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A chemical synapse transmits a signal by releasing neurotransmitter from a presynaptic neuron to act on postsynaptic receptors across a synaptic cleft. The two cell membranes remain separate. A typical fast transmission sequence is:

See Chemical Synapses and 3.1 Synapses. The fast pathway uses synaptic vesicles prepared near release sites; it does not require a distant synaptic vesicle to travel across the terminal after every action potential. See Basic Neurochemistry: calcium entry and vesicles at release sites. Some chemical synapses release neurotransmitter in response to graded presynaptic membrane potential changes rather than action potentials.

Neurotransmitter clearance limits how long released neurotransmitter remains available. Depending on the neurotransmitter and location, diffusion away, uptake through membrane transport proteins into the sending or neighboring biological cells, and breakdown by enzymes contribute. Receptor binding and unbinding also affect response duration. Recovering synaptic vesicle membrane is a separate recycling task; uptake of released neurotransmitter is not the return of an intact synaptic vesicle. See General Principles of Cell Communication.

This electrical-to-chemical-to-electrical sequence offers several places to adjust a signal: synaptic release probability, receptor availability and neurotransmitter clearance can change its strength or duration. It helps explain why synaptic plasticity can modify communication between neurons, and why an arriving action potential need not cause a postsynaptic action potential.

Otto Loewi's 1936 Nobel lecture describes his 1921 experiment: fluid from a frog heart whose nerves had been stimulated produced the corresponding response when transferred to another heart. The transferable substance established chemical mediation in that preparation. This was evidence for released chemical signaling, not yet an identification of synaptic vesicle fusion and not a claim that every synapse is chemical. See Loewi, The Chemical Transmission of Nerve Action, Nobel lecture, 12 December 1936.

A cellular projection whose structure and functions differ from the bacterial flagellar apparatus. Motile cilia can move fluid; a transport estimate must compare advection with diffusion.

A cloud chamber contains supersaturated vapor. Ionization along a charged particle's path supplies sites for nucleation; the resulting liquid drops scatter light and reveal the trail. The observed objects are drops, not magnified particles. Its beauty is that a spatially distributed microscopic interaction becomes a visible geometrical record.

Two ways of making supersaturated vapor must be distinguished. An expansion cloud chamber rapidly increases the volume of moist gas, producing a temporary sensitive interval through cooling by adiabatic expansion. A diffusion cloud chamber maintains a cold base and a warmer vapor supply, producing a continuously replenished sensitive layer through diffusion and a temperature gradient.

Cytosol dialysis is exchange between the cytosol and pipette solution through the opening used in conventional whole-cell recording. Diffusion can change ion concentrations and dilute soluble regulators of ion channels. A diminishing response can therefore reflect the recording's alteration of the biological cell, rather than the experimental treatment.

A direct motivating example is the 1988 article Muscarinic activation of ionic currents measured by a new whole-cell recording method. In rat lacrimal gland biological cells, the acetylcholine response in conventional whole-cell recording lasted about five minutes. With a pore-forming substance in the patch pipette instead of open break-in, the response remained undiminished after one hour. The readout was activation of calcium-activated potassium and ${\mathrm{Cl}}^{-}$ ion electric currents; preserving the response supported a role for retained cytoplasmic components. The authors also tested the substance extracellularly and observed reversible increases in cation electrical conductance. This comparison motivates perforated patch recording, but does not show that every intracellular constituent or every ion channel response is preserved. ^[ref]

Practical controls are matched pipette and bath compositions, recording elapsed time after break-in, time-matched untreated preparations, and tracking both patch-clamp series resistance and baseline leakage. Comparing conventional whole-cell recording with perforated patch recording can separate loss of soluble components from an intended treatment effect, but only if electrical access is characterized in both. These are controls suggested by the mechanisms above, not a new set of measured results.

Without rotational diffusion, $x=p_z$ follows $\dot{x}=(1-x^2)/B$, hence $x=(x_0+\tanh (t/B))/(1+x_0\tanh (t/B))$. Pushing forward the initial uniform orientation distribution gives $f=\alpha /[\pi (1+\alpha -(1-\alpha )x{)}^2]$, $\alpha =e^{-2t/B}$, per unit solid angle. Integrating $2\pi xf$ yields the displayed mean, interpreted continuously as zero at $t=0$. Its late deficit is $(4t/B-2)e^{-2t/B}+O((t/B)e^{-4t/B})$. At each finite time the density is smooth and normalized; the infinitely late distribution is a point mass at the upward pole.

