One of the central results in Einstein’s theory of Brownian motion is that the meansquare displacement of a randomly moving Brownian particle scales linearly with time. Overthe past few decades sophisticated experiments and data collection in numerous biological,physical and financial systems have revealed anomalous sub-diffusion in which the meansquare displacement grows slower than linearly with time. A major theoretical challengehas been to derive the appropriate evolution equation for the probability density functionof sub-diffusion taking into account further complications from force fields andreactions. Here we present a derivation of the generalised master equation for an ensembleof particles undergoing reactions whilst being subject to an external force field. Fromthis general equation we show reductions to a range of well known special cases, includingthe fractional reaction diffusion equation and the fractional Fokker-Planck equation.
