It is shown that for an pi-dimensional dynamical system admitting n-linearly independent Lie symmetry vector fields, the probability density function can be found analytically in terms of these symmetries. We also show that for a dynamical system with a vector held which has constant divergence and possesses a first integral the probability density function can be written in terms of these. Moreover, the rate of entropy change can be calculated analytically for these systems. Illustrative examples are also provided. (C) 1997 Elsevier Science B.V.