Bounds for Fisher information and its production under flow

We prove that two well-known measures of information are interrelated in interesting and useful ways when applied to nonequilibrium circumstances. A nontrivial form of the lower bound for the Fisher information measure is derived in presence of a flux vector, which satisfies the continuity equation. We also establish a novel upper bound on the time derivative (production) in terms of the arrow of time and derive a lower bound by the logarithmic Sobolev inequality. These serve as the revealing dynamics of the information content and its limitations pertaining to nonequilibrium processes.