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${)}_{\max }$=1/6I(t)d/dt〈${\mathit{r}}^2$(t)〉, where I(t) is the Fisher information about the centroid of the system, and 〈${\mathit{r}}^2$(t)〉 is the time-dependent mean-square particle position. A corollary is that, for classical particle flow obeying 〈r〉=0, positional uncertainty σ(t) must ever increase with time.