The gradient flow of the polarization measure. With an appendix

The polarization measure is the probability that among 3 individuals chosen at random from a finite population exactly 2 come from the same class. This index is maximum at the midpoints of the edges of the probability simplex. We compute the gradient flow of this index that is the differential equation whose solutions are the curves of steepest ascent. Tools from Information Geometry are extensively used. In a time series, a comparison of the estimated velocity of variation with the direction of the gradient field should be a better index than the simple variation of the index.