= Solution
Only the separable instrument and mechanical tasks are treated here; biological operation and stepping energetics remain deferred.
For generic <optical tweezers>, a tightly focused laser and objective with high <numerical aperture> create an intensity gradient that attracts a suitable dielectric bead toward the focus. <Momentum transfer> also creates a scattering <force>; stable three-dimensional trapping requires sufficient restoring gradient <force>. A condenser and photodetector or camera track displacement. Near a stable focus, $F_x=-kx$ defines the stiffness and requires both a position calibration and a <force> calibration.<Observation of a single-beam gradient force optical trap for dielectric particles by Arthur Ashkin and others>[Single-beam gradient trapping of dielectric particles]{ref}
At thermal equilibrium in a harmonic well, the <equipartition trap calibration> is $k\langle(x-\langle x\rangle)^2\rangle=k_BT$. Alternatively, fitting the <overdamped harmonic-trap power spectrum> gives $f_c=k/(2\pi\gamma)$, hence $k=2\pi\gamma f_c$. Detector noise, finite sampling and exposure time, wall corrections to drag, and drift must be controlled rather than attributed to stiffness.
For an isolated sphere in a Newtonian fluid, define $\mathrm{Re}=\rho_fv(2a)/\eta$ using its diameter. The supplied <viscosity> converts to $10^{-3}\,\mathrm{Pa\,s}$. The generic <Stokes drag> calculation gives
$$
\boxed{F_D=6\pi\eta av=1.88\times10^{-14}\,\mathrm N=0.0188\,\mathrm{pN},\qquad\gamma=6\pi\eta a=1.88\times10^{-8}\,\mathrm{N\,s\,m^{-1}}.}
$$
Here $\mathrm{Re}=2\times10^{-6}[\rho_f/(10^3\,\mathrm{kg\,m^{-3}})]$; this retains the unspecified fluid <density> as a parameter rather than treating a <density> as supplied data. <Stokes drag> assumes an isolated no-slip sphere; nearby surfaces change the coefficient. TODO: the requested assessment of biological motor operation is deferred.
TODO: the biological stepping-energy and step-detection estimate is deferred in its entirety; no biological energy source or motor mechanism has been researched.
Ignoring biological coupling, the mechanical thermal model is the <Langevin equation> for a harmonic trap,
$$
m\ddot x+\gamma\dot x+kx=\xi(t),\qquad\langle\xi(t)\rangle=0,\qquad\langle\xi(t)\xi(t')\rangle=2\gamma k_BT\delta(t-t').
$$
For independent repeats with the specified initial conditions, the ensemble <mean> $X=\langle x\rangle$ obeys the homogeneous damped-oscillator equation. An average over $N$ independent zero-<mean> <force> samples reduces its <variance> by $N$, with a finite detector bandwidth needed to define samples of white noise. \b[High bandwidth alone does not remove random forces; averaging independent records or sufficiently separated samples is the additional operation.] A single trajectory retains <thermal fluctuations>. For stationary overdamped position fluctuations, a long time average over duration $\mathcal T\gg\tau_k=\gamma/k$ has approximately $\operatorname{Var}(\bar x)\simeq2(k_BT/k)\tau_k/\mathcal T$.
The <initial-value solution of a damped harmonic trap> has characteristic roots $r_\pm=[-\gamma\pm\sqrt{\gamma^2-4mk}]/(2m)$. For distinct roots,
$$
\boxed{X(t)=\frac{v_0-r_-x_0}{r_+-r_-}e^{r_+t}+\frac{r_+x_0-v_0}{r_+-r_-}e^{r_-t}.}
$$
For critical damping, $\gamma^2=4mk$,
$$
X(t)=e^{-\gamma t/(2m)}\left[x_0+\left(v_0+\frac{\gamma x_0}{2m}\right)t\right].
$$
For the underdamped case the same real solution can be written $e^{-\gamma t/(2m)}[x_0\cos\omega_dt+(v_0+\gamma x_0/(2m))\sin\omega_dt/\omega_d]$, with $\omega_d^2=k/m-\gamma^2/(4m^2)$. These describe the deterministic response or conditional <mean>, not the entire noisy trajectory.
For <overdamped trap relaxation after an inertial boundary layer>, let $\epsilon=km/\gamma^2\ll1$. The slow and fast roots are
$$
r_s=-\frac k\gamma[1+O(\epsilon)],\qquad r_f=-\frac\gamma m[1+O(\epsilon)],\qquad\tau_m=\frac m\gamma\ll\tau_k=\frac\gamma k.
$$
After $t\gg\tau_m$, the fast exponential is negligible and the surviving amplitude is $x_0+(m/\gamma)v_0+O(\epsilon x_0,\epsilon mv_0/\gamma)$. If also $|(m/\gamma)v_0|\ll|x_0|$, then to leading overdamped order
$$
\boxed{X(t)\simeq x_0e^{-kt/\gamma}.}
$$
Small <Reynolds number> justifies the linear viscous <force>; it does not alone prove the separate small inertial-trap parameter $\epsilon$. Neglecting $v_0$ requires both rapid <momentum> relaxation and a negligible slow-mode amplitude shift. If $x_0=0$, a nonzero initial <velocity> can leave a small slow displacement even after its own fast transient decays. The result concerns generic bead mechanics only.
\Image[/past-exam-of-the-soft-matter-and-biological-physics-course-of-the-university-of-cambridge/2015/paper-1-trap-relaxation.png]
{title=Fast momentum relaxation leaves a slower trap mode, whose amplitude can still remember a sufficiently large initial velocity}
{source=paper-1-trap-relaxation.py}
Back to article page
