Zusammenfassung
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 non-linear
equation and contains a $d d$ matrix as input
(quasi-species dynamics first introduced by M. Eigen and P. Schuster).
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, for instance to diagonal or
triangular matrices, then the entropy distribution is maximal near zero, and
the dynamics typically becomes purifying.
We also consider the quantum-mechanical analogue of this evolution
and demonstrate that - in contrast to the classical case - the quantum
evolution is generally purifying. As an example we study the evolution in
two and three dimensional Hilbert spaces. These results are also compared to
analogous results for the Lindblad dynamics, which provides a consistent
description of a dissipative quantum system.
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