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Machine compliance is the extra load-path displacement per applied force attributed to the apparatus in a specified measurement model. Its units are $\mathrm{m}{\mathrm{N}}^{-1}$, the reciprocal units of a spring constant. Here $C_m$ includes the frame, load cell, adapters and grips insofar as their deformation contributes to the recorded crosshead displacement $x$. It is a property of that configuration and readout, rather than a universal constant of the testing machine.

Assume a uniform specimen with initial length $L$, constant $A_0$ and Young's modulus $E$, small axial elastic deformation, quasistatic loading, no slip, and a linear, length-independent apparatus response over the chosen force window. Equilibrium makes the same force act through specimen and apparatus, while their extensions add. The two springs in series therefore giveThe offset $x_0$ describes the displacement origin after seating; the slope $q$ describes total compliance. Fitting an intercept avoids forcing an arbitrary zero through the origin, but cannot repair slip or a curved response.

If all of $x-x_0$ is incorrectly attributed to specimen extension, the inferred modulus isFor positive $C_m$, this is too small. The fractional bias is most serious for a short, thick, stiff specimen because its own compliance $L/(E{A}_0)$ is small. A local extensometer instead has $d(\Delta L_g)/dF=L_g/(E{A}_0)$ in this ideal model, excluding deformation outside its contacts.

A shaped specimen requires more care. With varying initial cross-sectional area $A(s)$ and uniform $E$, its axial compliance is $\int ds/[EA(s)]$ in the one-dimensional approximation. Shoulder and end deformation must be included explicitly or remain identical between tests and be absorbed into an effective intercept. Changing only the extensometer gauge length on an unchanged specimen does not change its total compliance and cannot perform a two-length machine compliance separation.

For fixed spring constant $k$ and linear damping coefficient $b$, replacing mass $m$ by $\lambda m$ changes the exponential envelope rate from $\gamma =b/(2m)$ to $\gamma /\lambda$. The damped angular frequency becomes ${\omega }_{d,\lambda }^2=({\omega }_d^2+{\gamma }^2)/\lambda -{\gamma }^2/{\lambda }^2$. Thus envelope decay times scale exactly as mass, while the familiar inverse-square-root frequency scaling is a weak-damping approximation. A constant gravitational force shifts equilibrium without changing either rate.

A device combining micromachined mechanical elements with electrical actuation or detection.

For equal masses $m$ at spacing $d$ joined by nearest-neighbour springs of spring constant $K$, Newton's second law gives $m{\ddot{u}}_j=K(u_{j+1}+u_{j-1}-2u_j)$. A normal mode $u_j=u_0{e}^{i(kjd-\omega t)}$ givesThis single acoustic phonon branch is linear at small wavevector and has zero group velocity at the first Brillouin zone edge.

Two frictionless beads of positive masses $m_1,m_2$ on a circular hoop are joined by two identical springs of spring constant $k$, constrained to follow the two arcs. At their diametrically opposite equilibrium, each spring has natural length equal to half the circumference. For signed arc length displacements in the same rotational sense, the potential energy is $U=k(x_1-x_2{)}^2$. The normal mode of common rotation has mode shape $(1,1)$ and zero angular frequency, while the other normal mode has mode shape $(1,-m_1/m_2)$ and the positive angular frequency in the title.

Adding the equations of motion shows that $X=(m_1{x}_1+m_2{x}_2)/(m_1+m_2)$ satisfies $\ddot{X}=0$. Subtracting them gives $\ddot{\delta }+{\omega }_1^2\delta =0$ for $\delta =x_1-x_2$. Thus the complete zero mode is $X=C+Dt$, including uniform rotation, and the relative motion is a harmonic oscillator. In the limit $m_2/m_1\to \infty$ at fixed $m_1,k$, the oscillatory motion of the heavier bead vanishes for fixed lighter-bead amplitude, and ${\omega }_1\to \sqrt{2k/m_1}$. A common drift still survives if its initial velocity is held fixed.

