Robert Alicki and Mark Fannes
- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198504009
- eISBN:
- 9780191708503
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504009.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter defines and characterizes various degrees of ergodicity — both for classical and quantum systems — in terms of the Hilbert space formalism, Koopman formalism for classical systems, and ...
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This chapter defines and characterizes various degrees of ergodicity — both for classical and quantum systems — in terms of the Hilbert space formalism, Koopman formalism for classical systems, and GNS-representation for quantum systems. It presents mixing and asymptotic Abelianness. It then discusses a number of examples with non-trivial algebraic structures: quasi-free Fermionic automorphisms, highly anti-commutative systems, and Powers–Price shifts. It is shown that under certain ergodic assumptions the fluctuations around ergodic means can be modelled by Bose fields in quasi-free states (Gaussian distributions), the other extreme cases lead to a free probability scheme with semi-circular distributions. Expanding maps and Lyapunov exponents for classical dynamics are briefly discussed and a possible quantum analog, horocyclic actions, is presented for a quantum cat map.Less
This chapter defines and characterizes various degrees of ergodicity — both for classical and quantum systems — in terms of the Hilbert space formalism, Koopman formalism for classical systems, and GNS-representation for quantum systems. It presents mixing and asymptotic Abelianness. It then discusses a number of examples with non-trivial algebraic structures: quasi-free Fermionic automorphisms, highly anti-commutative systems, and Powers–Price shifts. It is shown that under certain ergodic assumptions the fluctuations around ergodic means can be modelled by Bose fields in quasi-free states (Gaussian distributions), the other extreme cases lead to a free probability scheme with semi-circular distributions. Expanding maps and Lyapunov exponents for classical dynamics are briefly discussed and a possible quantum analog, horocyclic actions, is presented for a quantum cat map.
Robert Alicki and Mark Fannes
- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198504009
- eISBN:
- 9780191708503
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504009.003.0010
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
A non-zero Kolmogrov–Sinai entropy for a classical dynamical system is a signature of dynamical instability. This chapter presents an approach to quantifying randomizing dynamical behaviour in ...
More
A non-zero Kolmogrov–Sinai entropy for a classical dynamical system is a signature of dynamical instability. This chapter presents an approach to quantifying randomizing dynamical behaviour in deterministic quantum systems based on a spin chain model. The starting point is an operational partition that is refined in the course of time. To each partition corresponds a correlation matrix and the dynamics lead eventually to a shift-invariant state on a quantum spin chain with its associated entropy. General properties and bounds are proved, which allow for the computation of the entropy in a number of simple model systems such as finite systems, shift dynamics on a quantum spin chain, free shifts, and Powers–Price shifts.Less
A non-zero Kolmogrov–Sinai entropy for a classical dynamical system is a signature of dynamical instability. This chapter presents an approach to quantifying randomizing dynamical behaviour in deterministic quantum systems based on a spin chain model. The starting point is an operational partition that is refined in the course of time. To each partition corresponds a correlation matrix and the dynamics lead eventually to a shift-invariant state on a quantum spin chain with its associated entropy. General properties and bounds are proved, which allow for the computation of the entropy in a number of simple model systems such as finite systems, shift dynamics on a quantum spin chain, free shifts, and Powers–Price shifts.
