Nonlinear quantum evolution with maximal entropy production

Abstract

We derive a well-behaved nonlinear extension of the nonrelativistic Liouville--von Neumann dynamics driven by maximal entropy production with conservation of energy and probability. The pure-state limit reduces to the usual Schr\"odinger evolution, while mixtures evolve toward maximum entropy equilibrium states with canonical-like probability distributions on energy eigenstates. The linear, near-equilibrium limit is found to amount to an essentially exponential relaxation to thermal equilibrium; a few elementary examples are given. In addition, the modified dynamics is invariant under the time-independent symmetry group of the Hamiltonian, and also invariant under the special Galilei group provided the conservation of total momentum is accounted for as well. Similar extensions can be generated for, e.g., nonextensive systems better described by a Tsallis q entropy.