Understanding the Dynamics of Embedded Objects in Inelastic Particle Systems

When a small object embedded in a liquid or a gas, it experiences bombardment byfast?moving atoms or molecules. This leads to random motion of the embedded object.Such random motion is called Brownian motion. Brownian motion has been widely usedto model phenomena that involve random behavior. Brownian motion is the mostfundamental building block in the mathematics of stochastic analysis. Over more than acentury of development, many theoretical tools have been established to study and toanalyze the Brownian motion. These include Maxwell distribution, diffusion equation, Itocalculus and martingale. However, none of these theoretical tools is applicable when oneconsiders an object embedded in inelastic particle systems. Although phenomenologicallythe motion of the embedded object is also random due to the bombardment by theinelastic particles, the dynamics is very different from Brownian motion. Our preliminarystudy showed that even in the absence of external forces, the average net force exertedon the embedded object by the inelastic particles can be non?zero. This is in sharpcontrast to the corresponding cases in fluids or in gases. In fluids and gases, in theabsence of external forces, the averaging net force on the embedded objects is zero andthis property is independent of the shape of the object. In this proposal, we will studysuch surprising phenomena and investigate the fundamental mechanism that leads tosuch phenomenon.