We study an active Brownian run-and-tumble particle (ABRTP) model that consists of an active Brownian run state, during which the active velocity of the particle diffuses on the unit circle, and a tumble state, during which the active velocity is zero, both with exponentially distributed time. Additionally, we add a harmonic trap as an external potential. In the appropriate limits the ABRTP model reduces either to the active Brownian particle model or the run-and-tumble particle model. Using the method of direct integration the equation of motion, pioneered by Kac, we obtain exact moments for the Laplace transform of the time-dependent ABRTP in the presence or absence of a harmonic trap. In addition, we estimate the distribution moments with the help of the Chebyshev polynomials. Our results are in excellent agreement with the experiments.

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• Brownian motion
• Active Brownian particles
• Self-propelled particles
• Langevin equation
• Stochastic differential equations