Diffusing diffusivity model of a polymer moving on a spherical surface

The movement of a polymer is modeled by Brownian motion accompanied with a fluctuating diffusion coefficient when the polymer is in contact with a chemostatted monomer bath triggering the chain polymerization, which is called a diffusing diffusivity (DD) model. In this paper, we extend the DD model from three dimensional Euclidean space to a two dimensional spherical surface. The DD model on the spherical surface is described by a coupling Langevin system in the directions of longitude and latitude, while the diffusion coefficient is characterized by the birth and death chain. Then, the Fokker-Planck and Feynman-Kac equations for the DD model on the spherical surface, respectively, governing the probability density functions (PDFs) of the two statistical observables, position and functional, are derived. Finally, we use two ways to calculate the PDFs of some statistical observables, i.e., applying a Monte Carlo method to simulate the DD model and a spectral method to solve the Fokker-Planck and Feynman-Kac equations. In fact, the unification of the numerical results of the two ways also confirms the correctness of the built equations.