Translate the liquid past a bead held in a stationary harmonic optical trap. At steady mean position, take $u$ as the local fluid velocity relative to the fixed trap and $x$ positive downstream. Averaging the overdamped force balance givesFor the invented optical trap Langevin calibration example, $u=50.0\mu m/s$ predicts $⟨x⟩=23.562\mathrm{nm}$, $V-V_0=0.23562\mathrm{V}$, and $\zeta u=k⟨x⟩=0.47124\mathrm{pN}$. Reversing $u$ reverses the mean shift but leaves the ideal thermal variance unchanged. A symmetric speed sweep tests the offset and the linear force range; the slope determines $\zeta /k$ if position is calibrated.

Near a wall, $\zeta =6\pi \eta a$ is not the appropriate drag without a boundary correction; record bead height and direction of motion. Stage velocity must correspond to the local fluid velocity after transients have settled. Escape under increasing flow estimates a maximum holding force in that geometry, whereas small-displacement drag calibration estimates the local spring constant. They are different quantities. The atomic force microscopy similarly needs a calibrated spring constant to convert a measured deflection to force, but its cantilever is a physical spring rather than this optical field.

Optical tweezers hold and move a small particle using the force exerted by a tightly focused beam of light, usually from a laser. They can position transparent beads in a liquid and manipulate suitable biological cells. Attaching a molecule to a trapped bead turns the bead into a handle: a measured bead displacement, together with a calibrated spring constant, reveals the force exerted on the attachment. This makes microscopic manipulation and measurement possible without a mechanical tool touching the bead. Living biological cells were manipulated with infrared beams by Ashkin, Dziedzic and Yamane in 1987; Svoboda and Block used trapped beads to measure forces from individual molecules in 1994. ^{[ref] [ref]}.

Why can light pull a bead towards a focus? Light carries momentum. A transparent bead with a higher refractive index than the surrounding liquid bends transmitted light rays by refraction, changing their momentum; conservation of momentum requires an opposite transfer to the bead. In the geometrical optics picture, unequal illumination of a displaced bead produces a net restoring force towards the brighter region. For a particle much smaller than the wavelength, a physical optics description replaces that ray picture. The detailed force depends on particle size, refractive index, beam shape and wavelength; the qualitative attraction does not supply a universal formula for every particle.

The optical gradient force draws a suitable transparent bead towards high illumination, while the optical scattering force generally pushes along beam propagation. A stable three-dimensional optical trap needs a balance with a restoring force in each direction: transverse confinement alone does not ensure axial confinement. Strong focusing can make the backward axial gradient contribution large enough to balance the forward scattering contribution, with the equilibrium displaced from the focus. Higher refractive index alone is insufficient; absorbing particles, different index contrasts and different beam geometries can behave differently. ^[ref].

Historically, Ashkin's 1970 experiments accelerated and trapped particles using radiation pressure. The 1986 paper by Ashkin, Dziedzic, Bjorkholm and Chu demonstrated the single-beam gradient optical trap for transparent particles. These are distinct steps: the earlier work established optical manipulation, and the later work supplied the tightly focused single-beam arrangement. ^{[ref] [ref]}.

Near a stable equilibrium, a harmonic optical trap behaves like a spring; thermal calibration of an optical trap uses its Brownian motion to determine the effective spring constant. The later passage through a translating optical trap sections instead consider a moving localized potential energy, which need not remain harmonic throughout a passage.

An Ornstein-Uhlenbeck process combines mean reversion with continuing random fluctuations. In the scalar model, its stochastic differential equation iswhere $W$ is a standard Brownian motion. The drift points towards $\mu$: it is negative above $\mu$ and positive below it. The coefficient $\theta$ sets the speed of mean reversion, while $\sigma$ sets the strength of the random forcing. We consider a fixed or normal initial value independent of future Brownian increments, giving a Gaussian process and a Markov process.

For a stipulated example, use $\theta =1$, $\mu =2$, $\sigma =\sqrt{2}$ and the fixed initial value $X_0=6$, in arbitrary consistent state and time units. Then the expected value relaxes as $2+4e^{-t}$ and the variance grows as $1-e^{-2t}$. At time one they are about $3.4715$ and $0.8647$; the limiting variance is one. The sample paths keep fluctuating even after the expected value is close to two. The Ornstein-Uhlenbeck transition distribution describes this evolving uncertainty; Ornstein-Uhlenbeck stationary initialization instead starts with that limiting normal distribution already in place.

