Studying the Finiteness of a Tracking Technique with Random Distances and Velocities that Reduces the Collision Time Between a Brownian Particle and a Nanosensor in the Fluid

The tracking technique that is examined in this study considers the nanosensor’s velocity and distance as independent random variables with known probability density functions (PDFs). The nanosensor moves continuously in both directions from the starting point of the real line (the line’s origin). It oscillates while traveling through the origin (both left and right). We provide an analytical expression for the density of this distance using the Fourier-Laplace representation and a sequence of random points. We can take the tracking distance into account as a function of a discounted effort-reward parameter in order to account for this uncertainty. We provide an analytical demonstration of the effects this parameter has on reducing the expected value of the first collision time between a nanosensor and the particle and confirming the existence of this technique.