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Equipartition theorem

An optically trapped bead in thermal equilibrium moves randomly within the optical trap. Because of the optical force, the natural Brownian movement is confined in the trap region, near the objective focus.

Thermally driven position fluctuations can give us information on the shape of the trapping potential, and we can use it to calibrate the optical trap. In equilibrium we expect the probability density of the particle position to be established by Boltzmann statistics. If we consider a two-dimensional movement:

where kB is the Boltzmann constant, T is the absolute temperature, and C is a normalization constant. This means that Brownian particles spend more time in positions where the optical potential energy is lower. If we take an statistically significant number of position measurements, the histogram of positions will have a similar shape than the spatial probability density described in (1). Then the trap potential U(x,y) can be obtained and fitted to a model in order to obtain some trap characteristic parameters. Trap potential is usually modeled as a harmonic potential because the trapping force scales linearly with thebead position near the trap center (spring-like force).


The expected probability density is a Gaussian function:



where kx and ky, are the trap stiffness's in the axis directions, and (x0, y0) is the equilibrium position. If we know the temperature, the width of the Gaussian-shaped histogram is directly related with the trap stiffness:



This relation can be very useful to get a fast estimation of the trap parameters in real time. However, the results are not very accurate because the harmonic model (and thus the Gaussian probability distribution) fails for particle positions far from the center.

Another possibility is obtaining the trap potential from the histogram function, and fitting a parabola just in the central region of the trap potential, where we can actually say that the potential is harmonic.

The outer part of the potential cannot fit the harmonic model because the force looses its linearity with displacement away from the centre. However, this non-harmonic region of the optical potential is also interesting to know information about the force in the outer part of the trap. There the force is not linear with the distance so it cannot be characterized with just a stiffness parameter, but it can be known from the potential.

The optical potential reconstruction using Boltzmann statistics can be used to describe any continuous optical trapping landscape in the region accessible by thermal agitation, and the only necessary parameter is the sample temperature. The problem is that any Gaussian noise will make the experimental histograms wider, and that will give softer trap stiffness than the real ones. For more accurate trap stiffness measurments it's highly recommended to use the Power Spectrum analyisis.