A Comprehensive Derivation of the Diffusion Equation from a Microscopic Random Walk Model

This manuscript provides a detailed theoretical exposition of the random walk model for atomic diffusion in crystalline solids. Beginning from the atomistic description of thermally activated jumps between lattice sites, we derive the mean square displacement and establish its linear dependence on time a hallmark of normal diffusion. Using combinatorial and statistical methods, we obtain the probability distribution of atomic displacements after a given number of jumps, demonstrating its convergence to a Gaussian form in the long-time limit. Finally, we rigorously connect the stochastic random walk model to the deterministic diffusion equation, deriving the diffusivity 𝐷 in terms of microscopic parameters: jump frequency 𝑘jump and lattice spacing 𝑎. The work serves as a pedagogical yet rigorous bridge between discrete atomic kinetics and continuum mass transport.