  We show that an information theoretic distance measured by the relative
Fisher information between canonical equilibrium phase densities corresponding
to forward and backward processes is intimately related to the gradient of the
dissipated work in phase space. We present a universal constraint on it via the
logarithmic Sobolev inequality. Furthermore, we point out that a possible
expression of the lower bound indicates a deep connection in terms of the
relative entropy and the Fisher information of the canonical distributions.
