Abstract
The notion of a nonlinear quantum dynamical semigroup is introduced, and the existence and uniqueness of solutions of the corresponding nonlinear evolution equations are studied in a more abstract framework. The construction of nonlinear quantum dynamical semigroups is carried out for two different mean-field models. First a mean-field coupling between a system of noninteracting subsystems and the bath is investigated. As examples, a nonlinear frictional Schrödinger equation and a model for a quantum Boltzmann equation are discussed. Second, a many-body system with mean-field interaction coupled to a bath is considered. Here, again, the form of the generator is derived; however, it cannot be obtained rigorously, except for some particular examples. Finally, the quantum Ising-Weiss model is briefly studied.
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Alexander von Humboldt fellow. On leave of absence from Institute of Theoretical Physics and Astrophysics, Gdańsk University, Gdańsk, Poland.
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Alicki, R., Messer, J. Nonlinear quantum dynamical semigroups for many-body open systems. J Stat Phys 32, 299–312 (1983). https://doi.org/10.1007/BF01012712
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DOI: https://doi.org/10.1007/BF01012712