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For one spherical particle per representative volume $L^3$, a volume fraction $f$ implies $d=L(6f/\pi {)}^{1/3}$. A mean nearest-neighbor distance in a random population does not determine this number density exactly, and an atomic thermodynamic phase fraction requires a molar-volume conversion before it is used as $f$.

The Press-Schechter differential abundance is

For a Gaussian measured stellar parallax, the likelihood at distance $r$ approaches a nonzero constant as $r\to \infty$. An unbounded spatial-volume prior proportional to $r^2$ therefore gives an improper posterior. A finite survey/population model or a decaying density is needed. The illustrative proper prior displayed has normalizing constant $1/(2L^3)$; its scale is a model choice requiring justification, not a new measured datum. prior sensitivity is especially important at low parallax signal-to-noise.

Above a chemical quench level, rapid eddy mixing can keep a species' number mixing ratio $f_A$ approximately fixed at its deeper value, provided condensation, photochemical loss, and molecular diffusion remain negligible. The species' number density is then $n_A=f_{A}p/(k_{B}T)$; constant mixing ratio does not imply constant number density. In an isothermal atmosphere, $n_A$ decreases exponentially with altitude on the atmospheric scale height.

In a homogeneous isotropic one-component fluid, $g(r)$ is the conditional number density at separation $r$ from a tagged particle divided by the bulk number density $n$. Thus the displayed shell contains the mean number of neighbours. For an uncorrelated ideal gas $g(r)=1$ away from the tagged particle; repulsive cores suppress small separations, while liquid coordination shells can produce damped oscillations approaching one at large distance. This is a pair correlation, not a normalized one-particle radial probability density. The static structure factor measures its Fourier-space counterpart.

For a uniform solid sphere of radius $R$, the squared radius of gyration about its centre is the mass-weighted mean of $r^2$:Thus its volume is $(4\pi /3)(5/3{)}^{3/2}{R}_g^3$, about $9.01R_g^3$. A nonuniform object or a different shape need not obey this conversion. This distinction matters when estimating molecular number density from scattering measurements of a radius of gyration.

In a dilute nonmagnetic medium of isotropic independent polarizable particles, $n-1\simeq N\alpha /(2{ϵ}_0)$ and the Rayleigh scattering cross-section is $8{\pi }^3{\alpha }^2/(3{ϵ}_0^2{\lambda }^4)$. Eliminating polarizability and setting direct-beam attenuation coefficient to $N\sigma$ gives the title relation. It describes scattering out of a narrow beam, not absorption of all radiant energy. It assumes fixed number density, negligible correlations/local-field corrections and a scalar polarizability; molecular anisotropy, aerosols and stratification require additional information.

Weak equal-mass small-angle gravitational encounters give a single-star transverse kick $2Gm/(bu)$ and a relative-velocity kick twice as large. With encounter rate $2\pi bdbnu$, the single-star squared-velocity diffusion rate is the displayed $D_v$, while the relative rate is $4D_v$. A chosen reference squared velocity divided by either rate defines a relaxation estimate; specifying which rate and reference are used is essential to a numerical prefactor. If $u=\sqrt{2}\sigma$, the convention $3{\sigma }^2/(4D_v)$ gives $3{\sigma }^3/(16\pi \sqrt{2}n{G}^2{m}^2\ln \Lambda )$. The single-star convention $3{\sigma }^2/D_v$ is four times larger. Velocity-distribution averaging further changes numerical coefficients, while the scaling and Coulomb logarithm in stellar dynamics persist. The factor $n$ must be number density, not a dimensionless population count.

For reservoir kinetics ${\mathcal{R}}_t=-({\gamma }_R+R_{R}n)\mathcal{R}+P$, adiabatic elimination gives $\mathcal{R}=P/({\gamma }_R+R_{R}n)$. Expanding for $R_{R}n/{\gamma }_R\ll 1$ and inserting into the condensate equation gives a complex Ginzburg–Landau equation withFast reservoir response and weak depletion are separate assumptions. Without the number density expansion the gain is saturable, not cubic. The coefficients depend on position when the pump does.

In a dilute thermal helium gas with rapid formation–decay balance of beryllium-8, write its small equilibrium abundance schematically as $n_8\propto n_{\alpha }^2{T}^{-3/2}\exp [-{\Delta }_8/(k_{B}T)]$. A narrow isolated Hoyle state gives a second-step coefficient proportional to $T^{-3/2}\exp [-{\Delta }_C/(k_{B}T)]$. Multiplication yieldswith the stabilizing decay strength absorbed into the coefficient. Here $k_B$ is the Boltzmann constant. The two positive offsets count energy relative to their respective entrance thresholds, not relative to the carbon-12 ground state. A resonance enhances capture at the relevant energy; the Boltzmann factor still selects how many encounters reach it. This is a sequential narrow-level approximation with fixed nuclear widths, excluding plasma screening and changes of composition or mass density.

William Alfred Fowler lists the Hoyle state $7.654\mathrm{MeV}$ above the carbon-12 ground state, $0.2875\mathrm{MeV}$ above beryllium-8 plus an alpha particle, and $0.3794\mathrm{MeV}$ above three alpha particles.^[ref] Subtraction gives ${\Delta }_8=0.0919\mathrm{MeV}$ for the diagram.

