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A theoretical physicist who calculated the spectrum of resonance fluorescence from a strongly driven chemical atom modeled as a two-level system. His 1969 treatment obtains the scattered-light power spectrum from two-time atomic dipole correlations, including radiation damping through the quantized electromagnetic field. The resulting Mollow triplet connects spectral structure to coherent driving and relaxation of a single quantum emitter.^[ref]

A damped mode $\delta =e^{-k^2/k_D^2}(A\cos q+B\sin q)$ does not alone imply oscillations in its ensemble power spectrum. Averaging its modulus squared gives the displayed formula. Equal independent initial variances, $P_A=P_B$ and $P_{AB}=0$, erase the oscillatory variance through ${\cos }^{2}q+{\sin }^{2}q=1$. A coherent initial phase, as supplied by a regular growing adiabatic mode, selects one common oscillator phase and retains acoustic peaks. Its leading density and velocity oscillations are a quarter-period apart, and angular projection also preserves their correlated contribution.

Cosmic microwave background diffusion damping, or Silk damping, damps the photon-baryon acoustic oscillator on small scales because Thomson scattering gives photons a finite mean free path. Repeated scatterings mix hotter and colder regions. Departures from perfect tight coupling also appear as photon-baryon velocity slip and a photon quadrupole, producing heat-conduction and shear-viscosity damping.

In natural units, the positive conformal scattering rate is ${\Gamma }_T=a{n}_e{\sigma }_T$, with proper free-electron number density $n_e$. A random-walk estimate gives squared comoving diffusion distance of order $\int d\eta /{\Gamma }_T$. More precisely, the leading polarization-inclusive tight-coupling amplitude containswhere $R=3{\rho }_b/(4{\rho }_{\gamma })$ is the baryon loading parameter. The corresponding power spectrum suppression is $e^{-2k^2/k_D^2}$. The approximation applies while the expansion in scattering time is controlled; it cannot be extended through a transparent universe by simply integrating $1/{\Gamma }_T$ indefinitely.^[ref]

Diffusion accumulates before last scattering and becomes especially important as the free-electron population falls. The finite width of the cosmological visibility function can also average oscillating line-of-sight sources; that averaging is a distinct mechanism acting in the same epoch. Both effects help shape the small-angular-scale damping tail, whose scale probes the ionization and baryon history.

A sufficiently long adiabatic tensor mode acts locally as an anisotropic change of spatial coordinates. Write the spatial metric as $a^2(e^{\gamma }{)}_{ij}$, with ${\gamma }_{ij}={ϵ}_{ij}^s{\gamma }^s$ transverse and trace-free. Removing a constant long mode maps the short wavevector to ${\tilde{k}}_i=k_i-{\gamma }_{ij}{k}_j/2$ at first order. The trace-free deformation has no first-order volume Jacobian, so its effect on an isotropic scalar power spectrum isCorrelating this response with the long graviton givesThe prime removes the momentum-conserving $(2\pi {)}^3{\delta }^3(\sum \mathbf{k})$, and $P_{\gamma }$ is the power of the same polarization amplitude used in ${\gamma }_{ij}={\sum }_s{ϵ}_{ij}^s{\gamma }^s$. Thus polarization normalization must be consistent on both sides.^[ref]

This consistency relation concerns a squeezed bispectrum configuration: it relates a three-point correlation function to a coordinate response of a two-point function. It requires the long tensor to be adiabatic and the state/dynamics to respect that response. Additional physical long-wavelength effects or incompatible initial-state assumptions cannot simply be replaced by this identity.

For an exactly massless minimally coupled scalar field on flat-slicing de Sitter spacetime with constant $H$, use natural units and neglect its metric backreaction. With $a=-1/(H\tau )$ and the Bunch-Davies vacuum, its canonical mode isWith power spectrum convention $⟨\delta {\varphi }_{\mathbf{k}}\delta {\varphi }_{{\mathbf{k}}^{\prime }}⟩=(2\pi {)}^3{\delta }^3(\mathbf{k}+{\mathbf{k}}^{\prime })P_{\delta \varphi }(k,\tau )$, taking the squared mode amplitude givesThe constant part is the frozen contribution to a de Sitter scalar spectrum; the remaining part is the subhorizon vacuum contribution to a de Sitter scalar spectrum. A merely light but nonzero mass changes the exact mode equation, so the displayed massless solution is then an approximation rather than an exact identity. Nonminimal curvature coupling also changes it.

