Fisherʹs information measure in a Tsallisʹ nonextensive setting and its application to diffusive process

Fisherʹs information measures, as adapted to a nonextensive (Tsallis) environment, are discussed. For systems of particles that are in a general state of motion a lower bound to these information measures is derived with the help of a recently established upper bound to the entropy increase. This lower bound to the information measure is the basis for a variational principle devised to determine unknown probability distributions. In the important instance of diffusion process we show that trying to ascertain which is the probability distribution that maximizes Fisherʹs information sheds some light on the meaning of Tsallisʹ q-parameter. We discuss applications to cosmological models that seem to suggest that a non-extensive thermostatistics with q = −1 provides an adequate scenario for discussing gravitation.