Abstract
The adaptive dynamics has two aspects, one of which is the “observable-adaptivity” and another is the “state-adaptivity”.
Information Dynamics (ID) was proposed to find a common frame treating chaotic behaviors of several dynamical systems, which synthesizes the dynamics of state change and the complexity of system. ID enables us to attain a new concept “adaptivity” studying dynamics. In this paper, we briefly review the ID and the adaptive dynamics, and we discuss how they are used to understand chaos.
Dedicated to Professor G. G. Emch on his 70th birthday
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References
L. Accardi, Urne e Camaleoni: Dialogo sulla realta, le leggi del caso e la teoria quantistica. Il Saggiatore, Rome (1997)
L.Accardi, K.Imafuku, M.Refoli, On the EPR-Chameleon experiment, Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 5, No. 1 (2002) 1–2
L.Accardi, M.Ohya, A Stochastic limit approach to the SAT problem”, Proceedings of VLSI 2003, and Open systems and Information Dynamics (2004)
L.Accardi, M.Ohya, Compound channels, transition expectations, and liftings”, Appl. Math. Optim., Vol.39, 33–59 (1999)
L.Accardi, M.Ohya, N. Watanabe, Note on quantum dynamical entropies Reports on mathematical physics, vol.38 n.3 457–469 (1996)
H.Araki, Relative entropy of states of von Neumann Algebras, Publ.RIMS, Kyoto Univ. Vol.11, 809–833 (1976)
L.Accardi, R.Sabbadini, On the Ohya-Masuda quantum SAT Algorithm, in: Proceedings International Conference UMC’01, Springer (2001)
K.T.Alligood, T.D.Sauer, J.A.Yorke, Chaos-An Introduction to Dynamical Systems-, Textbooks in Mathematical Sciences, Springer (1996)
R. Alicki, Quantum geometry of noncommutative Bernoulli shifts, Banach Center Publications, Mathematics Subject Classification 46L87 (1991)
R. Alicki, M. Fannes, Defining quantum dynamical entropy, Lett. Math. Physics, 32, 75–82 (1994)
F. Benatti, Deterministic Chaos in Infinite Quantum Systems, Springer (1993)
A. Connes, H. Narnhofer, W. Thirring, Dynamical entropy of C*-algebras and von Neumann algebras, Commun.Math.Phys., 112, pp.691–719 (1987)
R.L. Devaney, An Introduction to Chaotic dynamical Systems, Benjamin (1986)
G.G. Emch, H. Narnhofer, W. Thirring and G.L. Sewell, Anosov actions on noncommutative algebras, J.Math.Phys., 35, No. 11, 5582–5599 (1994)
G.G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley (1972)
K. Inoue, M. Ohya, I.V. Volovich, Semiclassical properties and chaos degree for the quantum baker’s map, J. Math. Phys., 43–2, 734–755 (2002)
K. Inoue, M. Ohya, I.V. Volovich, On quantum-classical correspondence for baker’s map, quant-ph/0108107(2001)
K. Inoue, M. Ohya and K. Sato, Application of chaos degree to some dynamical systems, Chaos, Soliton & Fractals, 11, 1377–1385 (2000)
R.S. Ingarden, A. Kossakowski, M. Ohya, Information Dynamics and Open Systems, Kluwer Publ. Comp. (1997)
K. Inoue, M. Ohya, A. Kossakowski, A Description of Quantum Chaos, Tokyo Univ. of Science preprint (2002)
A. Kossakowski, M. Ohya, Y. Togawa, How can we observe and describe chaos? Open System and Information Dynamics, 10(3):221–233 (2003)
A. Kossakowski, M. Ohya, N. Watanabe, Quantum dynamical entropy for completely positive maps, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2, No.2, 267–282 (1999)
N. Muraki, M. Ohya, Entropy functionals of Kolmogorov-Sinai type and their limit theorems, Letter in Mathematical Physics, 36, 327–335 (1996)
M. Ohya, D. Petz, Quantum Entropy and its Use, Springer-Verlag (1993)
M. Ohya, On compound state and mutual information in quantum information theory, IEEE Trans. Information Theory, 29, No.5, 770–774 (1983)
M. Ohya, Some aspects of quantum information theory and their applications to irreversible processes, Rep.Math.Phys., Vol.27, 19–47 (1989)
M. Ohya, Complexities and their applications to characterization of chaos, International Journal of Theoretical Physics,Vol.37, No.1, 495–505 (1998)
M. Ohya, State change, complexity and fractal in quantum systems, Quantum Communications and Measurement, Plenum Press, New York, 309–320 (1995)
M. Ohya, Complexity and fractal dimensions for quantum states, Open Systems and Information Dynamics, 4, 141–157 (1997)
M.Ohya, et al, Adaptive dynamics its use in understanding of chaos, TUS preprint
M. Ohya, Information dynamics and its applications to optical communication processes, Lecture Note in Physics, 378, 81–92 (1991)
M. Ohya, Entropy Transmission in C*-dynamical systems, J.Math. Anal. Appl., 100, No. 1, 222–235 (1984)
M. Ohya, N. Masuda, NP problem in quantum algorithm, Open Systems and Information Dynamics, Vol.7, No.1, 33–39 (2000)
M. Ohya, I.V. Volovich, New quantum algorithm for studying NP-complete problems, Rep.Math.Phys., 52, No. 1, 25–33 (2003) and Quantum computing and chaotic amplifier, J.Opt.B (2003)
M.Ohya, I.V.Volovich, Mathematical Foundations of Quantum Information and Quantum Computation, to be published in Springer-Verlag
A. Uhlmann, The ‘transition probability’ in the state space of a*-algebra.Rep.Math.Phys., 9:273–279 (1976)
H. Umegaki, Conditional expectation in operator algebra IV, Kodai Math. Sem. Rep., 14, 59–85 (1962)
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Ohya, M. (2007). Adaptive Dynamics and its Application to Chaos. In: Ali, S.T., Sinha, K.B. (eds) Contributions in Mathematical Physics. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-33-0_8
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DOI: https://doi.org/10.1007/978-93-86279-33-0_8
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