The massless infinite-volume two-point correlation function has an infrared divergence. A finite-volume or finite-duration infrared prescription is needed, and coincident products require ultraviolet regulation or smearing. A finite spectrum at each fixed nonzero $k$ does not by itself make the all-mode equal-position integral finite.

The equipartition theorem says that each independent quadratic term in a Hamiltonian described by classical mechanics has mean energy $k_{B}T/2$ in a canonical ensemble. Here $k_B$ is the Boltzmann constant and $T>0$ the absolute temperature. Count terms in the energy, rather than vaguely counting everything that can move: one harmonic oscillator has one position degree of freedom but TWO quadratic terms, one in momentum and one in displacement.

A precise sufficient setting is $H(y,\xi )=a{y}^2/2+H_0(\xi )$ with $a>0$, where $y$ is one of the canonical variables, $\xi$ denotes all remaining variables, and the phase space integration measure factors as $dyd\xi$. The range of $y$ is the whole real line, independently of $\xi$; there is no constraint coupling its allowed values to other variables. The partition function must be finite, and the relevant expected values must exist. The Boltzmann factor then givesThe probability density function ${\rho }_y$ is a normal distribution with variance $k_{B}T/a$. A larger coefficient narrows the fluctuations enough to cancel its greater energy cost. This is why the mean contribution is independent of mass or spring constant.^[ref]

For a proof outline, the one-variable Gaussian integral is $Z_y={\int }_{-\infty }^{\infty }{e}^{-\beta a{y}^2/2}dy=\sqrt{2\pi /(\beta a)}$, so $-{\partial }_{\beta }\ln Z_y=1/(2\beta )$. More generally, integration by parts in a canonical variable gives $⟨y{\partial }_{y}H⟩=k_{B}T$ when its boundary term $[y{e}^{-\beta H}]$ vanishes and the integrals converge. For $a{y}^2/2$ this is twice its mean energy. At a finite hard boundary the boundary term need not vanish; hard-wall correction to harmonic equipartition illustrates the resulting departure. A constraint, a nonquadratic potential energy, or a noncanonical integration measure therefore requires its own calculation. The theorem is an ensemble average, not equality of instantaneous energies, and applying it to a recorded trajectory requires equilibration and sufficient sampling.

For a classical one-dimensional harmonic oscillator, with positive mass $m$ and spring constant $k$,The origin is the equilibrium position. Its heat capacity is $k_B$ at fixed $m,k$, not $k_B/2$. Although the two mean energies agree, their measured amplitudes do not: $⟨p^2⟩=m{k}_{B}T$ and $⟨x^2⟩=k_{B}T/k$. In a harmonic optical trap, the position measurement observes the potential energy contribution; it does not measure both contributions by observing $x$ alone.^[ref]

For $N$ particles in a three-dimensional classical monatomic gas, neglect interactions and thermally accessible internal excitations. Its three translational momentum terms give internal energy $U=3N{k}_{B}T/2$, heat capacity at constant volume $C_V=3N{k}_B/2$, and heat capacity at constant pressure $C_p=C_V+N{k}_B=5N{k}_B/2$. Per mole, replace $N{k}_B$ by the molar gas constant $R$. For example, using stipulated $R=8.314\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, the molar heat capacity at fixed volume is $12.471\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. These results assume a dilute, nonrelativistic ideal gas with classical translational statistics, rather than a quantum degeneracy regime.

For an ideal gas of diatomic molecules, modeled as two chemical atoms with a bond, there are three translational quadratic terms, two accessible rotational kinetic energy terms for changes of bond orientation, and one bond-stretching harmonic oscillator with two quadratic terms. Rotation about the bond axis is not an additional active mode in this linear-molecule model. The three familiar limits of heat capacity at constant volume areCorresponding molar heat capacities are $3R/2$, $5R/2$ and $7R/2$. Counting the stretching harmonic oscillator only once as $k_{B}T/2$ would give the wrong last value.^[ref]

