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

Abstract

Stochastic processes such as diffusion can be analyzed by means of a partial differential equation of the Fokker-Planck type (FPE), which yields a transition probability density, or by a stochastic differential equation of the Langevin type (LE), which yields the time evolution of a statistical process variable. Provided the stochastic process is continuous and certain boundary conditions are met, the two approaches yield equivalent information. However, Brownian motion of radioactively decaying particles is not a continuous process because the Brownian trajectories abruptly terminate when the particle decays. Recent analysis of the Brownian motion of decaying particles by both approaches has led to different mean-square displacements. In this paper, we demonstrate the complete equivalence of the two approaches by 1) showing quantitatively and operationally how the probability densities and statistical moments predicted by the FPE and LE relate to one another, 2) verifying that both approaches lead to identical statistical moments at all orders, and 3) confirming that the analytical solution to the FPE accurately describes the Brownian trajectories obtained by Monte Carlo simulations based on the LE. The analysis in this paper addresses both the spatial distribution of the particles (i.e. the question of displacement as a function of diffusion time) and the temporal distribution (i.e. the question of first-passage time to fixed absorbing boundaries).

Mortal Brownian motion: Three short stories

“…The model of mortal Brownian motion is relevant for a number of practical issues, including diffusion of radioactive materials [5,7,8]. But it is also interesting in a more general context.…”

“…In the recent years there has been growing interest in the effects of a finite lifetime of particles on statistical properties of random walk and of its continuous limit, the Brownian motion [1][2][3][4][5][6][7][8]. Practical examples are found in physics, chemistry and biology and vary from diffusion of radioactive gases such as radon [5,7,8] to a variety of search problems such as the search for an oocyte by sperm [4]. Additional motivation comes from non-equilibrium statistical mechanics: The particle mortality breaks detailed balance and even makes any steady state impossible without a particle source.…”

Experimental and Modeling Investigation of the Mechanism for Preventing Readhesion via Zeta Potential in the Spin-Rinse Process

“…To take the effect of the particle behavior that depends on the zeta potential into account, the existence probability in the z direction of the particles after receiving the repulsive force was calculated. Given that the particle location due to Brownian motion is often expressed in terms of a probability density function, 20,21 the probability density of the particle after undergoing the repulsive force was estimated, as shown in Fig. 7, with the assumption that the particle distribution simply takes the form of a Gaussian distribution for the modeling.…”

Application value of the treatment of breast cancer bone metastases with radioactive seed 125I implantation under CT-guidance

“…With the increasing pressure of survival and the quickening of pace of life, the incidence rate of breast cancer is increasing. It mostly occurs in middle-aged and elderly people, and it is developing in a younger trend, which seriously endangers people's security of life [ 1 , 2 ]. In recent years, the treatment of breast cancer can be divided into seed implantation, radiotherapy and chemotherapy, surgical drug therapy and so on.…”

“…Although effective treatment can be achieved, there are many postoperative complications. Moreover, there are many untoward reactions after radiotherapy and chemotherapy, for example, bone marrow suppression, nausea and vomiting, which affect the quality of life of the patients [ 2 – 4 ]. With the constant improvement of science and technology and the advancement of medical technology, it has gradually become a mainstream topic to further improve the efficacy of treatment and reduce the adverse reactions of patients after treatment, which is also the current research direction of scholars.…”