Normalize the calculated rate at $T_0=10^8\mathrm{K}$ and $E_r=0.3794\mathrm{MeV}$. Holding alpha particle number density and the decay strength fixed gives the original sensitivity calculationAt $T_0$, a deliberately chosen shift $\delta E=+100\mathrm{keV}$ multiplies the rate by $9.12\times 10^{-6}$; $-100\mathrm{keV}$ gives $1.10\times 10^5$. These offsets are hypothetical perturbations, not alternative measured nuclear levels. The reference curve has local temperature exponent of a thermonuclear reaction $-3+E_r/(k_B{T}_0)=41.03$: raising temperature by $10\%$ gives a factor $41.12$ in this fixed-environment calculation. A real star changes its stellar structure and composition as it burns, so these factors do not directly predict final carbon abundance or a constraint on life.

In a flexible crosslinked polymer network, deformation restricts the configurations of chain segments between cross-links. The resulting decrease of entropy produces a restoring entropic force. For an ideal affine Gaussian network, the shear modulus is $G={\nu }_s{k}_{B}T$, where ${\nu }_s$ is the number density of elastically active strands. A nearly incompressible isotropic rubber has Young's modulus $E\simeq 3G$. At fixed network density within the rubbery regime, both increase approximately linearly with absolute temperature. This statement does not describe crossing the glass transition or chemical degradation.

For charge-neutral pure hydrogen in thermal equilibrium, write the Saha equation as $n_e{n}_p/n_{\mathrm{HI}}=S(T)$. A measured neutral number density fixes $n_e=n_p=\sqrt{S(T)n_{\mathrm{HI}}}$, so the neutral fraction is $f_{\mathrm{HI}}=n_{\mathrm{HI}}/(n_p+n_{\mathrm{HI}})=(1+\sqrt{S/n_{\mathrm{HI}}}{)}^{-1}$. At high ionization it scales as $\sqrt{n_{\mathrm{HI}}/S}$, rather than $n_{\mathrm{HI}}/S$. The latter mistake confuses measured neutral number density with total number density. Photoionization equilibrium with an external nonthermal field generally requires a rate equation instead of the Saha equation.

The second virial coefficient is the leading interaction correction to the ideal gas law: at fixed temperature, $B_2(T)={\lim }_{n\to 0}(Z-1)/n$, where $Z=p/(n{k}_{B}T)$ and $n$ is number density. For identical classical particles in a homogeneous three-dimensional gas, with an isotropic pair potential $u(r)$ and $r$ the centre-to-centre separation,The function $f$ is the Mayer f-function. The factor $4\pi r^{2}dr$ counts the volume of a thin spherical shell; the factor $1/2$ counts each unordered pair once. The result has units of volume. Repulsive regions, $u>0$, contribute positively, while attractive regions, $u<0$, contribute negatively. ^[ref]

This is a striking bridge from a microscopic pair potential to a macroscopic pressure reading. At sufficiently small number density, $B_2>0$ raises pressure relative to an ideal gas at the same temperature and number density, while $B_2<0$ lowers it. A single value of $B_2$, however, cannot determine the entire pair potential: many different functions have the same weighted integral. Measurements over several temperatures supply more information.

The integral must exist. For example, a bounded short-range attraction outside a hard core has no difficulty. If instead $u(r)\sim A/r^s$ as $r\to \infty$, then $1-e^{-u/(k_{B}T)}\sim u/(k_{B}T)$ and the radial tail behaves like $\int r^{2-s}dr$: it converges only for $s>3$. A bare unscreened Coulomb potential therefore falls outside this formula's ordinary short-range setting. This condition on the tail does not by itself establish the stability or convergence of the full virial expansion.

For an exactly integrable toy attraction, choose the following square-well pair potential, with diameter $\sigma >0$, attraction depth $ϵ>0$, and range multiplier $a>1$:Set $T^{\ast }=k_{B}T/ϵ$ and $b_{\mathrm{HS}}=2\pi {\sigma }^3/3$, the second virial coefficient of a hard-sphere gas. Integrating the core and attractive shell separately gives the exact toy resultThe core contributes $b_{\mathrm{HS}}$; the shell contributes $-b_{\mathrm{HS}}(a^3-1)(e^{1/T^{\ast }}-1)$. Heating weakens the relative Boltzmann factor enhancement of attractive configurations, so $B_2$ approaches the positive hard-core value as $T^{\ast }\to \infty$. For the illustrative choice $a=3/2$, the zero occurs atAt this temperature, $Z=1+B_3{n}^2+\cdots$: cancellation removes the linear correction, not every correction. At low $T^{\ast }$, the negative shell contribution becomes large, requiring a correspondingly smaller number density for a useful truncated virial expansion. The square well is a deliberately chosen model, not a fit to the carbon dioxide experiment.