The frozen regimes require quantum mechanics. In the simple molecular rotation model, $E_j=k_B{\theta }_{r}j(j+1)$ with state degeneracy $2j+1$, ${\theta }_r={\hslash }^2/(2I{k}_B)$, and moment of inertia $I$. For the bond-stretching quantum harmonic oscillator, ${\theta }_v=\hslash \omega /k_B$ with angular frequency $\omega$ and $\hslash =h/(2\pi )$ from the Planck constant. When ${\theta }_r\ll {\theta }_v$, the three limits above correspond to $T\ll {\theta }_r$, ${\theta }_r\ll T\ll {\theta }_v$, and $T\gg {\theta }_v$, provided translational motion remains classical. They are smooth crossovers, not abrupt switches. Real molecules can have different allowed rotational populations; at high temperature, thermal dissociation, internal excitations and nonharmonic bond motion can invalidate the fixed-molecule model before its last limit is reached.^[ref]

For a quantum harmonic oscillator with fixed $\omega$, put $z={\theta }_v/T$. Its exact mean energy and heat capacity areAt $T\gg {\theta }_v$, the mean energy approaches $k_{B}T$ and $C_{\mathrm{osc}}$ approaches $k_B$. At $T\ll {\theta }_v$, thermal excitation is exponentially suppressed and $C_{\mathrm{osc}}$ tends to zero, although the zero-point energy $\hslash \omega /2$ survives. Frozen thus means little thermal response, not zero motion or zero energy.^[ref]

A motivating measurement is thermal calibration of an optical trap. A focused laser through a microscope objective confines a bead in water; imaging or a quadrant photodetector records a position time series. Gieseler and colleagues describe a polystyrene bead of radius $1.03\mu \mathrm{m}$ at $293.15\mathrm{K}$, a $1064\mathrm{nm}$ trapping laser, and photodetector acquisition at $10^5\mathrm{Hz}$. Their recorded bead continues undergoing Brownian motion; higher trapping power produces a narrower equilibrium position distribution. Infer the centered variance, not a squared distance from an arbitrary camera origin:The normal distribution position histogram and agreement with dynamical calibration methods support the local harmonic optical trap interpretation. For a stipulated example, $T=300\mathrm{K}$ and ${\sigma }_x=20\mathrm{nm}$ imply $k\simeq 1.04\times 10^{-5}N/m=0.0104pN/nm$. This calculated scale is not an experimental reading.^[ref]

Position calibration is essential: if detector output $v$ has conversion $x=s(v-v_0)$, its physical variance is $s^2\mathrm{Var}(v)$. An unknown scale $s$ cannot be determined together with $k$ from that one variance. A complementary calibration demonstrated by Tolić-Nørrelykke and colleagues combines the power spectrum of thermal motion with the bead response to known sinusoidal motion of a translation stage. The driven response supplies an additional length scale, and their method does not require bead size, liquid dynamic viscosity or wall distance as inputs. It still uses a thermal dynamical model; agreement between methods is a consistency check, not a theory-free proof of equipartition theorem.^[ref]

The readout itself can hide fluctuations. Wong and Halvorsen used a $1064\mathrm{nm}$ optical trap and adjustable-exposure CCD imaging of a bead in water. They varied exposure and trapping power, held the bead away from the nearest wall, adjusted illumination to maintain comparable images, and measured tracking noise using a stuck bead. The measured variance fell as exposure increased; corrected results agreed with their time-averaging model. For a stationary overdamped harmonic optical trap, with optical trap relaxation time $\tau$, an exposure of length $W$ measures ${\bar{x}}_W=W^{-1}{\int }_0^{W}x(t)dt$. Its ideal noise-free variance isFor $\alpha >0$, $S(\alpha )<1$: using the blurred variance in the uncorrected formula overestimates $k$ by $1/S(\alpha )$. At $W=\tau$, the calculated factor is $e/2\simeq 1.36$. Additive tracking noise instead raises the measured variance and biases the inferred $k$ downward if left uncorrected.^[ref]

Useful controls are to record over many optical trap relaxation times, compare blocks for drift, calibrate the position scale, measure fixed-bead noise, vary exposure, and check the position histogram for departures from a normal distribution. A sparse sequence of accurate instantaneous positions can estimate an equilibrium variance; a long exposure averages away motion even if many frames are collected. Flow, appreciable laser heating, escape from the harmonic region or nonconservative forces can break the assumed thermal equilibrium. Inferring $k$ by imposing equipartition theorem is a calibration, whereas testing the relation requires an independently established stiffness and position scale. The beautiful practical connection is that microscopic randomness can set a force scale: once calibrated, a small constant applied force displaces the mean by $f/k$. The same quadratic-term counting predicts macroscopic heat capacity and makes microscopic fluctuations useful measurements.

