Brownian Motion of Decaying Particles: Transition Probability, Computer Simulation, and First-Passage Times

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.

Bending of a Tapered Rod: Modern Application and Experimental Test of Elastica Theory

“…From the numerical solution ( ) θ σ , the corresponding elastica curve ( ) η ξ was generated parametrically by integration of Equation 12using what is termed an update equation [25]. Figure 5 and Figure 6 show the solutions ( ) η ξ , corresponding respectively to the conditions of Figure 3…”

“…where use has been made of Equation (12). While it is often advantageous to have closed-form solutions to a problem, it should be noted that an alternative and computationally more efficient way to obtain the elastica coordinates ( ) , ξ η from the derivative (39) is to apply again the iterative algorithm (25). It takes much more computer time to evaluate the elliptic integrals (41) and (42) than to process the update algorithm even for N on the order of 1000.…”

Brownian Motion of Radioactive Particles: Derivation and Monte Carlo Test of Spatial and Temporal Distributions

“…In contrast to the FPE, the Langevin equation (LE) expresses the time-variation of a process random variable such as displacement or velocity, rather than the transition probability density. In the case of 1D Brownian motion of a decaying particle, the LE derived in [6] takes the form of an update stochastic differential equation [7] [8] based on a Wiener process [9] [10]…”

“…Thus, s p defined in Equation 4, is the probability of surviving the interval dt. The analytical solution to Equations ((3) and (4)) for a decaying particle undergoing k discrete one-dimensional Brownian displacements in time t at intervals of dt was shown in [6] to be ( ) ( )…”

“…Note that the expression in (5) is not a function, but a statistical distribution (Gaussian) whose variance Σ that was derived and explained in [6] (and will be used in Section 2.2).…”