LIMITATION ON ENTROPY INCREASE IMPOSED BY FISHER INFORMATION

Consider a system obeying conservation of flow, as in classical particle flow or in relativistic quantum mechanics. In such cases a probability density function p(r\t) may be used to describe the system, where r is particle position and t is time. Let H(t) be the Shannon form of the Boltzmann entropy corresponding to p (r\t). It is found that (dH/dt)max = 1/6 I(t)d/dt [r2(t)], where I(t) is the Fisher information about the centroid of the system, and [r2(t)] is the time-dependent mean-square particle position. A corollary is that, for classical particle flow obeying [r] = 0, positional uncertainty sigma(t) must ever increase with time.