For isotropic three-dimensional diffusion with coefficient $D$, the vector mean displacement is zero, the radial RMS is $\sqrt{6Dt}$ and the mean radial magnitude is $4\sqrt{Dt/\pi }$. The radial distribution is the Maxwell speed distribution arising from three independent Gaussian coordinate increments. State the statistic before calling a quantity an average distance; it is not the cumulative path length.

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.

For a free one-dimensional particle following Langevin dynamics of mass $m$ and drag coefficient $\zeta$, write $\tau =m/\zeta$ and $D=k_{B}T/\zeta$, where $T$ is the temperature and $k_B$ is the Boltzmann constant. Prepare the initial velocity in thermal equilibrium, independently of future noise. The stationary velocity autocorrelation function is $\mathbb{E}[V_s{V}_u]=(k_{B}T/m)e^{-|s-u|/\tau }$. Twice integrating this covariance gives the mean squared displacementFor $t\ll \tau$ this is $(k_{B}T/m)t^2+O(t^3)$: the particle initially keeps its thermal velocity. For $t\gg \tau$ the leading term is $2Dt$, recovering ordinary diffusion. A particle prepared with zero initial velocity instead has the transient in inertial Brownian displacement with zero initial velocity. The distinction is initial preparation, rather than a change in the long-time diffusion coefficient.^[ref]

For electronic $J=1/2$, an extremal total projection has only one product state rather than a two-dimensional spin block. With dimensionless angular-momentum operators and nuclear Zeeman coefficient $g_I$ in Bohr-magneton units, its energy is $AI/2\pm {\mu }_{B}B(g_J/2+g_{I}I)$. Extending both signs of a two-state Breit-Rabi formula square root to each stretched projection would invent an extra state.

A Branching process with mean offspring number one. In a continuous-time birth–death example with equal per-cell birth and death rates $\beta$, one initial individual has Survival probability $1/(1+\beta t)$ and mean size $1+\beta t$ conditional on survival.

An inclusive branching fraction sums many exclusive final states. A clean reconstruction mode can be experimentally attractive without dominating the total decay rate. For example, the J/psi has useful electron-pair and muon-pair decays, but predominantly decays to hadrons inclusively. Detector acceptance, branching fraction and background rejection jointly determine the usefulness of a channel.^[ref]

A quantum particle above a hard reflecting wall in a linearly increasing potential $V(x)=Fx$ for $x>0$. The Schrödinger equation reduces to the Airy equation. Decay at infinity selects $\mathrm{Ai}$, while the wall requires $\mathrm{Ai}(-z_n)=0$. Thus ${\psi }_n\propto \mathrm{Ai}(x/\ell -z_n)$ and the energies are the displayed values. The $n$th state has $n-1$ interior nodes and a decaying tail beyond its classical turning point. A gravitational realization has $F=mg$; gravitational neutron bound-state transmission tests such confinement experimentally.

A one-dimensional well $V=-V_0$ on $0<x<a$, with an infinite wall at zero and $V=0$ beyond $a$. For $-V_0<E<0$, the interior wavefunction is proportional to $\sin kx$ and the exterior decays as $e^{-qx}$. Continuity gives $k\cot ka=-q$, where $k^2=2m(E+V_0)/{\hslash }^2$ and $q^2=-2mE/{\hslash }^2$. Its half-line finite-well bound-state count has a nonzero minimum depth, unlike a full-line attractive one-dimensional well.

For a nonzero instantaneous homogeneous vector-potential step, scalar-field mixing has $|v_{\mathbf{k}}{|}^2\sim [e{\mathbf{A}}_0\cdot \hat{\mathbf{k}}{]}^2/(4{\hslash }^2{k}^2)$ at large wavevector. Its three-dimensional momentum integral diverges. Per-mode squeezed-state formulas remain meaningful, but at finite quantization volume the mode product needs an ultraviolet regulator or a sufficiently smooth physical pulse. Infinite-volume representations require a separate limit.

For a two-mode quadratic Hamiltonian, a transformation can remove $AB+A^{†}{B}^{†}$ while leaving $A^{†2}$ or $B^{†2}$ terms. Such terms create quanta from the proposed vacuum, so their absence must also be checked before identifying that vacuum with an energy eigenstate. Coordinate coupling $kxy$ contains both pair and number-conserving terms and supplies an explicit example.

For a translation-invariant quadratic bosonic quantum field, anomalous terms couple $\mathbf{k}$ to $-\mathbf{k}$. A full-momentum sum uses half the pair-creation coefficient in each term, because both labels describe the same anomalous pair. After a canonical Bogoliubov transformation, however, the two members carry distinct quasiparticle occupations; each contributes its own oscillator energy. The vacuum correction is $\frac{1}{2}{\sum }_{k\ne 0}(E_k-A_k)$ in a full-zone sum. Switching to one representative from each pair requires changing both the occupation sum and constant consistently.

The thermal sphere-of-influence scale $r_B=GM/c_s^2$ compares gravitational potential per unit mass with the ambient gas sound-speed squared. A convention based on escape speed instead introduces a factor of two. The scale is not a capture cross-section or an accretion rate without a flow and dissipation model.

A single photon thermalized in a macroscopic body adds energy $hc/\lambda$. If its heat capacity is sufficiently large that the temperature change is negligible, the entropy increase is $hc/(\lambda T)$. The Boltzmann entropy then gives the displayed ratio of accessible microstates. The exponential can be enormous even though the deposited energy is macroscopically tiny.

BLOSUM62 is a BLOSUM amino-acid substitution matrix constructed using a 62-percent sequence-identity clustering threshold for the contributing blocks. It is widely used in protein sequence alignment. Its substitution scores form only one part of a complete alignment scheme: gap costs and local versus global objectives also affect the result. The number 62 describes construction of the matrix rather than a required identity of the query and target. ^[ref]

A fit of a thermal accretion-disk spectrum that infers the inner emitting radius and identifies it with the innermost stable circular orbit. A relativistic thin-disk model requires independently constrained mass, distance and inclination, together with spectral hardening and absorption. Use of a suitable disk-dominated state is an essential physical check, not merely a statistical cut.

An inference from radiation reflected by an irradiated accretion disc, including iron-line and continuum features. Doppler effects, gravitational redshift and ray propagation determine its profile. Associating the innermost reflecting radius with the innermost stable circular orbit can constrain spin, conditional on illumination, ionization, abundance, inclination and disk structure.

Connecting segments into chains reduces their independent translational mixing entropy per segment by factors $1/N_A$ and $1/N_B$. Consequently the tolerable positive contact penalty tends to zero as both chains become long. Equal chain lengths have ${\chi }_c=2/N$; favorable unlike interactions can still produce a miscible polymer pair.

A representation $n={\sum }_{i\ge 0}{a}_i{2}^i$ with $a_i\in \{0,1,2\}$ and only finitely many nonzero coefficients. If $h(n)$ counts these representations, then $h(0)=1$, $h(2m+1)=h(m)$, and $h(2m)=h(m)+h(m-1)$ for $m\ge 1$. The integer parity of $n$ determines whether the coefficient of $2^0$ must be 1 or may be 0 or 2.

Bilayer graphene with a relative rotation between the layers. Hybridization of displaced Dirac cones creates angle-dependent bands and optical resonances.

An opioid receptor ligand, also called oliceridine, studied for differences between G protein activation and arrestin recruitment. Apparent signalling preference must be distinguished from low intrinsic efficacy.