Narrow Results

We consider birth-and-death stochastic particle systems in continuum which are under a self-regulation mechanism controlling configurations of particles via a pairwise interaction between them. The latter is reflected in a potential perturbation of the free generator. We show that the ground state renormalization scheme in the considered model leads to an invariant measure, a renormalized generator and resulting equilibrium birth-and-death stochastic dynamics for the system. The proof is based on the Gibbs-type representation for related path space measure. This measure has OS-positivity property and is constructed via the cluster expansion method.

A homopolar motor-like system is investigated in this work, where a cylindrical battery is attached with button magnets of different diameters at both ends, and the system is placed on an aluminium foil, resulting in a circular motion. The physics behind the system is discussed and elaborated qualitatively and quantitatively in this paper, including the electromagnetic field generated by the magnets and the battery, and the relevant rotational and translational dynamics. Due to the complicated vector calculus and visualization required, electromagnetism is often considered counter-intuitive and challenging even for undergraduate students to comprehend. This paper aims to enhance the understanding of such systems without relying on very complex mathematics or visualizations of the various fields and forces.