Dialysis exchanges small solutes between a sample and an external bath through a semipermeable membrane, while retaining suitably large macromolecules, such as proteins. The transport is passive diffusion: random motion carries permeant molecules both ways, with net transport down their concentration gradient. The bath is the dialysate. At equal permeant solute concentrations, microscopic exchange continues but net exchange stops.^[ref]

For example, a protein sample containing excess sodium chloride can lose the small ions to a low-salt dialysate without deliberately transferring the protein. Conversely, components of a new buffer solution can enter from the bath. This is useful before studying enzyme kinetics: changing the surrounding chemical solution can remove an interfering small solute while keeping the enzyme available for the next experiment.

The striking feature is that the retained protein does not have to be captured and released again to change its surroundings. However, retention is conditional on the semipermeable membrane and the particular macromolecule, and dialysis is poorly suited to separating proteins of nearly the same size. The molecular weight cut-off is a nominal retention rating, not a sharp geometrical boundary.

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 spin echo refocuses a nucleus's fixed detuning. A nucleus that moves through a magnetic field gradient can have different detunings during the two intervals. To see the distinction without assuming that all noise is reversible, let $\Delta (t)$ vary during free evolution. With the same pulse convention, its residual phase at $2\tau$ isStatic detuning cancels because $\int y(t)dt=0$. A detuning ${\Delta }_a$ during the first interval and ${\Delta }_b$ during the second leaves $\Phi =\tau ({\Delta }_a-{\Delta }_b)$, which need not vanish. The final component signal is $e^{i\Phi }$ before relaxation; changing the overall receiver convention changes its conjugation but not its magnitude.

For an original quantitative model, take a constant magnetic field gradient $G=d{B}_z/dx$ and unrestricted one-dimensional Brownian motion $x(t)=x_0+\sqrt{2D}{W}_t$, where $D$ is the diffusion coefficient and $W_t$ is standard Brownian motion. Then $\Delta (t)={\Delta }_0+\gamma Gx(t)$. The constant terms cancel, and the Gaussian distribution of the remaining phase follows from the Brownian covariance kernel $\mathbb{E}[W_s{W}_t]=\min (s,t)$. Writing $Q(r)={\int }_r^{2\tau }y(t)dt$ givesThe characteristic function of a zero-mean Gaussian distribution now supplies the extra factor $\mathbb{E}[e^{i\Phi }]=e^{-\mathrm{Var}\Phi /2}$, soHere $\gamma$ is in radians per second per tesla, $G$ in tesla per metre, and $D$ in square metres per second: the extra exponent is dimensionless. The figure chooses $\gamma /(2\pi )=42.58MHz/T$, $G=0.010T/m$ and $D=2.0\times 10^{-9}{\mathrm{m}}^2/s$. With the earlier $\tau$ and $T_2$, diffusion reduces the echo from $0.60653$ to about $0.32930$. These are illustrative model parameters, not values extracted from Hahn's experiment.

This result assumes instantaneous perfect pulses, a constant magnetic field gradient, unrestricted diffusion with constant diffusion coefficient, and independent homogeneous transverse relaxation. Restricted motion, changing gradients, finite pulses and interactions between spins can require different attenuation laws. Even a fixed Hamiltonian is refocused only for the terms whose sign the pulse reverses: a simultaneous 180° rotation of two spins leaves their $J_{z,1}{J}_{z,2}$ product unchanged, because both factors change sign. A recovered echo therefore demonstrates a particular recoverable phase history, rather than reversal of every microscopic process.

The scale $\sqrt{Dt}$ characterizes diffusion over time $t$. For unrestricted isotropic diffusion, the rms displacement in one coordinate is $\sqrt{2Dt}$ and the total three-dimensional rms displacement is $\sqrt{6Dt}$. Stating the convention avoids spurious disagreement over a numerical prefactor.

The continuous-time diffusion model on a finite undirected graph uses the Graph Laplacian in the ordinary differential equationAcross each edge, the higher value sends an amount proportional to the difference to the lower value. The two contributions cancel in the sum over vertices, giving conservation of mass:If the entries represent masses, start with nonnegative entries; signed entries can instead describe deviations from a common level. For the three-vertex path graph, the initial vector $x(0)=(1,0,-1{)}^T$ is an eigenvector with eigenvalue one, so the exact solution isMore generally, an eigenvector with eigenvalue $\lambda$ decays by $e^{-\lambda t}$. The spectral theorem for real symmetric matrices supplies an orthonormal basis of these eigenvectors, and the kernel of a graph Laplacian identifies the surviving terms. On a nonempty connected graph,The limiting value is the original arithmetic mean. On a disconnected graph, each connected component of a graph converges to its own original arithmetic mean. The same matrix therefore measures neighbour disagreement, detects disconnected regions and governs relaxation toward agreement.

