Diffusion as a Series of Discrete Jumps

In general, a particle migrates in a material by a series of thermally activated jumps between positions of local energy minima. Macroscopic diffusion is the result of all the migrations executed by a large ensemble of particles. The spread of the ensemble due to these migrations connects the macroscopic diffusivity to the microscopic particle jumping. 

If a particle jumps with average frequency T in a sequence of displacements fi (fi gives the magnitude and direction of the ith jump), then after a period of time, r, the particle will execute Nr = Tr individual jumps.4 The position relative to its starting point is the sum of the individual displacements, 

The random walk process can be characterized by the distribution of total displacements for either a large set of noninteracting walkers or for repeated trials of an isolated walker. The average displacement is a vector (R Nr))a.nd the mean-square displacement (R(NT) R(NT) = (R2(Nr)) is a scalar that characterizes the spread or diffuseness of the distribution of total displacements about its average. 

The square displacement for a given sequence of random steps is 

4r represents the average total jump rate during the sequence, which generally consists of jumps of different lengths and directions. This contrasts with T, which represents the jump rate between two specified sites. 

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