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Vigorous convection keeps the interior nearly uniform, so the upper thermal boundary layer carries a temperature drop of order $\theta$. A boundary layer becomes unstable when its local Rayleigh number reaches an order-fixed critical value $R_c$:This is marginal-stability thermal boundary-layer scaling using the Local Rayleigh-number closure: boundary-layer buoyancy overcomes viscous and thermal-diffusive stabilization. Equivalently, a buoyancy-driven velocity scale $u\sim {\rho }_{0}g\alpha \theta {\delta }^2/\eta$ makes $u\delta /\kappa$ order $R_c$. ThusThere is no $d$ in this closure, as required for a thin boundary layer controlled by its own stability. The exponent is fixed by scaling; its absolute coefficient is not. Here $R_c$ represents the geometry and boundary conditions, together with any order-one boundary-layer coefficient. The question supplies no numerical value for it. The assumption requires $\delta \ll d$, positive $\theta$, and properties sufficiently constant over the temperature contrast.

Oceanic magnetic anomalies. Newly formed oceanic crust contains magnetic minerals. As it cools through their magnetic blocking temperatures, it acquires thermoremanent magnetization recording the ambient field polarity. Continuing seafloor spreading carries that record away from the ridge. Geomagnetic reversals therefore generate approximately paired marine magnetic stripes parallel to the ridge, with older crust farther from its axis. A magnetometer measures the superposed field above this magnetized material; subtracting a reference field reveals magnetic anomalies. The anomaly sign and shape depend on magnetization direction, latitude, source depth and geometry, so a positive measured anomaly is not universally identical to normal rock polarity.^[ref]

Matching a sequence of stripes to an independently dated reversal chronology gives plate ages and spreading histories. A stripe of width $w$ formed over interval $\Delta t$ implies half-spreading rate $v=w/\Delta t$ under steady symmetric spreading; full separation speed is $2v$. Unequal widths on opposite flanks can reflect asymmetric spreading or later deformation. Alteration, remagnetization and overlapping magnetic sources complicate inversion. The sequence correlation is stronger evidence than an isolated stripe or a field-strength peak.

Rayleigh-Bénard convection. Consider a horizontal fluid layer of thickness $d$, hotter below by $\Delta T>0$. The conductive state is motionless with a linear temperature gradient. A rising warm parcel is buoyant, but viscosity resists its motion and thermal diffusion erases its temperature anomaly. In the Boussinesq approximation, density variations are retained in buoyancy while the flow is otherwise incompressible. The competing effects are measured by the Rayleigh number and Prandtl number,For stress-free impermeable, fixed-temperature boundaries, linear stationary disturbances with horizontal wavenumber $q/d$ giveMinimizing gives $q_c=\pi /\sqrt{2}$ and the critical Rayleigh number for an isothermal layer, ${\mathrm{Ra}}_c=27{\pi }^4/4\simeq 657.5$. Two rigid no-slip, fixed-temperature boundaries instead give approximately $1708$; the critical value is not independent of boundary conditions.^[ref]

Above onset, buoyancy can sustain overturning rolls and heat transfer exceeds pure conduction, with Nusselt number $\mathrm{Nu}>1$. Nonlinear advection eventually limits the disturbance; larger driving can produce time-dependent or turbulent states. The ideal Boussinesq problem does not require hexagonal cells: planform selection depends on boundaries and non-Boussinesq effects. Convection driven by surface tension at a free surface is a distinct mechanism. An adverse temperature gradient supplies driving, but overcoming diffusion and viscosity requires a finite threshold.

Oceanic thermal subsidence. Newly formed oceanic lithosphere is hot and buoyant. Cooling increases its column density, causing subsidence by isostasy as it moves away from a ridge. The half-space cooling model has profile isWith volumetric thermal expansion coefficient $\alpha$, the excess column mass relative to hot mantle isCompensation with seawater density ${\rho }_w$ givesThus young-to-intermediate-age seafloor generally deepens approximately as the square root of age. At old ages, finite thermal-plate thickness and basal heat supply flatten this trend; indefinite half-space deepening is inappropriate. Sediments, variable crust, volcanic swells and dynamic topography also perturb bathymetry. Ocean-floor depth and heat flow jointly constrain cooling models.^[ref]

Effective elastic thickness. This is the thickness of a homogeneous elastic plate with the observed effective flexural rigidity, not necessarily the crustal or thermal lithospheric thickness:Infer it by modelling deflection around known loads, such as seamounts or trenches, or fitting gravity-topography admittance and gravity-topography coherence across wavelengths. A simple Fourier balance is $(D{k}^4+\Delta \rho g)w_k=q_k$ for a signed transverse load. The transition from buoyancy-dominated to bending-dominated behaviour constrains $D$; then assumed Young's modulus $E$ and Poisson's ratio $\nu$ give $T_e$.

