Non-Hamiltonian and Dissipative Systems

Difference of the suggested book from others is consistent use of the functional analysis and operator algebras. To read the text, preliminary knowledge of these sections of mathematics is not required. All the necessary information, which is beyond usual courses of the mathematical analysis and linear algebra, is included.

To describe the theory, we use the fact that quantum and classical mechanics are connected not only by limiting transition, but also realized by identical mathematical structures. A common basis to formulate the theory is an assumption that classical and quantum mechanics are different representations of the same totality of mathematical structures, i.e., the so-called Dirac correspondence principle. For construction of quantum theory, we consider mathematical concepts that are the general for Hamiltonian and non-Hamiltonian systems. Quantum dynamics is described by the one-parameter semi-groups and the differential equations on operator spaces and algebras. The Lie–Jordan algebraic structure, Liouville space and superoperators are used. It allows not only to consistently formulate the evolution of quantum systems, but also to consider the dynamics of a wide class of quantum systems, such as the open, non-Hamiltonian, dissipative, and nonlinear systems. Hamiltonian systems in pure states are considered as special cases of quantum dynamical systems.

The closed, isolated and Hamiltonian systems are idealizations that are not observable and therefore do not exist in the real world. As a rule, any system is always embedded in some environment and therefore it is never really closed or isolated. Frequently, the relevant environment is in principle unobservable or is unknown. This would render the theory of non-Hamiltonian and dissipative quantum systems to a fundamental generalization of quantum mechanics. The quantum theory of Hamiltonian systems, unitary evolution, and pure states should be considered as special cases of the generalized approach.

Usually the quantum mechanics is considered as generalization of classical mechanics. In this book the quantum mechanics is formulated as a generalization of modern nonlinear dynamics of dissipative and non-Hamiltonian systems. The quantization of equations of motion for dissipative and non-Hamiltonian classical systems is formulated in this book. This quantization procedure allows one to derive quantum analogs of equations with regular and strange attractors. The regular attractors are considered as stationary states of non-Hamiltonian and dissipative quantum systems. In the book, the quantum analogs of the classical systems with strange attractors, such as Lorenz and Rössler systems, are suggested. In the text, the main attention is devoted to non-Hamiltonian and dissipative systems that have the wide possibility to demonstrate the complexity, chaos and self-organization.

The text is self-contained and can be used without introductory courses in quantum mechanics and modern mathematics. All the necessary information, which is beyond undergraduate courses of the mathematics, is presented in the book.

From Preface of "Non-Hamiltonian and Dissipative Systems"
by Vasily E. Tarasov.