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An initially resting, along-ridge-uniform shallow water layer with height disturbance ${\zeta }_0(x)$ conserves $q-f\zeta /h=-f{\zeta }_0/h$. Its balanced remainder satisfies the displayed modified Helmholtz equation, with $R_D=\sqrt{gh}/f$. On the infinite line,For each Fourier transform component, $\hat{\zeta }(t)={\hat{\zeta }}_s+({\hat{\zeta }}_0-{\hat{\zeta }}_s)\cos (\sqrt{f^2+gh{k}^2}t)$. The oscillatory part is inertia-gravity waves. For a localized Gaussian function on the infinite line those waves spread away, leaving the steady component locally; a reflecting finite basin generally retains oscillations.

For box-normalized plane waves in volume $\mathcal{V}$, a potential has matrix element $\tilde{W}({\mathbf{k}}_f-{\mathbf{k}}_i)/\mathcal{V}$. The final-state density per energy and solid angle is $\mathcal{V}m{k}_f/[(2\pi {)}^3{\hslash }^2]$. Dividing the golden-rule rate by incident flux $\hslash k_i/(m\mathcal{V})$ gives $(m^2/4{\pi }^2{\hslash }^4)(k_f/k_i)|\tilde{W}{|}^2$. Elastic scattering has $k_f=k_i$; the arbitrary box volume cancels.

A historical unit of mass, equal to approximately 64.8 milligrams. The unit must not be confused with a minim, which measures volume.

At fixed temperature, volume, and chemical potential, the grand canonical ensemble permits both energy and particle number to fluctuate and weights a state by $e^{-\beta (E-\mu N)}$.

Differentiating the grand partition function at fixed temperature and volume gives $⟨N⟩=-{\partial }_{\mu }\Phi$ and $\mathrm{Var}N=-k_{B}T{\partial }_{\mu }^2\Phi =k_{B}T{\partial }_{\mu }⟨N⟩$. It follows because derivatives of the logarithm of the state sum generate cumulants of particle number. Nonnegative variance makes the grand potential concave in chemical potential in a stable grand-canonical description.

If an axisymmetric thin-film equation is deliberately closed using only the meridional curvature $\kappa =-h^{\prime \prime }$, its regular equilibrium obeys $h^{\prime \prime \prime }=h^{\prime }/l^2$, with capillary length $l=\sqrt{\gamma /(\rho g)}$. The boundary conditions $h^{\prime }(0)=0$, $h(R)=0$, $h^{\prime }(R)=-\tan {\theta }_c$ give the displayed profile and $\alpha =\tan {\theta }_c/\sinh (R/l)$. Its volume is $2\pi \alpha l^3[s^2\cosh s/2-s\sinh s+\cosh s-1]$, $s=R/l$. This model omits the azimuthal contribution of axisymmetric free-surface curvature. Retaining that term instead gives $h=l\tan {\theta }_c[I_0(R/l)-I_0(r/l)]/I_1(R/l)$ in terms of modified Bessel functions.

For identical touching atoms represented by hard spheres of radius $r$, a face-centered cubic lattice has conventional side $a_F=2\sqrt{2}r$ and four atoms per conventional cell. A body-centered cubic lattice has side $a_B=4r/\sqrt{3}$ and two atoms per conventional cell. Their primitive unit cell volumes are respectivelyEach primitive cell contains one atom in the monatomic model, so its mass density is atomic mass divided by $\Omega$. These are contact-geometry estimates; a fixed hard-sphere radius does not encode thermal expansion or phase-dependent electronic bonding.

A hard-sphere gas is a classical model of identical impenetrable spheres, each of diameter $\sigma$ and radius $\sigma /2$. Its pair potential is $u(r)=+\infty$ for centre separation $r<\sigma$ and $u(r)=0$ for $r\ge \sigma$; the convention at exact contact has no effect on the integral. It isolates the effect of particle size without an attractive potential energy.

The upper-left panel is a cross-section of a three-dimensional forbidden region: the second centre cannot enter a sphere of radius $\sigma$ around the first centre. The particles themselves have radius $\sigma /2$. Consequently, with particle volume $v_{\mathrm{s}}$ and pair excluded volume $v_{\mathrm{ex}}$,The last equality follows directly from the Mayer f-function: $f=-1$ throughout the forbidden volume and zero outside it. Confusing particle radius with contact separation changes the answer by a factor of eight; omitting the unordered-pair factor changes it by a further factor of two.

