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 a static medium to a uniformly and even nonuniformly expanding medium. The dynamic mechanism of the anomalous diffusion in the nonuniformly expanding medium has only been investigated by the approach of continuous-time random walk. To study more physical observables and to supplement the physical models of the anomalous diffusion in the expanding mediums, we characterize the nonuniformly expanding medium with a spatiotemporal dependent scale factor a ( x , t ) and build the Langevin picture describing the particle’s motion in the nonuniformly expanding medium. Besides the existing comoving and physical coordinates, by introducing a new coordinate and assuming that a ( x , t ) is separable at a long-time limit, 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. Different forms of the scale factor a ( x , t ) are considered to uncover the combined effects of the particle’s intrinsic diffusion and the nonuniform expansion of medium. The theoretical analyses and simulations provide the foundation for studying more anomalous diffusion phenomena in the expanding mediums.