Good inference includes load geometry and density, buried interfaces, sediment and internal loading, boundary conditions and data resolution. These can trade off with rigidity. Plastic yielding, faulting, mechanical layering and creep make $T_e$ an effective, timescale- and stress-dependent quantity. Older, colder lithosphere is often stronger, but age at loading and subsequent relaxation matter; a single universal age-to-$T_e$ conversion is unjustified.

Assume a stable, lossless elastic body with the usual symmetry of the elasticity tensor and boundary conditions that eliminate the surface terms in integration by parts, such as a traction-free outer surface. Its elastic stiffness operator is then self-adjoint. When density varies spatially, the precise eigenproblem is generalized:Here $\alpha$ abbreviates the pair of mode labels. Equivalently, ${\rho }^{-1}\hat{H}$ is a self-adjoint operator in the mass inner product. Stability makes ${\omega }_{\alpha }^2$ real and nonnegative. Modes at different eigenvalues are orthogonal, and an orthonormal basis can be selected within each degenerate eigenspace. Completeness permits expansion of a suitable displacement field in these mass-weighted elastic normal modes.

Insert the expansion and project onto ${\mathbf{u}}_{\alpha }^{\ast }$. Mass normalization diagonalizes both the acceleration term and the restoring term. For a symmetric stress glut tensor supported away from the boundary,The symmetry of the stress glut tensor removes the antisymmetric part of the displacement gradient, leaving the strain. The result is one independent forced oscillator per mode, with the stated excitation integral. If the modes instead have mass norm $N_{\alpha }$, that integral is divided by $N_{\alpha }$. A source touching the boundary requires the corresponding surface forcing to be retained rather than silently dropping it.

Rigid translations and rotations can be zero-frequency modes of a free body. Their strain vanishes, so an internal symmetric seismic moment tensor does not excite them through this coupling. Attenuation, radiative boundaries or other non-self-adjoint effects require modifications of the simple conservative expansion.

For a bulk bond, $K_{n+1}=K_n(1-2n_n)$ and $c_n(1-2n_n)=-c_n$. Reordering the remaining different-site operators then givesThe periodic spin boundary condition requires separate attention. Set $P={\prod }_{n=1}^N(1-2n_n)=(-1{)}^{N_f}$. The end bond becomes $S_N^{+}{S}_1^{-}=-P{c}_1^{†}{c}_N$, with the conjugate relation for its adjoint. Consequently the exact finite-ring Hamiltonian isfermion parity is conserved, so diagonalize in a sector $P=\pm 1$. There the end bond can be written like every other bond provided $c_{N+1}=-P{c}_1$. This parity-dependent boundary condition of the Jordan-Wigner chain is antiperiodic for even fermion number and periodic for odd fermion number. Simply assuming periodic fermions in both sectors changes the finite-size spectrum.

For a normal-ordered bosonic Hamiltonian, insert coherent-state resolutions between short imaginary-time evolution factors. The overlaps generate $\bar{\psi }{\partial }_{\tau }\psi$ and the Hamiltonian symbol is obtained by replacing creation and annihilation operators by $\bar{\psi }$ and $\psi$. Including the chemical potential gives, in the stated $\hslash =1$ convention,The trace imposes periodic imaginary-time boundary conditions,so bosonic Matsubara frequencies are ${\omega }_n=2\pi n/\beta$. The two complex coherent-state integration variables are treated independently before imposing the physical saddle contour. There is no extra zero-point ordering term: the supplied interaction is already normal ordered.

The spatial confinement is not specified beyond the volume. The uniform bulk results below use a homogeneous box with periodic spatial boundaries, or the interior of a large vessel where boundary effects can be neglected. A hard wall instead requires a vanishing field at that wall and an inhomogeneous saddle near it. Temporal periodicity follows from the thermal trace and does not by itself impose periodic spatial boundaries.

