= Thermal calibration of an optical trap
{title2=$\sigma^2=k_BT/k$}
A bead in a liquid keeps undergoing <Brownian motion> even when an <optical trap> holds its average position fixed. For a stationary <harmonic optical trap> in <thermal equilibrium> at absolute <temperature> $T$, the <equipartition theorem> assigns mean <potential energy> $k_BT/2$ to this one quadratic coordinate. With $k_B$ the <Boltzmann constant> and $x=0$ at <equilibrium>,
$$
\langle x\rangle=0,\qquad \sigma^2=\operatorname{Var}(x)=\langle x^2\rangle=\frac{k_BT}{k}.
$$
At the same <temperature>, quadrupling the <spring constant> divides the position <variance> by four and the <standard deviation> by two. The <Boltzmann factor> gives the complete equilibrium <probability density function>, a <Gaussian distribution>:
$$
p(x)=\sqrt{\frac{k}{2\pi k_BT}}\exp\!\left(-\frac{kx^2}{2k_BT}\right),\qquad -\infty<x<\infty.
$$
The prefactor makes $\int_{-\infty}^{\infty}p(x)\,dx=1$; its second moment is $\int_{-\infty}^{\infty}x^2p(x)\,dx=k_BT/k$. The <probability density function> has units of inverse length, and its area gives <probability>. This density is exact for the ideal quadratic model. In a finite <optical trap> it approximates the occupied well when thermal excursions remain within the harmonic region and escape is negligible during observation. Driven motion, appreciable nonconservative <forces> or <laser> heating need not give this equilibrium density.
\Image[/optical-trap-thermal-position.png]
{title=Thermal force calibration model: exact Gaussian position densities for relative trap stiffnesses 1, 2 and 4 at a common temperature}
Here $\ell=\sqrt{k_BT/k_0}$, $q=x/\ell$ and $\kappa=k/k_0$. The plotted density per unit $q$ is $p_q(q)=\ell p(\ell q)=\sqrt{\kappa/(2\pi)}e^{-\kappa q^2/2}$. Each curve has total area one; their dimensionless <standard deviations> are $1$, $1/\sqrt2$ and $1/2$. The curves are exact model densities, not noisy experimental data.
<Brownian motion> becomes a measuring tool: a calibrated position measurement supplies $\sigma^2=\langle(x-\langle x\rangle)^2\rangle$, and the known <temperature> supplies $k=k_BT/\sigma^2$. Once $k$ is known, a slowly applied constant external <force> $f$ shifts the mean by $\langle x\rangle=f/k$, so $f=k\langle x\rangle$ in the same harmonic range. Fluctuations therefore calibrate the conversion from <displacement> to <force>. Svoboda and Block used this <equipartition theorem> method when calibrating their bead traps. https://www.andrew.cmu.edu/course/42-620/handouts/svoboda_cell_94.pdf[Svoboda and Block, 1994, trap stiffness calibration]{ref}.
For the example stiffness $0.01\,\mathrm{pN/nm}$ and $T=300\,\mathrm K$, $k_BT\approx4.142\,\mathrm{pN\,nm}$ and $\sigma\approx20.35\,\mathrm{nm}$. Using a <variance> measured in uncalibrated detector units would not determine that physical stiffness. Measurement noise, finite exposure time, drift and insufficient sampling of the correlated <Brownian motion> can bias the estimate; record positions over many <optical trap relaxation times> and account for the detector response. The deterministic relaxation curve above describes relaxation without thermal forcing; an individual thermal trajectory continues to fluctuate.
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