A narrow population band undergoing uniform electrophoresis and ordinary diffusion has mean position $x_0+\mu Et$ and variance ${\sigma }_0^2+2Dt$. This follows from the advection-diffusion equation by changing to the coordinate $x-\mu Et$. For two species with mobility difference $\Delta \mu$, their centre separation is $|\Delta \mu |Et$, whereas diffusion widths grow only as $\sqrt{t}$. Starting from point-like bands with common diffusion coefficient $D$, a required separation of $q$ standard deviations needsA prescribed distance between centres alone is a resolution convention; finite loading widths and other dispersion mechanisms must be included when interpreting actual electrophoresis data.

diffusion by hopping of an interstitial particle between interstitial sites without displacing a host lattice particle. Direction sums over neighbour vectors determine the conductivity tensor in a dilute independent-jump model.

A cooperative change of atomic positions and lattice shape without the long-range compositional diffusion required by a diffusional transformation. Tetragonal-to-monoclinic transformation in zirconia is a nonmetallic example.

Damping removes random kinetic energy, but the discrete photon kicks cause diffusion in momentum. Their balance gives a nonzero temperature. Define the momentum diffusion coefficient $D_p$ by the noise contribution $d⟨p_x^2⟩/dt=2D_p$. With atom mass $m$ and linear damping coefficient $\alpha$,at stationary balance, where $k_B$ is the Boltzmann constant and mean momentum is zero.

For ideal three-dimensional optical molasses with three identical independent beam pairs and isotropic spontaneous emission, the $x$-pair absorption rate at rest is $R_x=\Gamma s/(1+4d^2)$. Absorption from that pair contributes $(\hslash k{)}^2{R}_x$ to the noise growth of $⟨p_x^2⟩$. Emission from all three pairs contributes the same amount: its total rate is $3R_x$ and each emitted photon has mean squared $x$-momentum $(\hslash k{)}^2/3$. Hence $D_p=(\hslash k{)}^2{R}_x$. Using the Doppler cooling force coefficient givesPut $a=-d>0$. Then $(1+4d^2)/(-d)=a^{-1}+4a\ge 4$, with equality at $a=1/2$. The ideal weak-field Doppler cooling limit is therefore $T_D=\hslash \Gamma /(2k_B)$ at $\delta =-\Gamma /2$. It is the minimum of this specified diffusion-and-damping model, rather than a universal lower bound for laser cooling. The diffusion approximation also assumes recoil energy much smaller than $\hslash \Gamma$. In a one-pair geometry, emission patterns and transverse recoil change the diffusion bookkeeping. ^[ref].

Diffusion-ordered spectroscopy, separating NMR signals by molecular translational diffusion.

A drift current is an electric current caused by the response of mobile charge carriers to an electric field. It is distinct from the diffusion contribution due to concentration gradients. At equilibrium in a p-n junction, the two contributions cancel although each can be nonzero.

An electrolyte is a conducting medium in which mobile ions carry electric current, according to IUPAC Gold Book: electrolyte. In introductory chemistry, a substance that yields ions when dissolved in water is also called an electrolyte. Both usages point to the same mechanism: an electric field gives mobile cations and anions directed drift velocities.

Mobility matters as much as the presence of electric charge. In a simple ideal model of solid sodium chloride, ions are confined to their sites and do not carry a sustained transport electric current. Molten sodium chloride has mobile ions; dissolved sodium chloride gives mobile ${\mathrm{Na}}^{+}$ and ${\mathrm{Cl}}^{-}$ ions in water. Real solids can have some ionic transport through imperfections; the immobile-solid picture is an introductory limit, rather than a claim that all solids are perfect insulators.

A strong electrolyte yields essentially all ions from the dissolved substance, whereas a weak electrolyte yields only a fraction under the specified conditions. Dissolved sodium chloride is a strong electrolyte. Sucrose is a nonelectrolyte: it dissolves as neutral molecules, so dissolving something does not automatically create mobile ions. Water's own small population of hydronium and hydroxide remains possible. See 11.2 Electrolytes (§11.2: strong, weak, and nonelectrolytes) and Flowers et al., Chemistry 2e, §11.4, Table 11.3: sucrose and sodium chloride.

Acetic acid is a weak electrolyte and a weak acid. In water, some neutral acetic acid molecules transfer a proton to water, producing hydronium and the negatively charged partner:Both sides have net electric charge zero. At chemical equilibrium, neutral acetic acid coexists with these ions; the fraction converted depends on the conditions. Electrolyte strength describes this extent of ion production, while concentration describes how much substance is present per specified quantity of chemical solution. A dilute strong electrolyte can contain fewer ions per volume than a concentrated weak electrolyte. Neither the words strong and weak nor a conductivity comparison alone specifies concentration.