Divide the infinitesimal variation by its constant parameter $\alpha$, so the transformation generator is $(\delta {\varphi }_1,\delta {\varphi }_2)=({\varphi }_2,-{\varphi }_1)$. The variation of the kinetic term isand that of ${\varphi }_1^2+{\varphi }_2^2$ is $2{\varphi }_1{\varphi }_2-2{\varphi }_2{\varphi }_1=0$. Thus the Lagrangian density is invariant pointwise, without a nonzero boundary current. The finite transformation is a rotation of the two real scalar fields, giving an SO(2) global symmetry in field theory.

Noether's theorem supplies the parameter-independent Noether currentIts divergence is ${\partial }_{\mu }{j}^{\mu }={\varphi }_2\square {\varphi }_1-{\varphi }_1\square {\varphi }_2$, since the first-derivative terms cancel. Both fields obey the same Klein-Gordon equation, so this vanishes on solutions. With spatial boundary conditions giving zero flux at infinity, the Noether charge isA finite box with periodic boundary conditions works as well. The spatial integral is conserved because the local continuity equation converts its time derivative into a boundary flux; neither vanishing local current nor constant individual fields is required. Equivalently, $\Phi =({\varphi }_1+i{\varphi }_2)/\sqrt{2}$ transforms as $\Phi \mapsto e^{-i\alpha }\Phi$, exhibiting the equivalent global U(1) symmetry and fixing the sign of the generator.

In the position representation the Time-dependent Schrödinger equation isFor a normalized state define probability density and probability current byMultiplying the equation by $\overline{\psi }$ and its complex conjugate by $\psi$, then subtracting, cancels the real potential. The resulting identity isThus $\boxed{{\partial }_t\rho +{\partial }_{x}j=0}$. Integrating over an interval gives $d{\int }_b^c\rho dx/dt=j(b)-j(c)$: probability changes only through flux across the endpoints. This is local conservation of quantum probability.

Continuity with the zero exterior imposes zero endpoint values on the infinite square well. Solving the stationary equation with these boundary conditions giveswith zero extension outside. The integral of ${\sin }^2(n\pi x/a)$ is $a/2$, which fixes the normalization. The initial coefficients are $3/5$ and $4/5$, whose squared moduli sum to one. Each energy component acquires its stationary phase:At the midpoint, ${\Psi }_1=\sqrt{2/a}$, ${\Psi }_1^{\prime }=0$, ${\Psi }_2=0$, ${\Psi }_2^{\prime }=-(2\pi /a)\sqrt{2/a}$. HenceTaking its imaginary part proves

Inside the infinite square well, the time-independent Schrödinger equation and Dirichlet boundary conditions areand the wavefunction vanishes outside the interval. For a nonzero state, integration by parts givesThe strict inequality follows because a function with zero derivative and zero endpoint values is zero. Put $k=\sqrt{2mE}/\hslash$. The general solution is $A\sin kx+B\cos kx$; the first boundary condition gives $B=0$, and the second gives $ka=n\pi$, $n=1,2,\dots$. Since ${\int }_0^a{\sin }^2(n\pi x/a)dx=a/2$, the normalized energy eigenstates and eigenvalues areThe normalization fixes the magnitude of the coefficient; its arbitrary global phase has been chosen real and positive.

The time-independent Schrödinger equation is the energy eigenvalue equation for the Hamiltonian operator:Here $\hslash$ is the reduced Planck constant. A corresponding stationary wavefunction evolves as $\Psi (x,t)=\psi (x)e^{-iEt/\hslash }$, which follows by substitution in the Time-dependent Schrödinger equation. Its position probability density is time independent because the time factor has unit modulus. Physical energy eigenstates must also satisfy the system's boundary conditions; for a bound state, wavefunction normalization requires ${\int }_{-\infty }^{\infty }|\psi (x){|}^{2}dx=1$.

