Emergence of order in selection-mutation dynamics.

We characterize the time evolution of a d-dimensional probability distribution by the value of its final entropy. If it is near the maximally possible value we call the evolution mixing, if it is near zero we say it is purifying. The evolution is determined by the simplest nonlinear equation and contains a d x d matrix as input. Since we are not interested in a particular evolution but in the general features of evolutions of this type, we take the matrix elements as uniformly distributed random numbers between zero and some specified upper bound. Computer simulations show how the final entropies are distributed over this field of random numbers. The result is that the distribution crowds at the maximum entropy, if the upper bound is unity. If we restrict the dynamical matrices to certain regions in matrix space, to diagonal or triangular matrices, for instance, then the entropy distribution is maximal near zero, and the dynamics typically becomes purifying.