Trajectory versus probability density entropy - art. no. 016223

We show that the widely accepted conviction that a connection can be establ ished between the probability density entropy and the Kolmogorov-Sinai (KS) entropy is questionable. We adopt the definition of density entropy as a f unctional of a distribution density whose time evolution is determined by a transport equation, conceived as the only prescription to use for the calc ulation. Although the transport equation is built up for the purpose of aff ording a picture equivalent to that stemming from trajectory dynamics, no d irect use of trajectory time evolution is allowed, once the transport equat ion is defined. With this definition in mind we prove that the detection of a time regime of increase of the density entropy with a rate identical to the KS entropy is possible only in a limited number of cases. The proposals made by some authors to establish a connection between the two entropies i n general, violate our definition of density entropy and imply the concept of trajectory, which is foreign to that of density entropy.