_s⟨\mathbf{p}⟩)-D\nabla C]=0$. Impermeable boundaries have zero normal total cell flux. For constant cell volume $v$, the volume fraction is $vC$ and obeys the same equation, but its buoyancy coefficient has no additional factor $v$.

The cell membrane, also called the plasma membrane, is the selective boundary separating a biological cell's cytoplasm from its surroundings. Both prokaryotes and eukaryotes have a cell membrane. Where a cell wall is present, it is a distinct structure outside the cell membrane; the two names are not synonyms. This separation lets a biological cell maintain internal conditions while exchanging material through membrane transport. See NHGRI: cell membrane.

The familiar architecture is a lipid bilayer containing membrane proteins. In a phospholipid-based lipid bilayer, two membrane leaflets face opposite ways: hydrophilic heads face the water on each side, while hydrophobic tails face one another in the interior. The heads are therefore exposed both outside and inside the biological cell. This arrangement combines a water-facing surface with a barrier to many water-soluble solutes; the schematic represents organization rather than an atomistic structure. See Alberts et al., Molecular Biology of the Cell: the lipid bilayer.

Selective permeability means that different substances cross at different rates under comparable conditions. Small nonpolar molecules can pass through a lipid bilayer by diffusion, while ions encounter a strong barrier without a suitable route. An ion channel supplies a selective water-filled pathway when open; a carrier protein binds a solute and changes shape to expose it to the other side. These membrane proteins therefore add specific routes to the barrier. An open ion channel permits passive transport, whereas a carrier protein can mediate either passive transport or active transport. See Alberts et al.: principles of membrane transport.

The useful combination is retention plus controlled exchange: a completely impermeable boundary would prevent uptake, whereas unrestricted mixing would erase differences in concentration and membrane potential. The fluid mosaic model connects these functions to an organized but mobile lipid bilayer containing proteins.

Chaotic advection is complicated, sensitive material motion generated by a smooth velocity field, often even when the Eulerian flow is periodic and simple. Transverse intersections of stable manifolds and unstable manifolds create stretching, folding and lobe transport across former separatrices. Incompressible flow preserves parcel area and advected scalar values in the absence of diffusion, so chaotic motion can create fine filaments without independently producing molecular homogenization.

A localized ionization signal can be collected or induced on adjacent strips through diffusion and the weighting fields of the readout electrodes. Combining their analogue signals estimates the impact position within a strip pitch, improving on binary strip position resolution when noise and calibration permit.

The chemical potential measures how a thermodynamic potential changes when matter is added under specified constraints. It plays for matter exchange the role that temperature plays for heat exchange: a difference supplies a direction, and equality characterizes thermodynamic equilibrium when the corresponding exchange is allowed. This connects diffusion, phase coexistence and chemical equilibrium through one quantity.

For a single component, the particle convention iswhere $U$ is internal energy, $N$ counts particles, $S$ is entropy and $V$ is volume. This partial derivative holds entropy and volume fixed; it measures an incremental change rather than the average energy per particle. See David Tong, Statistical Physics, section 1.4.

In chemistry the molar convention instead uses amount of substance $n$, measured in moles:Here $G$ is Gibbs free energy, $T$ is temperature, $p$ is pressure and $N_{\mathrm{A}}$ is the Avogadro constant. The units are $\mathrm{J}$ per particle versus $\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}$. For several chemical species, define each chemical potential by varying only its own particle count or amount of substance, with all other components' counts or amounts fixed. Many texts use $\mu$ for either convention, so the units and the variable being differentiated must identify it.

For a simple compressible system with variable component inventories, the fundamental differentials areThey describe changes between states in thermodynamic equilibrium with only pressure-volume work. If every component inventory is fixed, the sums vanish. The Gibbs free energy form follows by differentiating $G=U-TS+pV$ and using $d{N}_i=N_{\mathrm{A}}d{n}_i$; this is why the two definitions describe the same chemical potential in different units.

A chemical potential can be negative. Holding entropy fixed while adding matter need not increase internal energy; Tong explains this explicitly for a classical ideal gas in Statistical Physics, section 2.2.4. Its absolute sign also depends on the reference energy: replacing $U$ by $U+{ϵ}_{0}N$ shifts ${\mu }_{\mathrm{particle}}$ by ${ϵ}_0$. A common reference shift cancels from differences for the same chemical species, which are what determine exchange. Thus “energy cost” means a signed increment, not an always positive amount that must be supplied.