On the interpretation of mean-square displacement in heterogeneous systems

Abstract When dealing with heterogeneous systems, interpreting the mean-square displacement (MSD) data appropriately is crucial. This is due to the limitation that the various interpretations of stochastic differential equations with state-dependent diffusivity may yield the same MSD data, leading to erroneous conclusions. Aided by analytical MSD solutions supported by Langevin simulations, we explore this limitation in diffusion processes exhibiting power-law state-dependent diffusivity, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>L</mml:mi> <mml:msup> <mml:mo stretchy="false">|</mml:mo> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mi>μ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , with diffusion constant D 0 and scaling index µ . We find that distinct interpretations leading to same MSD data possess different D 0 values. A method is introduced to discern these D 0 -dependent interpretations. When applying the method to subdiffusion MSD data, we observe interpretation transitions depending on the value of the anomalous diffusion coefficient. Furthermore, we demonstrate that these D 0 -dependent interpretations respond diversely to memory effects, which may also be used to decide on the interpretation of MSD data. We believe that these findings offer insights for interpreting MSD data in heterogeneous systems.