A physical motivation is the velocity of a particle undergoing Brownian motion: linear drag removes a persistent initial velocity, while random forcing continually renews it. The original calculation gives a normal distribution for the velocity after subtracting its exponentially decaying initial contribution, with decay rate equal to the friction coefficient divided by the particle's mass.^[ref] A position coordinate in an overdamped harmonic optical trap gives another interpretation of the same scalar equation, with $\theta =k/\zeta$ for spring constant $k$ and drag coefficient $\zeta$. Its physical assumptions and connection to measurements are developed in thermal calibration of an optical trap.

Use the deterministic Hookean dumbbell model, with a zero-rest-length spring and negligible connector drag. For $\mathbf{R}={\mathbf{r}}_1-{\mathbf{r}}_2$, its elastic forces on the two beads areHere $H>0$ is the spring constant. No independent thermal forcing from Brownian motion or bead inertia is included. The unperturbed linear velocity field is $\mathbf{Lr}$, with $\mathrm{tr}\mathbf{L}=0$ by incompressibility.

The fluid velocity used in each bead's drag law is the imposed flow plus the disturbance from the other bead, with its own singular field omitted. At leading order in $a/R$, the evenness of the Oseen tensor, $\mathsf{G}(-\mathbf{R})=\mathsf{G}(\mathbf{R})$, givesThese are the pair hydrodynamic mobility matrix relations. Their signs follow from balancing each spring force with $-\zeta ({\dot{\mathbf{r}}}_i-{\mathbf{u}}_{\mathrm{ext},i})$, the fluid drag on that bead.

Adding the equations cancels both the direct spring slips and the cross-induced velocities, since ${\mathbf{F}}_1+{\mathbf{F}}_2=0$. For the midpoint ${\mathbf{r}}_c=({\mathbf{r}}_1+{\mathbf{r}}_2)/2$,The midpoint therefore follows the undisturbed linear velocity field. Subtracting the equations instead gives the hydrodynamic interaction of a Hookean dumbbell:The dimensionless interaction matrix isIts transverse and longitudinal eigenvalues are $3a/(4R)$ and $3a/(2R)$, respectively. Since the spring force is longitudinal, the separation equation can also be writtenThe relative correction to the free-draining spring relaxation rate is therefore $3a/(2R)$, explicitly proportional to $a/R$. Entrainment by the other bead reduces their relative approach for given opposing spring forces.

This is a far-field hydrodynamic interaction, valid for $R\gg a$; it is not a near-contact law and omits higher reflections and lubrication. The printed target also fixes the spring's rest length implicitly. For an ordinary spring obeying Hooke's law with nonzero rest length $R_0$, the elastic force would be $-H(R-R_0)\hat{\mathbf{R}}$, and every spring term above would acquire the factor $1-R_0/R$. The displayed result uses the zero-rest-length spring convention appropriate to the ideal Hookean dumbbell model.

Write $u=x/L$ and $A=k_{B}T{L}_p$. The worm-like chain force-extension interpolation isA fit to an elastic rising branch estimates its contour length $L$ and, if not fixed independently, its persistence length $L_p$. Comparing successive fitted contour lengths gives the length released by a domain and helps distinguish whole-repeat unfolding from partial events. $L_p$ estimates the flexibility of the unfolded chain and the bending rigidity $A$; at a stated temperature, $A=k_{B}T{L}_p$.

For small extension, expansion gives $f\simeq [3k_{B}T/(2L_{p}L)]x$, so the effective entropic spring constant is $3k_{B}T/(2L_{p}L)$. Close to full extension, $f\sim k_{B}T/[4L_p(1-x/L{)}^2]$, giving strong strain stiffening. These regimes constrain the fit differently and correlate its parameters if the measured interval is short. The fit does not by itself identify a unique folding barrier or unfolding kinetic rate.

