  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.