Apply the Frobenius method at the origin. The lowest power in $f^{\prime \prime }-\ell (\ell +1)f/r^2$ is $r^{c-2}$, with coefficientThe terms $-2\alpha f^{\prime }$ and $2mKZf/({\hslash }^{2}r)$ start at power $r^{c-1}$ and therefore cannot cancel a nonzero $r^{c-2}$ coefficient. Because $a_0\ne 0$, the indicial equation isIts roots are $c=\ell +1$ and $c=-\ell$. Since $\psi =r^{-1}f{e}^{-\alpha r}$, their leading behaviors are respectively $\psi \sim r^{\ell }$ and $\psi \sim r^{-\ell -1}$. The physical choice isThis is the Coulomb radial origin condition. For $\ell \ge 1$, the other branch is not even locally normalizable: its radial probability integral behaves as ${\int }_0^{\epsilon }{r}^2{r}^{-2\ell -2}dr={\int }_0^{\epsilon }{r}^{-2\ell }dr$, which diverges.

For $\ell =0$, the singular behavior $1/r$ is locally square-integrable with the measure $r^{2}dr$, so local wavefunction normalization alone would not reject it. The ordinary three-dimensional Coulomb potential problem additionally requires the regular boundary condition $u(0)=0$. A $1/r$ branch violates that condition and has divergent kinetic energy integral ${\int }_0^{\epsilon }{r}^2|{\partial }_r(1/r){|}^{2}dr={\int }_0^{\epsilon }{r}^{-2}dr$. Thus it is excluded from the physical energy domain as well. The regular choice gives $u\sim r^{\ell +1}$ in every case.

In a frame rotating with angular velocity $\mathbf{\Omega }$, the Coriolis force is ${\mathbf{F}}_C=-2m\mathbf{\Omega }\times {\mathbf{v}}_{\mathrm{rot}}$. In the northern hemisphere horizontal flow is deflected to the right: balanced circulation is counterclockwise about a low-pressure center and clockwise about a high-pressure center. The deflection reverses in the southern hemisphere. Jupiter's counterclockwise southern Great Red Spot is consequently anticyclonic, a high-pressure system. These statements assume a large-scale rotating atmosphere; they do not explain tiny vortices dominated by local boundary conditions.

For the upper half-plane use $0<\theta <\pi$, and interpret the source's arctangent as this argument. Then $V=(V_0/\pi )\mathrm{Im}(z\mathrm{Log}z)$ is harmonic, so $\mathbf{E}=-\nabla V$ has zero curl and divergence where there is no charge. On the positive boundary $\theta \to 0$ and $y\log r\to 0$; on the negative boundary $\theta \to \pi$. Hence the two boundary values are respectively zero and $V_{0}x$.

Branch qualification: the principal real arctangent alone gives zero on both sides of $y=0$, contradicting the stated negative-axis boundary condition, for example at $x=-1$. The continuous upper-half-plane argument repairs this notation; it is not an optional additive constant.

The full equation rearranges to $L\psi =-U\psi$. Since a bound state has no incoming free wave, convolution with the decaying inverse gives $\boxed{\psi (x)=-m\int e^{-\kappa |x-x^{\prime }|}U(x^{\prime })\psi (x^{\prime })d{x}^{\prime }/(\kappa {\hslash }^2)}$. Applying $L$ to this identity recovers the differential equation and the decay boundary conditions.

For the amplitude continue to $k=i\kappa$ on the decaying physical sheet. Then $\zeta =q$ and $D=\cos (qa)+({\kappa }^2-q^2)\sin (qa)/(2\kappa q)$. The double-angle identity factors its zero condition into the even and odd matching conditions $\kappa =q\tan (qb)$ and $\kappa =-q\cot (qb)$. Thus physical upper-half-plane amplitude poles are exactly the bound energies.

The literal statement for the probability requires correction. The displayed real-axis expression continued algebraically is $1/[D(k)D(-k)]$, and includes poles of the opposite-boundary-condition amplitude too. A concrete counterexample uses ${\hslash }^2/(2m)=1$, $a=2$, $q=1$, $\kappa =\cot 1$ and $U_0=1+{\cot }^{2}1$. Its continued probability denominator vanishes at $k=i\cot 1$, because $D(-k)=0$. But neither $\kappa =\tan 1$ nor $\kappa =-\cot 1$ holds, so $E=-{\cot }^{2}1$ is not a bound energy. Keeping the amplitude and the decaying boundary condition removes this spurious identification.

On the physical sheet the bound-state amplitude poles lie at $k=i{\kappa }_j$, ${\kappa }_j>0$, and map to real negative $E_j=-{\hslash }^2{\kappa }_j^2/(2m)$. The square-root map has two sheets meeting at the continuum threshold. Analytic continuation can also have virtual-state poles on the negative imaginary $k$ axis and resonance poles in the lower half-plane; these are not normalizable bound states. In the energy plane the resonances belong to the nonphysical sheet, rather than additional negative-axis bound levels.

