Diffusion and random walks

Diffusion describes the spreading of particles or quantities through random motion; random walks are mathematical models in which successive steps are chosen probabilistically.

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Random walk: A stochastic process formed by successive steps whose directions or sizes are chosen randomly. It is the basic discrete model whose accumulated steps produce spreading.
Probability distribution: A mathematical description assigning probabilities to the possible values of a random variable. Diffusion tracks how probability distributions change across space and time.
Brownian motion (physics): The irregular motion of microscopic particles suspended in a fluid, driven by molecular collisions. It is the observable particle motion that helped establish molecular explanations of diffusion.
Anomalous diffusion: Transport in which mean squared displacement grows with time at a rate different from linear. It contrasts with ordinary diffusion's linear mean-squared-displacement growth.
Brownian motion: A continuous-time stochastic process with continuous paths and independent Gaussian increments. It models the continuous limit of random walks and particle motion.
Stochastic process: A collection of random variables indexed by time, space, or another parameter. Random walks and Brownian motion are stochastic processes.
Heat equation: A partial differential equation describing the evolution of temperature by thermal conduction. Heat spreads mathematically like a diffusing quantity.
Lévy flight: A random walk whose step lengths follow a heavy-tailed distribution and can include exceptionally long jumps. Long jumps replace the short-step assumptions behind ordinary diffusion.
Diffusion equation: A partial differential equation describing how a concentration or probability density spreads over time. It captures the macroscopic evolution generated by random microscopic motion.
Markov property: The property that a process's future, given its present, is independent of its past. Many random-walk models choose each next step using only the current state.