For a planar elastic filament under positive constant extension force $F$, the quadratic force-controlled energy isThe higher-order Euler-Lagrange equation is $B{\zeta }^{\prime \prime \prime \prime }-F{\zeta }^{\prime \prime }=B{\kappa }_0^{\prime \prime }$. On a periodic interval or in a bulk Fourier transform treatment, every nonzero mode obeys the displayed transfer relation. The length deficit is $\frac{1}{2}\int ({\zeta }^{\prime }{)}^{2}dx$, so its ensemble value is determined by the power spectrum of the quenched intrinsic curvature. Boundary conditions must be specified for a finite filament.

The second term in the equal-time two-point function of a de Sitter scalar is independent of time for constant $H$. It dominates on superhorizon scales and has constant dimensionless power spectrum, giving a scale-invariant inflationary power spectrum. The infrared qualification of a scale-invariant covariance still applies to the formally infinite-volume coincident variance.

Assume all relevant modes begin outside the horizon at common conformal time ${\tau }_i$, grow as ${\tau }^2$ until radiation-era horizon entry, then freeze until matter-radiation equality, and again grow as ${\tau }^2$ in matter domination. With horizon entry ${\tau }_h=2\pi /k$, $k_{\mathrm{eq}}=2\pi /{\tau }_{\mathrm{eq}}$ and final time ${\tau }_0$, direct multiplication of the growth factors gives the displayed transfer with $G=({\tau }_0/{\tau }_i{)}^2$. A primordial density power spectrum $P_i=Ak$ becomes $P_0=A{G}^{2}k$ below equality wavenumber and $A{G}^2{k}_{\mathrm{eq}}^4{k}^{-3}$ above it. This approximation neglects the true logarithmic radiation-era matter growth, baryonic acoustic structure and late acceleration; a conventionally normalized transfer divides out $G$.

Galaxy bias describes the statistical relation between fluctuations in galaxy counts and the underlying matter density contrast. In the simplest large-scale limit, ${\delta }_g=b_1{\delta }_m$, so the corresponding deterministic power spectrum is $b_1^2{P}_m$. Different selections can have different $b_1$ because they sample different formation environments. More generally, the bias expansion includes nonlinear, tidal and stochastic terms, so a single multiplicative constant is an approximation.^[ref]

The galaxy bispectrum measures the connected three-point correlation of the Fourier modes of the galaxy density contrast. For a mean-zero field, statistical homogeneity givesThe delta function enforces a closed wavevector triangle. With the additional assumption of statistical isotropy, one can write $B_g(k_1,k_2,k_3)$ as a function of triangle size and shape. Statistical homogeneity alone leaves possible orientation dependence.

The galaxy bispectrum adds information beyond the power spectrum. For example, take a zero-mean Gaussian random field $\delta$ and a local tracer model ${\delta }_g=b_1\delta +(b_2/2)({\delta }^2-⟨{\delta }^2⟩)$. Two Wick contractions at each quadratic insertion cancel its factor of two, yielding at leading orderThus nonlinear galaxy bias can generate a three-point signal even from Gaussian matter fluctuations. Nonlinear gravitational evolution supplies a further matter-bispectrum contribution. Both effects must be modeled before interpreting a measured galaxy three-point signal as primordial non-Gaussianity.^[ref]

Sampling with a homogeneous Poisson point process of mean comoving number density $\bar{n}$ adds a constant $1/\bar{n}$ to the dimensional galaxy power spectrum. Subtracting its expectation leaves sampling uncertainty. Correlated tracer populations and survey weights modify this simple formula; survey depth alone does not specify $\bar{n}$.

For the Fourier transform convention $C(\mathbf{s})=\int \Phi (\mathbf{\kappa })e^{i\mathbf{\kappa }\cdot \mathbf{s}}{d}^2\kappa /(2\pi {)}^2$, angular integration of an isotropic power spectrum gives the displayed Hankel transform. A purely radial Hankel transform without the prefactor corresponds to defining its spectrum as $\Phi /(2\pi )$.

The $P_{13}$ contribution correlates a linear density field with a third-order field:Both orderings contribute $2P_{13}$ to the power spectrum.

A power spectrum describes the variance of Fourier amplitudes as a function of wavenumber. In a statistically homogeneous continuum convention,The paper's $⟨|{\Delta }_M(k){|}^2⟩$ is the corresponding finite-volume or normalization-suppressed statement. For statistical isotropy only $k=|\mathbf{k}|$ remains. The dimensionless variance per logarithmic interval is $k^3{P}_M(k)/(2{\pi }^2)$, a different quantity.