For the algebraically continued probability, include the mirror poles contributed by $D(-k)$ as explained in part d; the two $k$ signs map to the same numerical energy but need not impose the same boundary condition. A sketch must label the sheet or the amplitude being plotted, rather than treating every singularity as a bound state.

For an infinitely deep isothermal atmosphere the bounded solution is $\boxed{I(\tau )=\pi B}$. A more general formal solution contains $C{e}^{\tau }$, excluded by the bounded deep boundary condition. Emission exactly replenishes absorption, so the emergent spectrum is blackbody in this idealized wavelength-by-wavelength thermal model.

Let $x_{<}=\min (x,x^{\prime })$ and $x_{>}=\max (x,x^{\prime })$. A homogeneous solution vanishing at the left wall is $u(x)=\sin [k(x+a/2)]$; one vanishing at the right wall is $v(x)=\sin [k(a/2-x)]$. Their Wronskian is $u{v}^{\prime }-u^{\prime }v=-k\sin ka$. Continuity and the source derivative jump $G_{+}^{\prime }-G_{-}^{\prime }=-2m/{\hslash }^2$ therefore giveThis is exactly the printed form because $x_{<}=(x+x^{\prime }-|x-x^{\prime }|)/2$ and $x_{>}=(x+x^{\prime }+|x-x^{\prime }|)/2$, with a minus sign from reversing the argument of the second sine.

The true poles occur at $k_n=n\pi /a$, $n=1,2,\dots$, corresponding toThe apparent zero at $k=0$ is removable: the numerator and $k\sin ka$ both vanish quadratically. Negative values of $k_n$ duplicate the same energies, rather than describing extra levels. The boundary conditions and source jump uniquely determine the inverse whenever $E$ is not an eigenvalue.

DLVO theory adds attractive Van der Waals forces and electric double layer repulsion to obtain an effective colloid pair potential. For two identical spheres of radius $b$ at a surface gap $h\ll b$, a useful schematic form isHere $A_H$ is the Hamaker constant. For a symmetric monovalent electrolyte with number density $n$ of each ionic species, ${\kappa }_D^2=2n{e}^2/(ϵk_{B}T)$. The exponential is a simplified screened repulsion; its amplitude depends on electric potential at the surface, charge regulation and geometry. Constant-charge and constant-potential boundary conditions need not produce the same close-approach law.

A primary attractive minimum near contact may be separated from dispersed configurations by an energy barrier; a shallower secondary minimum can produce loose reversible aggregation. A barrier much larger than $k_{B}T$ can give long-lived kinetic stabilization without making the dispersed state the global free-energy minimum. Adding salt shortens the Debye–Hückel screening length and can lower the barrier, allowing colloidal coagulation. Changing pH can alter surface ionization, so screening alone does not determine stability.

One can vary salt concentration while holding particle concentration, temperature and pH fixed, then follow particle aggregation using light scattering or microscopy. A solvent-only control and an independently measured surface-charge indicator help distinguish a change of screening from adsorption or charge reversal. Classical DLVO theory omits steric stabilization, hydration forces and strong ionic correlations; the formal attraction diverging at contact is cut off by molecular-scale physics.

An amphiphile has a solvent-affine part and a solvent-averse part within one molecule. In water these are commonly called hydrophilic head and hydrophobic tail. Two nonbiological examples are sodium dodecyl sulfate, with a twelve-carbon tail and an anionic sulfate head, and cetyltrimethylammonium bromide, with a sixteen-carbon tail and a cationic quaternary-ammonium head. Their counterions balance the ionic head-group charge; the head and tail do not have interchangeable interactions with water.

Aggregation reduces unfavorable exposure of hydrocarbon tails to water but loses molecular translational entropy. Interfacial free energy, head-group solvation and electrostatic repulsion, chain packing and conformational entropy determine the preferred aggregate. The hydrophobic effect is a solvent-mediated free-energy balance, rather than a literal attractive bond between tails.