Persistence-length convention. In a three-dimensional isotropic worm-like chain the standard definition is $⟨\mathbf{t}(s)\cdot \mathbf{t}(0)⟩=e^{-s/L_p}$. Thus at $s=L_p$, $\arccos ⟨\cos \theta ⟩=\arccos (e^{-1})\simeq 68.4^{\circ }$. This is not literally the mean angle $⟨\theta ⟩$. For a simple counterexample, equally probable angles $0^{\circ }$ and $90^{\circ }$ have mean angle $45^{\circ }$ but $\arccos ⟨\cos \theta ⟩=60^{\circ }$. The source's angle description is shorthand for the tangent-correlation convention.

The AFM pulls a tethered molecule through its cantilever. Deflection, multiplied by a calibrated cantilever spring constant, measures force; support motion must be corrected for that deflection and any linker compliance to obtain molecular extension. Initially folded structure can resist extension, then contacts break and more polypeptide polymer contour length becomes available. The readable trace rises with increasing extension, with fluctuations and a strongly rising large-extension part. It should not be interpreted as a direct image of individual amino acids or as establishing a unique microscopic unfolding pathway.

An ideal inextensible freely jointed chain with the stated segment count and length hasIts mean extension is strictly below $14\mathrm{nm}$ at every finite force. Thus the quoted $15\mathrm{nm}$ maximum does not establish ideal FJC behaviour: if both numbers are taken literally, it exceeds the model's polymer contour length by about $7\%$. Finite segment stretch, inaccurate effective segment parameters, additional linkers or an extension-zero/calibration error can account for such a difference. If the numerical descriptions are approximate, the discrepancy may be within their uncertainty; even exact agreement with a polymer contour length would not prove freely jointed statistics. Low-force fluctuations and the full force–extension relation are separate tests.

The coefficient in the linear force law isStretching suppresses the many disordered orientations available to the chain. The Gaussian chain free energy cost $\Delta \mathcal{F}\simeq 3k_{B}T{R}^2/(2N{b}^2)$ has gradient equal to this restoring force. This is entropic elasticity; individual covalent bonds need not stretch to produce the weak-force spring.

At moderate force, even an ideal FJC becomes nonlinear as its orientations align, so failure of a linear fit alone need not imply nonideal chemistry. But failure of the full FJC curve can reflect backbone spring constant and correlated segment angles, nonuniform effective links, remaining folded contacts, force-induced conformational changes, energetic bond compliance, or tether/calibration effects. An extensible FJC or an appropriate semiflexible model includes some of these effects; their parameters require independent justification. The stated inextensible $14\mathrm{nm}$ polymer contour length already excludes a literal $15\mathrm{nm}$ extension.

The first panel of the original model figure below shows the full orientational response and weak-force tangents without tracing the experimental data.

If the ideal-chain segment count and segment length remain fixed, the entropic spring constant is proportional to absolute temperature. ThereforeThis is an increase by about $11.7\%$. The factor $k_{B}T$ multiplies the free energy penalty for the loss of chain configurations; a larger force is required to impose the same normalized weak extension at higher temperature.

Not degrading does not guarantee that a real protein's folded state, effective spring constant or solvent interactions remain unchanged. Thermal unfolding and changes in persistence length can occur without covalent degradation. Those changes modify the experimental curve and may invalidate a fixed-$N,b$ comparison. The boxed value is the explicit ideal-FJC prediction under fixed segment geometry, not an unconditional thermal-stability prediction for the protein.

The weak-force spring constant isUsing the literal printed $N=186$, $b=7\mathrm{nm}$ at $25^{\circ }\mathrm{C}=298.15\mathrm{K}$ givesThis is a calculation with the quoted parameters, not a self-consistent fit to the plotted $26\mu \mathrm{m}$ contour. If instead $N=186$ and $L_c=26\mu \mathrm{m}$ are retained, then $b=L_c/N$ and $k=3.40\times 10^{-9}\mathrm{N}{\mathrm{m}}^{-1}$. Retaining $b=7\mathrm{nm}$ and $L_c=26\mu \mathrm{m}$ gives $k=6.79\times 10^{-8}\mathrm{N}{\mathrm{m}}^{-1}$. These distinct alternatives quantify the unresolved parameter inconsistency rather than silently choosing a correction.