The response changes around $2k^2\sim 9{\mathcal{H}}^2$, or $k\sim 2.1\mathcal{H}$: the velocity contribution to the relativistic constraint and the effective growth equation change relative importance across the horizon-scale transition. Large-scale ${\Delta }_M$ is approximately constant for its regular mode, while short-scale density perturbations grow gravitationally. A smooth initial potential spectrum can therefore acquire a different $k$ dependence in the density spectrum. For illustration, a constant matter-era potential gives ${\Delta }_M=-(5+k^2{\eta }^2/6)\Psi$, so sufficiently small-scale $P_M$ has four additional powers of $k$ relative to $P_{\Psi }$, while the very large-scale limits have the same power dependence.

This identifies a dynamical transition, not a universal prediction of the primordial spectrum or an exact spectral break at $2\mathcal{H}$ for arbitrary initial data. The familiar late-time matter-spectrum turnover also retains the history of horizon entry around radiation–matter equality; it is not generically tied to the current Hubble scale. Initial conditions and the intervening expansion history are needed for an actual spectrum.

Galaxy formation combines gravitational assembly of dark-matter haloes with dissipative baryonic physics. Primordial density fluctuations supply a statistical seed distribution. In linear theory, each Fourier mode evolves through a growth factor and a scale-dependent transfer function; the power spectrum describes the variance by scale. Radiation pressure, free streaming and the time of horizon entry influence the small-scale transfer, so one should not replace the spectrum by a single universal growing power of time.

When fluctuations become nonlinear, overdense regions collapse, virialize and build larger structures through accretion and mergers. Cold dark matter permits a broad hierarchy of bound haloes. A spherical-collapse threshold and a mass-variance prescription, such as the Press-Schechter formalism, organize an approximate halo abundance; departures from spherical collapse and correlated environments limit its literal interpretation.

Gas falling into a halo is compressed and may be shock-heated toward a virial scaleUnlike collisionless dark matter, it can radiate away thermal energy. A rough cooling time iswhere the denominator specifies a chosen optically thin astrophysical cooling function convention. Cooling is effective for condensation when this time is short compared with the relevant dynamical or supply time. Atomic transitions, molecular processes, ionization state, density and metal abundance matter. A stable virial shock is not obligatory for all accretion: cold dense inflows can supply gas in regimes where shock-heated gas would cool rapidly.

The distinction between a gravitationally assembled massive halo and a radiatively condensing luminous component is fundamental. In the classic two-stage picture, non-dissipative clustering builds potential wells while cooling allows gas to concentrate much further than the halo component.^[ref]

Angular momentum inherited from tidal torques and later accretion helps gas settle into a rotationally supported disc after cooling; disc size depends on angular-momentum retention and transport. Locally self-gravitating gas fragments and forms stars. Stellar radiation, winds and supernovae then heat gas, drive turbulence, enrich it with heavy elements and can expel it from shallow potential wells. Metal enrichment can also accelerate later cooling, so feedback is more than a single positive or negative energy term.

Galaxy growth is usefully expressed as a gas-budget competition:Here $\mathcal{R}$ is the promptly returned stellar mass fraction; unlike the simplification in Question1, a general evolutionary model need not set it to zero. The equation separates newly supplied, converted, expelled and recycled gas rather than assuming that halo mass uniquely fixes stellar mass.

Accreting central black holes can provide radiative and mechanical feedback. Heating a halo, ejecting gas or maintaining a hot atmosphere can reduce fresh fuel; removal of a satellite's gas supply and ram pressure in a surrounding medium provide additional environmental routes. Quiescent galaxy assembly can still add stellar mass through dry mergers, so cessation of star formation is not identical to cessation of all mass assembly.

Mergers and disc instabilities can rearrange morphology and trigger bursts, while smooth accretion can rebuild discs. Cosmological expansion changes the supply and merger environment; accelerated late-time expansion also slows linear growth. A predictive model must connect these mechanisms to the stellar mass function, gas fractions, star-formation histories, sizes, chemical abundances and clustering. Those observables test different pieces of the assembly-and-feedback picture; none alone implies a one-to-one instantaneous relation between galaxy luminosity and total halo mass.

