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Dissipative Dynamics in Classical and Quantum Conservative Systems

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Mathematical Systems Theory in Biology, Communications, Computation, and Finance

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 134))

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Abstract

In this paper, which is largely review, I will discuss dissipative behavior in mechanical systems which preserve energy. The paper will encompass both the classical and quantum domains. In the classical context I will consider almost Poisson systems; systems which which have a Poisson braket which fails the Jacobi identity. This class of systems includes nonholonomic mechanical systems: systems with nonintegrable constraints such as rolling constraints. Such systems may either preserve or fail to preserve a natural measure. I will discuss also pure Hamiltonian systems such as the Toda lattice which can exhibit dissipative behavior in certain contexts as well as infinite-dimensional systems exhibiting radiative damping. In the quantum context I will discuss systems of quantum oscillators coupled to a heat bath, which also exhibit natural dissipative behavior.

Research partially supported by NSF grants DMS-9803181 and DMS-0103895.

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  1. Department of Mathematics, University of Michigan, 48109-1109, Ann Arbor, MI, USA

    Anthony M. Bloch

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  1. Department of Mathematics, University of Notre Dame, 46556, Notre Dame, IN, USA

    Joachim Rosenthal

  2. Department of Mathematics and Statistics, Texas Tech University, 79409, Lubbock, TX, USA

    David S. Gilliam

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Bloch, A.M. (2003). Dissipative Dynamics in Classical and Quantum Conservative Systems. In: Rosenthal, J., Gilliam, D.S. (eds) Mathematical Systems Theory in Biology, Communications, Computation, and Finance. The IMA Volumes in Mathematics and its Applications, vol 134. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21696-6_4

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  • DOI: https://doi.org/10.1007/978-0-387-21696-6_4

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