Quantum dynamical semigroups and the neutron diffusion equation

D.S. Bobker

Anton Arnold, Christof Sparber

Sergio A Albeverio, Raphael J. Høegh-Krohn

Yu. N. Orlov, Vsevolod Zhanovich Sakbaev

We investigate a model that associates random walks in a finite-dimensional Euclidean coordinate space of a classical system with random quantum walks, i.e. random transformations of the set of states of a quantum system arising from quantization of a classical system. As is known, the convolution semigroup of Gaussian measures on a coordinate space admits a representation by a semigroup of self-adjoint contractions in the space of square-integrable functions described by the heat equation. We obtain a representation of the convolution semigroup of Gaussian measures on a coordinate space by a quantum dynamic semigroup in the space of nuclear operators. We give a description of the quantum dynamic semigroup by solutions of the Cauchy problem for a degenerate diffusion equation. We establish the generalized convergence in distribution of a sequence of quantum random walks to an operator-valued random process with values in the Abelian algebra of shift operators by a vector with a normal distribution.

Robert Alicki, K. Lendi

Anton Arnold, Christof Sparber

We consider a class of evolution equations in Lindblad form, which model the dynamics of dissipative quantum mechanical systems with mean-field interaction. Particularly, this class includes the so-called Quantum Fokker-Planck-Poisson model. The existence and uniqueness of global, mass preserving solutions is proved, thus establishing the existence of a nonlinear conservative quantum dynamical semigroup.