Write the growing density contrast as ${\delta }_{m,\mathbf{k}}=A_{\mathbf{k}}{\tau }^2$, with $A_{\mathbf{k}}$ having the indicated $k^{n/2}$ spectral amplitude. In the dust-comoving synchronous gauge, $h^{\prime }=-2{\delta }_m^{\prime }$, and the supplied $h_s^{\prime }=h^{\prime }$ therefore implies $h_s^{\prime \prime }=-4A_{\mathbf{k}}$. On the same comoving density slicing, the cosmological Poisson equation isbecause $\mathcal{H}=2/\tau$. This is the comoving-density form of the relativistic scalar constraint, so it can be used on these large scales; replacing its density by the Newtonian-gauge density would not be justified there. Combining the two results givesThus the large-angle Sachs-Wolfe effect has source $\Phi /3$. The potential is constant for this matter-era growing mode, and its power spectrum obeys $P_{\Phi }(k)\propto k^{n-4}$.

For subhorizon pressureless matter during radiation domination, expansion is too rapid for significant self-gravitating growth. Neglecting the small matter-gravity term gives ${\ddot{\delta }}_m+t^{-1}{\dot{\delta }}_m\simeq 0$, hence a constant plus a logarithm of time; the exact coupled radiation problem supplies the detailed amplitudes. This is the Mészáros effect. Modes entering the horizon during radiation domination are suppressed relative to modes entering later, while after equality ordinary matter growth resumes.

The matter power spectrum consequently has a turnover near the equality horizon scale $k_{\mathrm{eq}}\sim a_{\mathrm{eq}}{H}_{\mathrm{eq}}$ in units with $c=1$. Shorter-scale modes retain the radiation-era suppression. This statement concerns cold pressureless matter; baryons coupled to photons also show acoustic effects before recombination.

For a general colored medium and unit plane-wave input $E(0,\mathbf{s})=1$, let $U_W(x,0)$ be the propagator of ${\partial }_x=i{\Delta }_{\perp }/(2k)+ik\mu W(x,\mathbf{s})$. Then the solution isA time-ordered operator exponential defines $U_W$. Its expectation depends on the full longitudinal covariance; the initial Gaussian and transverse-stationarity assumptions do not specify a universal closed expression.

With the stated longitudinal Gaussian white noise and transverse statistical homogeneity, the moment depends only on $\mathbf{\rho }={\mathbf{s}}_1-{\mathbf{s}}_2$. Hence $({\Delta }_1-{\Delta }_2)\Gamma =0$. The closed equation in the previous part and initial value $\Gamma (0)=1$ giveIn particular $\Gamma (x,0)=1$: the ensemble mean intensity remains constant for plane-wave input, even though coherent attenuation in a white-noise random medium reduces the mean complex field.

Use a consistent transverse power spectrum convention,For an isotropic spectrum, angular integration gives the isotropic transverse covariance spectrum, $C(\rho )=(2\pi {)}^{-1}{\int }_0^{\infty }\Phi (\kappa )J_0(\kappa \rho )\kappa d\kappa$. ThereforeThe supplied two-dimensional Fourier transform definition has angular factor $2\pi$, whereas the displayed radial transform pair omits it. The latter is a normalized Hankel transform pair. If its radial spectrum is denoted ${\Phi }_H=\Phi /(2\pi )$, use ${\Phi }_H$ in the integral and omit the prefactor $1/(2\pi )$. These are equivalent conventions; mixing them changes the attenuation rate.

Introduce a constant filament bending modulus $B$, which sets the mechanical force scale. The filament has intrinsic curvature of an elastic filament ${\kappa }_0(s)$, and its actual signed curvature in a small-slope Monge representation is $\kappa \simeq {\zeta }_{xx}$. To quadratic order one may identify material arc length $s$ with $x$ in ${\kappa }_0$, whileAt fixed positive extension force, the mechanical potential is bending energy minus $FL$. Replacing $L$ by $\mathcal{L}-(\mathcal{L}-L)$ leaves, up to a force-dependent constant,Its first variation isThe higher-order Euler-Lagrange equation is thereforeFor a Fourier transform relation, take a periodic bulk segment, or a long-filament bulk approximation with negligible endpoint contributions; these are necessary boundary conditions because the question specifies no filament end conditions. Using $f(x)=\int \hat{f}(q)e^{iqx}dq/(2\pi )$, every nonzero wavenumber obeysA zero displacement mode is an arbitrary translation. In a periodic representation a constant intrinsic-curvature mode contributes a constant bending cost but no bulk slope, so it does not contribute to the following deficit. The Parseval identity givesFor clarity about finite normalization, on a periodic interval of length $L$ define ${\kappa }_n=L^{-1}{\int }_0^L{\kappa }_0(x)e^{-i{q}_{n}x}dx$, $q_n=2\pi n/L$. Then the equivalent exact quadratic-order formula isThis is force-extension of an intrinsically curved filament, determined by its particular fixed shape rather than by a thermal orientation entropy.

