Langevin picture of subdiffusion in nonuniformly expanding medium

Anomalous diffusion phenomena have been observed in many complex physical and biological systems. One significant advance recently is the physical extension of particle's motion in static medium to uniformly (and even nonuniformly) expanding medium. The dynamic mechanism of particle's motion in the nonuniformly expanding medium has only been investigated in the framework of continuous-time random walk. To study more physical observables and supplement the theory of the expanding medium problems, we characterize the nonuniformly expanding medium with a spatial-temporal dependent scale factor $a(x,t)$, and build the Langevin picture describing the particle's motion in the nonuniformly expanding medium. By introducing a new coordinate, besides of the existing comoving and physical coordinates, we build the relation between the nonuniformly expanding medium and the uniformly expanding one, and further obtain the moments of the comoving and physical coordinates. Both exponential and power-law formed scale factor are considered to uncover the combined effects of the particle's intrinsic diffusion and the nonuniform expansion of medium. Our detailed theoretical analyses and simulations provide the foundation for studying more expanding medium problems.