%0 Book Section %1 statphys23_0753 %A Posch, H.A. %A Thirring, W. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K biological computer dynamics evolution lindblad mechanical quantum quasi-species simulation statphys23 topic-10 %T Emergence of order in selection-mutation dynamics %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=753 %X 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.

@incollection{statphys23_0753, abstract = {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 $d \times d$ matrix as input ({\em 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.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Posch, H.A. and Thirring, W.}, biburl = {https://www.bibsonomy.org/bibtex/245c01908dc73e032252284e72e2005a5/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {0b53e146acbaf4a4ab755987974f83c9}, intrahash = {45c01908dc73e032252284e72e2005a5}, keywords = {biological computer dynamics evolution lindblad mechanical quantum quasi-species simulation statphys23 topic-10}, month = {9-13 July}, timestamp = {2007-06-20T10:16:28.000+0200}, title = {Emergence of order in selection-mutation dynamics}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=753}, year = 2007 }