We study the problem of counting the number of particles in a closed volume where the particles motion are modeled by a Brownian motion process. This problem arises in many biological and chemical sensing experiments, e.g., counting the number of analytes in blood sample, air pollutant measurement, etc. Finding the exact count of particles is challenging in these systems and one generally relies on an estimate based on the sample readouts. We study the statistical properties of the counting process in equilibrium and present the fundamental detection and estimation limitations. In particular, we demonstrate that the count process is inherently noisy and has a quantum-limit signal to noise ratio.
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