Now specify an electrolytic cell with a positive anode on the left, a negative cathode on the right, and an approximately uniform electric field pointing right in the bulk electrolyte. With a steady external source and neglecting diffusion and convection, positive cations drift right toward the negative cathode; negative anions drift left toward the positive anode. Yet both populations contribute a rightward conventional current! For number densities $n_{+},n_{-}$ and drift velocities ${\mathbf{v}}_{+},{\mathbf{v}}_{-}$, the transport current density isThe product of electric charge and velocity determines conventional current: for the anion, both the charge and its rightward velocity component are negative, making its rightward contribution positive. This is the same sign rule used for electron conventional current. It follows directly from the existing charge carrier formula.

For an independent numerical example, stipulate $n_{+}=n_{-}=10^{25}{\mathrm{m}}^{-3}$, $v_{+,x}=2\times 10^{-4}\mathrm{m}{\mathrm{s}}^{-1}$, and $v_{-,x}=-3\times 10^{-4}\mathrm{m}{\mathrm{s}}^{-1}$. With the exact elementary charge $e=1.602176634\times 10^{-19}\mathrm{C}$, the contributions areEqual number densities of singly charged cations and anions give zero bulk charge density, while their motion gives nonzero current density. This is why a neutral electrolyte can conduct. These numbers are chosen toy inputs, not measurements of sodium chloride; maintaining sustained cell electric current also requires suitable electrode reactions. Blocking electrodes can instead lead to charge accumulation and decay of the electric current.

An electrolyte connects the microscopic motion of ions to observable electrical conductivity. A counterion-only model keeps just the mobile counterions needed to oppose a fixed electric charge; Poisson-Boltzmann equations use ideal Boltzmann distributions to model their equilibrium distributions, rather than the drift example above.

For a point-like object with constant electrophoretic mobility, deterministic traversal is the integral of inverse local drift speed. In a conical frustum of length $L$ at pore voltage $V$, the one-dimensional field is $E(z)=Vab/[Lr(z{)}^2]$. Integrating gives $t=L^2(b/a+1+a/b)/(3\mu V)$ for positive $\mu V$. For two symmetric frusta of total length $2l$ and the same total pore voltage, replace $L$ by $2l$. The inequality $b/a+a/b\ge 2$ shows that this nonuniform field cannot produce a shorter deterministic point-particle transit than a uniform field at fixed total length and voltage. Reservoir forces, diffusion, adsorption and finite-object conformation are additional physics.

If both diffusion coefficients equal $D$, each spatial mode has matrix $J-D{q}^{2}I$. Its eigenvalues are those of $J$ shifted by the negative real number $D{q}^2$. Thus a reaction-stable homogeneous state remains stable under this equal-diffusion linear mechanism.

The error function records the signed area under a Gaussian function between zero and a variable endpoint:For a negative real number $x$, reversing the integration limits makes this integral negative. The normalization uses the Gaussian integral: the full area of $e^{-t^2}$ is $\sqrt{\pi }$, so its positive half contributes one after multiplication by $2/\sqrt{\pi }$. The same definition extends to a complex number $z$ along any path from zero to $z$, giving an entire function.^[ref]

For example, $\mathrm{erf}(0)=0$, $\mathrm{erf}(1)\approx 0.842701$, and $\mathrm{erf}(-1)\approx -0.842701$. A finite positive endpoint captures only part of the positive Gaussian integral; $\mathrm{erf}(x)$ approaches one as $x\to +\infty$ but never reaches it at a finite real $x$.

The useful idea is to name a recurring integral once instead of carrying it through every calculation. The error function and the normal distribution express probabilities and quantiles through this area, while an error-function diffusion front describes how diffusion smooths a sharp jump. The complementary error function retains the small area left in the tail.

The fundamental theorem of calculus givesThus the real-axis error function is strictly increasing, with values in $(-1,1)$. Its integrand is an even function, so the error function is an odd function, $\mathrm{erf}(-x)=-\mathrm{erf}(x)$. The derivative also gives ${\mathrm{erf}}^{\prime \prime }(x)=-4x{e}^{-x^2}/\sqrt{\pi }$: the curve is concave for $x>0$ and convex for $x<0$. These real-axis properties do not imply boundedness throughout the complex plane; the existing quadrant asymptotics of the error function describe a different behavior in other directions.