The amphiphile packing parametercompares tail volume $v$, effective head-group area $a_0$ and maximum tail length ${\ell }_c$. A spherical core of radius $R$ obeys $v/a_0=R/3$, and a long cylindrical core obeys $v/a_0=R/2$. The condition $R\le {\ell }_c$ gives the conventional spherical window $P\lesssim 1/3$ and cylindrical window $1/3\lesssim P\lesssim 1/2$. Less strongly curved bilayers become compatible for $1/2\lesssim P\lesssim 1$, with planar packing near one; values above one can favor inverse curvature in an appropriate nonpolar surrounding medium. These windows are geometric guides rather than exact phase boundaries.

A larger hydrated or electrostatically repelling head increases $a_0$ and favors positive curvature. Screening of charged heads or a smaller head can lower $a_0$ and favor rods or sheets. Multiple tails increase the tail volume relative to head area and can favor bilayers. Tail length, branching, temperature and solvent selectivity alter both geometry and interaction free energy, so one cannot predict a unique aggregate from hydrophobicity alone.^[ref]

In the supplied chemical-potential expression, $\mu$ is the equilibrium chemical potential per amphiphile, ${\mu }_{\alpha }^0$ the standard free energy per amphiphile within an aggregate of size $\alpha$, and $x_{\alpha }$ the dimensionless molecular fraction belonging to that size class. The aggregate molecular fraction versus aggregate number fraction distinction explains the logarithm: the aggregate count fraction is $x_{\alpha }/\alpha$, and dividing its translational chemical potential by $\alpha$ gives a per-molecule value. ThusHere $k_B$ is the Boltzmann constant, $T$ absolute temperature, and $\alpha$ a positive integer. The logarithm acts on a dimensionless ideal activity; a concentrated nonideal solution would require activity coefficients.

equilibrium exchange of amphiphiles gives the same per-molecule $\mu$ for every size. For monomers, $\mu ={\mu }_1^0+k_{B}T\ln x_1$. Substitution yields the finite-micelle aggregation-number distributionThis formula by itself does not prescribe a preferred micelle size: the size dependence of ${\mu }_{\alpha }^0$ must include the competing packing and interaction terms. If their balance favors a finite size ${\alpha }_{\ast }$, a monomer population and a peak around that size can coexist. Dividing by $\alpha$ would instead plot the distribution of aggregate counts.

At very low $x_{\mathrm{tot}}$, the factors $x_1^{\alpha }$ suppress large aggregates and most molecules remain monomers. As $x_1$ rises, a favorable aggregate class grows extremely rapidly because of this high power. The critical micelle concentration from standard chemical potentials has the large-size estimateMore precisely, the illustrative crossover condition $x_{{\alpha }_{\ast }}=x_1$ givesAdded material above the crossover enters aggregates predominantly. Monomer concentration then increases slowly rather than being exactly pinned: its logarithm enters $\mu$, while aggregate populations respond with a large factor ${\alpha }_{\ast }$. For finite micelles this is a sharp association crossover, not an absolute prohibition on aggregates below one exact concentration.

The drawing uses a declared illustrative standard-free-energy model favoring aggregates near $\alpha =40$. It is not an inferred distribution or critical concentration for either named surfactant. The separate monomer peak is at $\alpha =1$; a distribution of molecular fractions weights each aggregate by its number of molecules.

For the general physics of a nearly planar fluid sheet, let $h(x,y)$ be a small-slope displacement, $\kappa$ membrane bending rigidity and $\sigma$ mechanical tension. At fixed topology and symmetric reference curvature, the quadratic Helfrich energy isThe bending term is the small-slope curvature-elastic energy; additional spontaneous-curvature or boundary terms require extra information.^[ref]

For Fourier amplitudes $h_{\mathbf{q}}=A^{-1}{\int }_{A}h(\mathbf{x})e^{-i\mathbf{q}\cdot \mathbf{x}}{d}^{2}x$, equipartition gives the membrane fluctuation spectrum under tension,Higher temperature increases fluctuations, while greater membrane bending rigidity or tension suppresses them. Short-wavelength modes are predominantly bending-controlled; longer wavelengths can be tension-controlled. Finite sheet size, supports, confinement and imposed area set boundary conditions and infrared cutoffs; continuum elasticity also needs a short-distance cutoff. solvent viscosity affects relaxation rates, although it does not enter this equilibrium amplitude at fixed elastic parameters.

TODO: user-deferred biology. The requested importance of these fluctuations for bodily function is deferred. The preceding result covers only the independent continuum and thermal physics.