At a transparent nonmagnetic dielectric interface, the Fresnel reflection coefficients differ for s and p linear polarization. At Brewster incidence the reflected p amplitude vanishes and reflected unpolarized light becomes s polarized. For a horizontal reflecting surface this is horizontal glare. Away from Brewster incidence it is generally partial polarization. Refraction and internal reflection in droplets also select the two components differently, explaining polarization of rainbow light without a biological mechanism.

For $\dot{\Omega }=-K(t){\Omega }^q$, differentiation gives the displayed braking index. Inferring the torque power $q$ from a measured index therefore requires an effectively constant coefficient. Field evolution, changing inclination and moment-of-inertia changes can all contribute to the difference.

The exact three-dimensional oscillator ground energy is $3\hslash \omega /2$ above the potential minimum. Subtracting it sets the condensate energy to zero. The commonly used $g(ϵ)={ϵ}^2/[2(\hslash \omega {)}^3]$ is the leading continuum term; it does not include the finite-level corrections from shifting the energy origin.

With the ground energy set to zero, the ideal-gas chemical potential approaches zero from below at condensation and remains there to leading thermodynamic order. Above onset it is determined by the trap density of states. The shape differs from that for a uniform gas.

At ground-relative chemical potential zero, the leading semiclassical excited population is $N_{\mathrm{ex}}=\zeta (3)(k_{B}T/\hslash \omega {)}^3$. Additional particles occupy the ground state once this capacity is reached. The continuum description requires many excited trap levels to be thermally accessible.

The ideal isotropic-trap transition scale is $T_0=\zeta (3{)}^{-1/3}(\hslash \omega /k_B)N^{1/3}$. The numerical prefactor $\zeta (3{)}^{-1/3}\simeq 0.9405$ follows from the Bose integral ${\int }_0^{\infty }{x}^2/(e^x-1)dx=2\zeta (3)$.

Expand a weakly depleted Bose gas around its macroscopically occupied mode. Replacing a condensate operator by an amplitude of order $\sqrt{N}$ gives quadratic, cubic and quartic fluctuation interaction scales proportional to $N{N}_{\mathrm{ex}}$, $\sqrt{N}{N}_{\mathrm{ex}}^{3/2}$ and $N_{\mathrm{ex}}^2$, with the same interaction-volume factor. Thus the depletion fraction supplies formal amplitude counting. This is not a uniform bound on infrared modes or a substitute for checking the interaction regime. At fixed particle number, expansion of the condensate interaction must precede the final quadratic truncation; the resulting cancellation is the number-conserving Bogoliubov quadratic Hamiltonian.

A normal mode of linearized condensate dynamics with particle and hole components. Quantizing a suitably normalized mode gives a Bogoliubov quasiparticle. The density and phase components of one mode share a frequency; that fact alone does not establish a twofold excitation degeneracy.

With growing total matter mode, constant relative offset, initial ${\Delta }_B=0$, and $x=R/R_r$, the exact ratio is ${\Delta }_B/{\Delta }_D=f_D(x-1)/(f_{D}x+f_B)$. It reduces to $1-x^{-1}$ only in the dark-matter-dominated limit.

In the dark-matter-dominated limit, imposing both ${\Delta }_B(t_r)=0$ and ${\dot{\Delta }}_B(t_r)=0$ requires a relative decaying mode. With $x=R/R_r$, the ratio becomes $1-3x^{-1}+2x^{-3/2}$. It has a different early-time approach from the constant-offset branch and the same late-time common growth.

Scale-factor ratios involve $(1+z)/(1+z_r)$ exactly. Replacing this by $z/z_r$ is accurate only when both redshifts are large; it cannot justify an exact zero-redshift identity.

The atmospheric boundary layer over an ocean surface. A box-model ozone budget can include in-layer production, chemical loss, dry deposition, entrainment from the free troposphere and horizontal Advection. Fixed-depth local steady-state assumptions must be checked against transport and day–night variation.

An equilibrium construction for a supported particle that modifies the Wulff construction using particle-support and support-medium interfacial energies. The support truncates the corresponding free-particle shape.

For a uniform dilute repulsive gas, the periodic complex-field integral has uniform saddle $n_0=\mu /g$ for $\mu >0$. The paired Gaussian fluctuation kernel has determinant $(\hslash {\omega }_n{)}^2+{\epsilon }_k({\epsilon }_k+2g{n}_0)$, giving the Bogoliubov spectrum. In three dimensions the physical low-energy coupling is $g=4\pi {\hslash }^2{a}_s/m$ and the weak-depletion parameter is $n{a}_s^3\ll 1$. Contact coupling beyond mean field needs ultraviolet matching to the scattering length.

One linear Bogoliubov branch $E_k=\hslash c_{s}k$ in three dimensions gives thermal energy ${\pi }^{2}V(k_{B}T{)}^4/(30{\hslash }^3{c}_s^3)$ and the displayed cubic heat capacity. There is one branch, rather than the three polarizations in a simple crystal's Debye model. The regime requires $\hslash c_s/L\ll k_{B}T\ll \mu$ for a box: the lower inequality permits a continuum integral, and the upper one selects phonons. Fixed finite boxes ultimately have discrete-level behaviour.

In a box of size $L$, confinement energy scales as ${\hslash }^2/(m{L}^2)$, transition temperature as that scale times $N^{2/3}$, and mean interaction energy as ${\hslash }^{2}Na/(m{L}^3)$. The displayed window therefore makes interaction energy large compared with confinement but small compared with $k_B{T}_c$. It simultaneously implies dilute-gas parameter $n{a}^3\ll 1$. Ground-state interaction dominance does not require a large relative shift of the condensation temperature.

Expand an interacting Bose field about a macroscopically occupied condensate and retain terms quadratic in its fluctuations. A Bogoliubov transformation diagonalizes the stable quadratic Hamiltonian. For contact repulsion this gives the Bogoliubov quasiparticle dispersion $E_k=\sqrt{{ϵ}_k({ϵ}_k+2g_B{n}_0)}$, a linear sound mode at small momentum and particle-like behavior at large momentum. Mixing creation and annihilation operators also produces quantum depletion of a Bose condensate. The approximation requires controlled fluctuations; it does not give an exact strongly interacting liquid or the full fluctuation-dominated critical region.

For a three-dimensional repulsive gas at zero temperature, leading quantum depletion is $(n-n_0)/n=8\sqrt{n{a}_s^3}/(3\sqrt{\pi })$. Small gas parameter controls the expansion; this expression does not apply unchanged in lower dimensions.

The relative phase accumulated by orthogonal eigenpolarizations is $\Delta =2\pi L(n_2-n_1)/{\lambda }_0$. A quarter-wave plate has relative phase $\pi /2$ modulo $2\pi$, while a half-wave plate has $\pi$ modulo $2\pi$.

Eliminate cloud radius using its mean density and the critical radius $r_c=4GM/(15c_s^2)$. The displayed threshold scales as $T^{3/2}{\bar{\rho }}^{-1/2}$ at fixed mean particle mass. Equivalently, the maximum confining pressure is $10125c_s^8/(1024\pi G^3{M}^2)$. The numerical coefficient depends on the assumed density profile and on whether the density refers to the mean or the boundary.