Extracting Active Fluctuating forces from Fluctuating motions of an active Brownian particle in a quadratic confinement

Fluctuating motions of active Brownian particles are produced by a combination of active and passive forces. Active forces carry the origin and the mechanisms of how the forces are generated. Extracting the active forces from the fluctuating motions of an active particle is nontrivial because the motions produced by the two different forces cannot be distinguished. However, if the particle is confined in a quadratic potential, we found the histogram of the fluctuating positions of the particle is a convolution of the histograms of the active and the passive positions. We hypothesize that if the histogram of the total fluctuating position and the histogram of the Brownian fluctuations are both known, one can extract the histogram of the pure active fluctuation by deconvolution. Because the convolution hypothesis is based on that the force on a particle is a vector sum of the passive and active forces, the validity of the deconvolution procedures will depend on whether they are time scale invariant – typical experimental sampling rates of the particle position are often much longer than that of the molecular fluctuations. Electrophoresis-driven Janus particles in an optical trap and Langevin equation-based numerical simulations were conducted to examine the hypothesis and the effects of sampling time to the deconvolution method. Both the experiments and the Langevin simulation show the deconvolution procedures work well and they are indeed time scale invariant. In conclusion, we extracted active forces from the histogram of the fluctuating position of an active particle in quadratic confinement by deconvolution.