Define the packing fraction $\eta =n{v}_{\mathrm{s}}$, where $n$ is number density. Then the virial expansion beginsFor example, at the toy value $\eta =0.01$, the leading correction predicts $Z\approx 1.04$; the omitted terms still matter if accuracy beyond this approximation is needed. The classical second virial coefficient is independent of temperature because the geometry of forbidden configurations does not change when the gas is warmed. The pressure itself still scales with temperature at fixed number density.

This model makes the positive correction particularly clear: eliminating overlapping configurations changes the dependence of the available configuration space on volume, even though the allowed configurations all have zero interaction energy. The figure's other panels add a toy attractive shell and show how its Boltzmann factor competes with that geometry. The hard-sphere gas is a useful reference model, while a low-order virial expansion remains a dilute-gas approximation rather than a dense sphere packing formula.

For a fixed-composition simple compressible system, a Maxwell relation gives $C_p-C_V=T(\partial p/\partial T{)}_V(\partial V/\partial T{)}_p$. At fixed volume, the differential equation of state gives $(\partial p/\partial T{)}_V={\alpha }_V/{\kappa }_T$, proving the displayed result. Positive absolute temperature and positive isothermal compressibility imply a nonnegative difference. It vanishes when thermal expansion does, so strict inequality requires a nonzero expansion coefficient. This stability argument includes nonideal internal-energy changes, unlike a purely ideal-gas expansion-work explanation.

Thermal energy generated per unit time, often specified per volume as $q^{\prime \prime \prime }$ in $\mathrm{W}{\mathrm{m}}^{-3}$. Integrating a volumetric source over a region gives its total thermal power.

Transfer of thermal energy between regions. In a constant-property thermal Boussinesq approximation, the advective heat flux per unit area is ${\rho }_0{c}_{p}T\mathbf{u}$ and transport across a section is its normal component integrated over that section. With zero net volume transport, adding a constant reference temperature does not change that integral. Otherwise the reference and the return-flow budget must be specified. Atmospheric energy transport also includes latent heat and gravitational potential energy.

A homogeneous deformation has the same deformation gradient at every material point. On a connected reference domain it is an affine map $y(X)=FX+b$ with constant matrix $F$ and translation $b$. Orientation preservation requires $\det F>0$. Its stretches and local volume ratio are spatially constant. A homogeneous deformation describes the kinematics; a homogeneous hyperelastic material instead describes spatial independence of a constitutive law.

Ice embryos form without a foreign nucleating surface once thermal fluctuations can overcome the free-energy barrier on the relevant timescale. An apparent freezing threshold depends on droplet volume and measurement duration.

Two convex bodies are homothetic convex bodies if one is a translation of a positive uniform dilation of the other. Their lengths scale by $s$ and their volumes by $s^n$. Closed disks of positive radii are homothetic convex bodies; axis-aligned positive-sided rectangles are homothetic convex bodies exactly when their aspect ratios agree. The Brunn-Minkowski theorem characterizes this relation by equality of root-volumes in Minkowski addition.

If the typical wall separation grows as cosmic time, surface energy per volume is of order wall tension divided by time. This decreases more slowly than the usual radiation-era total density and can lead to a wall-domination problem.

A composite containing two or more reinforcing fibre types. In the ideal longitudinal model its axial modulus is the volume-fraction-weighted sum of constituent moduli.

The hydraulic radius is cross-sectional flow area divided by wetted perimeter. A perimeter-averaged boundary shear stress $\tau$ exerts drag per unit fluid volume $\tau \ell /A=\tau /R$. It differs from both actual depth and the radius of a noncircular channel.

Kinetic helicity measures the alignment and linking of a velocity field and its vorticity. It is the volume integral of $\mathbf{u}\cdot \mathbf{\omega }$, with boundary conditions needed for its conservation.

The integral measures linkage and twisting of vortex lines through the velocity and vorticity fields.

An incompressible point source of volume flux $Q$ at height $a$ above an impermeable plane has velocity potential $\varphi =-Q(1/r_{+}+1/r_{-})/(4\pi )$, with an equal image source below the plane. On the wall, $u_s=Qs/[2\pi (s^2+a^2{)}^{3/2}]$. The Bernoulli equation gives $p-p_{\infty }=-\rho Q^2{s}^2/[8{\pi }^2(s^2+a^2{)}^3]$. Integrating the pressure deficit over the plane gives the displayed attraction force relative to ambient pressure. If source strength instead denotes the coefficient $Q/(4\pi )$ of radial inverse-square velocity, the force is $\pi \rho m^2/a^2$.