Probability distribution of ages of the nearest particle pairs

We define the age of a nearest particle pair as the difference between current time and the time when the particles becomes the nearest pair. Such defined age varies from pair to pair. A statistical method is used to study them. Similar to the human society, we then can compute the average age of the nearest particle pairs at a given time in a flow. We can also investigate the life expectancy of the nearest pairs, which is the average age when the nearest particle in the pair is replaced by another particle. Evolution equation of the age included nearest particle probability distribution function is derived. Under the assumption of random destruction of the nearest pairs, we find the age distribution of nearest particle pairs is exponential. As consequences, the average age, the life expectancy, and the decay time of the probability distribution are then the same.

This theoretical prediction is confirmed by particle resolved simulations of sedimentation flow under gravity. The relation between this age distribution of the nearest particle pairs and the well-known drafting-kissing-tumble motion will be discussed.