Abstract
Recent developments in the measurement of radioactive gases in passive diffusion motivate the analysis of Brownian motion of decaying particles, a subject that has received little previous attention. This paper reports the derivation and solution of equations comparable to the Fokker-Planck and Langevin equations for one-dimensional diffusion and decay of unstable particles. In marked contrast to the case of stable particles, the two equations are not equivalent, but provide different information regarding the same stochastic process. The differences arise because Brownian motion with particle decay is not a continuous process. The discontinuity is readily apparent in the computer-simulated trajectories of the Langevin equation that incorporate both a Wiener process for displacement fluctuations and a Bernoulli process for random decay. This paper also reports the derivation of the mean time of first passage of the decaying particle to absorbing boundaries. Here, too, particle decay can lead to an outcome markedly different from that for stable particles. In particular, the first-passage time of the decaying particle is always finite, whereas the time for a stable particle to reach a single absorbing boundary is theoretically infinite due to the heavy tail of the inverse Gaussian density. The methodology developed in this paper should prove useful in the investigation of radioactive gases, aerosols of radioactive atoms, dust particles to which adhere radioactive ions, as well as diffusing gases and liquids of unstable molecules.
Highlights
Two Approaches to Brownian MotionBrownian motion, one of the simplest examples of a random walk, is a nonequilibrium statistical process the mathematics of which serves to model a wide variety of stochastic processes throughout the physical and social sciences
This paper reports the derivation and solution of equations comparable to the Fokker-Planck and Langevin equations for one-dimensional diffusion and decay of unstable particles
Motivated by new experimental methods to measure radioactive atoms and ions diffusing in gas, on dust, or in liquids, this paper analyzed the Brownian motion of decaying particles, a process that has received relatively little prior attention
Summary
One of the simplest examples of a random walk, is a nonequilibrium statistical process the mathematics of which serves to model a wide variety of stochastic processes throughout the physical and social sciences. Because unstable particles are removed from the system at random instances during the time they are moving randomly within a specified sample space, the familiar standard equations (e.g. Langevin and Fokker-Planck) for Brownian motion must be re-derived on the basis of a conservation law that takes particle loss into account. This more general relation, presented, leads to significant differences in the analytical structure of the stochastic equations and statistical moments compared with corresponding expressions derived for the random walk of a stable particle.
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