For an ensemble with quenched intrinsic curvature, first minimize each realization and then average the last expression, replacing $|{\kappa }_n{|}^2$ by its ensemble mean. For stationary bulk disorder, let $S_{\kappa }(q)$ be the power spectrum, normalized so that its correlation is $⟨{\kappa }_0(x){\kappa }_0(x+r)⟩=\int S_{\kappa }(q)e^{iqr}dq/(2\pi )$. The measurable mean deficit per unit projected length isIf the curvature is smooth enough that the second spectral moment is finite, one can take the large-force limit under this integral, obtaining the smooth-curvature high-force extension lawFor a stationary ensemble the integral is $L⟨({\kappa }_{0,x}{)}^2⟩$. Equivalently, the bulk slope is ${\zeta }_x\simeq -(B/F){\kappa }_{0,x}$; a plot of the mean deficit against $F^{-2}$ measures the mean squared intrinsic-curvature gradient, if $B$ is known. The $F^{-2}$ law is a smooth-curvature bulk result, not a universal law for every random filament. For example, ideal white intrinsic-curvature disorder with $S_{\kappa }(q)=S_0$ has an infinite second spectral moment; retaining the full integral instead gives $⟨\mathcal{L}-L⟩/L=S_0\sqrt{B}/(8\sqrt{F})$ in that ideal continuum model.

Endpoint conditions also matter. If each end is moment-free and has no transverse applied force, the natural conditions from the variation areat each end. These can produce a free-end curvature boundary layer of width $\ell =\sqrt{B/F}$. For a constant intrinsic filament curvature $\kappa$, integration of the higher-order Euler-Lagrange equation with zero transverse force gives $B({\zeta }_x{)}_{xx}-F{\zeta }_x=0$ and henceThe last formula follows by integrating ${\sinh }^2[(x-L/2)/\ell ]$, and tends to ${\kappa }^2{\ell }^3/2\propto F^{-3/2}$, despite the vanishing bulk curvature gradient. Thus a finite free-ended measurement must include or suppress endpoint bending before interpreting a bulk $F^{-2}$ fit.

For the freely jointed chain, rigid links have independent orientations, so the restoring force is entropic. Let $T$ now denote absolute temperature, and $\beta =1/(k_{B}T)$ with Boltzmann constant $k_B$. In the usual three-dimensional chain, a segment at polar angle $\theta$ has force energy $-Fb\cos \theta$. Its angular canonical partition function and the full chain partition function areDifferentiating $\log Z_N$ with respect to $\beta F$ yields the exact mean projected displacementHere $\mathcal{L}$ is the Langevin function; this is force-extension of a three-dimensional freely jointed chain, in the fixed-force canonical ensemble. At high extension, $\coth \xi =1+2e^{-2\xi }+\cdots$, soThus the smooth quenched-curvature bulk deficit falls as $F^{-2}$, whereas the three-dimensional freely jointed chain deficit falls as $F^{-1}$ and depends explicitly on temperature. The distinction is mechanical straightening of a fixed intrinsic shape versus orientational entropy; thermal bending fluctuations of a stiff filament have not been included in the first calculation.

If the comparison chain is also restricted to two dimensions, it is a planar freely jointed chain rather than the three-dimensional model. In this case,The last identity follows directly by differentiating the integral defining $I_0$; $I_0$ and $I_1$ are Modified Bessel functions of the first kind. At large $\xi$ the angular weight is proportional to $e^{-\xi {\theta }^2/2}$, giving $⟨\cos \theta ⟩\simeq 1-⟨{\theta }^2⟩/2=1-1/(2\xi )$. Consequently $Nb-⟨L⟩\sim N{k}_{B}T/(2F)$: the same inverse-force exponent, with half the three-dimensional coefficient. Both versions assume inextensible links and forces below the regime where the links themselves stretch.