Take identical rigid spheres of radius $a$, particle density ${\rho }_p$ and displaced liquid density ${\rho }_f$, at uniform absolute temperature $T$. Let $z$ increase upwards from a reference layer and write $\Delta \rho ={\rho }_p-{\rho }_f>0$. The buoyancy correction gives the effective gravitational massThe actual grain mass is ${\rho }_{p}V$; the displaced liquid subtracts ${\rho }_{f}V$ from the effective gravitational mass. In a dilute, noninteracting particle suspension, the Boltzmann distribution therefore givesHere $n$ is number density and $k_B$ is the Boltzmann constant. Equivalently, the dilute osmotic pressure $\Pi =n{k}_{B}T$ supports the grains above it: $d\Pi /dz=-n{m}_{b}g$. Dividing by $n$ and integrating yields the same exponential function. No dynamic viscosity appears in this equilibrium argument. If ${\rho }_p<{\rho }_f$, the population instead increases upwards; at equal densities this gravitational model predicts uniform $n$.

A microscope counts a finite optical slab, not an infinitesimal plane. For equal viewed area $A$ and slab thickness $\delta$, an ideal uniform detection window centred at $z$ givesThus equal optical volumes preserve the exponential function count ratio exactly, even for finite $\delta$, provided the windows lie wholly inside the accessible chamber and have the same detection response. Unequal depth response or overlapping a boundary can corrupt this inference. Repeated observations estimate the mean count; closely spaced visits need not be independent because grains remain in view.

For an original chosen model, set $a=0.250\mu \mathrm{m}$, ${\rho }_p=1200\mathrm{kg}{\mathrm{m}}^{-3}$, ${\rho }_f=1000\mathrm{kg}{\mathrm{m}}^{-3}$, $T=298\mathrm{K}$ and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$. These are illustrative inputs, not Jean Baptiste Perrin's measurements. Using $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$ givesIf equal observation effort gives expected count $400$ in the reference layer, the expected counts $20$, $40$ and $60\mu \mathrm{m}$ higher are approximately $214.3$, $114.8$ and $61.49$. Fractions describe expected values over observations, not fractional grains. Using the actual grain mass instead of its buoyancy-corrected value would make the inferred Avogadro constant six times too small for these chosen densities.

A semiconductor has an electronic band gap and a conductivity strongly affected by temperature, impurities and carrier injection. Optical absorption can create mobile electrons and holes, enabling a photodiode.

The useful feature of a semiconductor is that its charge carrier population can be controlled. Semiconductor doping changes the supply of mobile electrons or electron holes; carrier injection changes their population away from thermal equilibrium. A transistor turns that control into an electronic amplifier: a small input signal controls a larger flow of supplied energy.

For a uniform semiconductor in a weak electric field, a simple drift model writes its electrical conductivity asHere $q>0$ is the elementary charge, $n,p$ are the number densities of mobile electrons and electron holes, and ${\mu }_n,{\mu }_p$ their positive charge carrier mobilities. The two contributions add because opposite electric charges drift in opposite directions and therefore carry electric current in the same direction. This model assumes fixed temperature and mobilities and omits diffusion from density gradients.

An original dimensionless example isolates the effect of carrier injection. Suppose initially $p\ll n=n_0$ and ${\mu }_n=2{\mu }_p$. Adding equal excess populations $\Delta n=\Delta p=0.1n_0$ gives $\Delta \sigma /{\sigma }_0=0.15$, even though the added electron holes remain a minority carrier population. A semiconductor can respond strongly to a small injected population without changing its underlying band gap.

In a stable-particle approximation with negligible later annihilation, comoving particle number is conserved and physical number density falls as $a^{-3}$. Replacing scale-factor ratios by photon-temperature ratios additionally assumes no change in the entropy degrees of freedom.

For an isothermal thin atmosphere with number density $n(z)=n_0{e}^{-z/H}$ and extinction cross-section ${\sigma }_{\lambda }$, a tangent ray near radius $R_p+z$ has height $z+{\ell }^2/(2R_p)$ at distance $\ell$ along the ray. The resulting Gaussian integral gives slant optical depthThe effective transit height is near ${\tau }_{\lambda }\sim 1$, so $z(\lambda )=H\log {\sigma }_{\lambda }+\mathrm{constant}$ if composition is constant. This explains the scattering slope of a transmission spectrum.

For an absorbing spherical target of radius $a$ in an infinite uniform reservoir, steady diffusion gives $c(r)=c_{\infty }(1-a/r)$. Integrating the inward flux yields $k=4\pi D_{\mathrm{rel}}a$, where $D_{\mathrm{rel}}=D_A+D_B$ if both partners diffuse. Here $k$ multiplies number densities; its units are volume per time. Molecular orientation, finite-domain transients and imperfect capture require corrections.

For proper number density $n(t)$ in an FLRW metric $d{s}^2=d{t}^2-a^2[d{r}^2/(1-k{r}^2)+r^{2}d{\Omega }^2]$, a radial shell has proper volume $4\pi a^3{r}^2|dr|/\sqrt{1-k{r}^2}$. A radial null geodesic obeys $|dr|=\sqrt{1-k{r}^2}dt/a$. Multiplication gives the displayed shell count on a local radial branch of the past light cone. Spatially uniform number density need not be time independent or have conserved comoving number density.