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