The restoring force is entropic. Stretching selects a small fraction of the many disordered orientations, reducing conformational entropy. The Gaussian chain free-energy cost is $\Delta \mathcal{F}\simeq 3k_{B}T{R}^2/(2N{b}^2)$, whose gradient gives the Hooke-law coefficient above. In an ideal inextensible chain, individual bonds need not change their length to generate this force.

At fixed $N$ and $b$, the entropic spring constant is proportional to absolute temperature, because the free-energy penalty for a given loss of conformational entropy is $-T\Delta S$. ThereforeFor the literal printed chain this is $\boxed{1.287\times 10^{-6}\mathrm{N}{\mathrm{m}}^{-1}}$. The same factor multiplies either alternative contour-consistent spring constant from part (d), provided its Kuhn length is held fixed. Real DNA's persistence length, base pairing and solvent interactions can themselves vary with temperature and salt; then $b(T)$ contributes through $k\propto T/b(T{)}^2$ at fixed segment count, or $k\propto T/[L_{c}b(T)]$ if the physical contour is instead held fixed. The simple linear temperature result belongs to the specified ideal-chain model.

The nontotally symmetric longitudinal modes are the two ${\Sigma }_u^{+}$ coordinates $t$ and $a$. Both have $q_3-q_2=0$, so the middle spring is not extended. Hence its spring constant cannot alter their energies:These results are exact for every positive $\gamma$, not just perturbative. The zero mode expresses overall translation invariance.

Set $l=m=g=1$. At $({\theta }_1,{\theta }_2)=(0,0)$ the spring has $d=r$, so its potential energy has zero gradient: ${\partial }_j{V}_s=k(d-r){\partial }_{j}d=0$. The gravitational potential energy has ${\partial }_j{V}_g=\sin {\theta }_j=0$ there. Zero velocities therefore satisfy the Euler-Lagrange equations, proving this is an equilibrium.

For fixed $r>0$, let $\epsilon =\max (|{\theta }_1|,|{\theta }_2|)$ and $\delta ={\theta }_2-{\theta }_1$. The Taylor series giveThus, near the downward equilibrium, the horizontal separation is positive andIn particular, $V_s=k{\delta }^2/2+O({\epsilon }^4)$ and $V_g=-2+({\theta }_1^2+{\theta }_2^2)/2+O({\epsilon }^4)$. Dropping the additive constant and terms beyond quadratic order gives the harmonic approximationFor physical spring constant $k\ge 0$, the quadratic potential energy is a positive-definite quadratic form. The downward position is therefore a stable equilibrium in Lagrangian mechanics, and its nearby motion admits small oscillations.

Substitute the traveling normal mode $u_j=u\exp [i(kja-\omega t)]$. Combining the opposite-neighbour phases givesso the positive-frequency branch isFor positive spring constants this is a stable acoustic branch. At long wavelengths,The next-nearest-neighbour bond spans twice the distance, hence contributes the factor four. Relative to $G=0$, positive $G$ increases the sound speed by $\sqrt{1+4G/K}$. The chain still has one acoustic branch and no optical branch.

The group speed near $k=0$ and the maximum frequency need not change in the same way. The zone-edge value is always ${\omega }_0(\pi /a)=2\sqrt{K/m}$, because the next-nearest-neighbour sites then move together. For $G>K/4$ the frequency maximum moves inside the zone, with $\cos (ka)=-K/(4G)$; the small-$k$ sound speed nevertheless keeps increasing.

Put $y=e^{-r/\xi }$. Differentiating the Morse potential givesThe finite-separation minimum has $y=B/(2A)$. ThusThe positive bond-dissociation energy is $D=B^2/(4A)=0.250\mathrm{eV}$, measured relative to the separated-atom value $V(\infty )=0$. This distinguishes the well's negative potential energy from the positive energy required to break the bond.

For the displacement $x=r-a$, the Taylor expansion isConsequentlyThe restoring force is $-\kappa x$ to leading order, identifying the bond's Hooke's law spring constant.