Final answer:
The generalized Stokes-Einstein equation with gradient diffusion incorporates the movement of particles due to a velocity gradient into the classic diffusional framework, expanding the traditional Stokes-Einstein relation to account for fluid flow. This adaptation makes it possible to accurately describe the dispersion of particles in a dynamic fluid environment.
Step-by-step explanation:
The generalized Stokes-Einstein equation with gradient diffusion is a fundamental concept in the field of physical chemistry and physics, particularly in the study of diffusion kinetics in fluids. The Stokes-Einstein relation correlates the diffusion coefficient D of a particle in a fluid to its size (typically expressed as the hydrodynamic radius r) and the temperature T of the fluid, incorporating the fluid's viscosity η. This equation is typically presented as D = kBT / (6πηr), where kB is Boltzmann's constant. With gradient diffusion, one must consider the contribution of a velocity gradient to the motion of particles, therefore introducing a convective derivative term to account for the movement of the particles which are attached at one end to a moving surface.
The evolution equation for a probability density function p(x, t), which describes the number of particles as a function of location and time, is then adjusted to include terms that account for convective movement due to velocity v. This inclusion allows for a more realistic description of how particles disperse in a moving fluid. For a complete and accurate solution, convolutions of the Green's function of the system with the internal force distribution also need to be considered.