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Fractional Dynamics of Open Quantum Systems QFTHEP 2010 Vasily E. Tarasov Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow [email protected]

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  • Fractional Dynamics of Open Quantum Systems

    QFTHEP 2010Vasily E. Tarasov

    Skobeltsyn Institute of Nuclear Physics,Moscow State University, Moscow [email protected]

    QFTHEP 2010

  • Fractional dynamicsFractional dynamics is a field of study in physics and mechanics, studying the behavior of physical systems that are described by usingintegrations of non-integer (fractional) orders,differentiation of non-integer (fractional) orders.

    Equations with derivatives and integrals of fractional orders are used to describe objects that are characterized by power-law nonlocality, power-law long-term memory, fractal properties.

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  • History of fractional calculusFractional calculus is a theory of integrals and derivatives of any arbitrary real (or complex) order.

    It has a long history from 30 September 1695, when the derivatives of order 1/2 has been described by Leibniz in a letter to L'Hospital

    The fractional differentiation and fractional integration go back to many great mathematicians such as Leibniz, Liouville, Riemann, Abel, Riesz, Weyl.

    B. Ross, "A brief history and exposition of the fundamental theory of fractional calculus", Lecture Notes in Mathematics, Vol.457. (1975) 1-36.

    J.T. Machado, V. Kiryakova, F. Mainardi, "Recent History of Fractional Calculus", Communications in Nonlinear Science and Numerical Simulations Vol.17. (2011) to be puslished*/42QFTHEP 2010

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  • Mathematics BooksThe first book dedicated specifically to the theory of fractional calculus

    K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Academic Press, 1974).

    Two remarkably comprehensive encyclopedic-type monographs:

    S.G. Samko, A.A. Kilbas, O.I. Marichev, Integrals and Derivatives of Fractional Order and Applications} (Nauka i Tehnika, Minsk, 1987); Fractional Integrals and Derivatives Theory and Applications (Gordon and Breach, 1993).

    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, 2006).

    I. Podlubny, Fractional Differential Equations (Academic Press, 1999).

    A.M. Nahushev, Fractional Calculus and Its Application (Fizmatlit, 2003) in Russian.

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  • Special Journals"Journal of Fractional Calculus";

    "Fractional Calculus and Applied Analysis";

    "Fractional Dynamic Systems";

    "Communications in Fractional Calculus".

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  • Physics Books and ReviewsR. Metzler, J. Klafter, "The random walk's guide to anomalous diffusion: a fractional dynamics approach" Physics Reports, 339 (2000) 1-77.

    G.M. Zaslavsky, "Chaos, fractional kinetics, and anomalous transport" Physics Reports, 371 (2002) 461-580.

    R. Hilfer (Ed.), Applications of Fractional Calculus in Physics (World Scientific, 2000).

    A.C.J. Luo, V.S. Afraimovich (Eds.), Long-range Interaction, Stochasticity and Fractional Dynamics (Springer, 2010) .

    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (World Scientific, 2010).

    V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, 2010).

    V.V. Uchaikin, Method of Fractional Derivatives (Artishok, 2008) in Russian.* /42QFTHEP 2010

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  • 1. Cauchy's differentiation formula* /42QFTHEP 2010

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  • 2. Finite difference* /42QFTHEP 2010

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  • Grunwald (1867), Letnikov (1868) * /42QFTHEP 2010

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  • 3. Fourier Transform of Laplacian * /42QFTHEP 2010

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  • Riesz integral (1936)*/ /42QFTHEP 2010

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  • 4. Fourier transform of derivative* /42QFTHEP 2010

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  • Liouville integral and derivative * /42QFTHEP 2010

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  • Liouville integrals, derivatives (1832)* /42QFTHEP 2010

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  • 5. Caputo derivative (1967)* /42QFTHEP 2010

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  • Riemann-Liouville and Caputo* /42QFTHEP 2010

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  • Physical ApplicationsFractional Relaxation-Oscillation Effects;

    Fractional Diffusion-Wave Effects;

    Viscoelastic Materials;

    Dielectric Media: Universal Responce.

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  • 1. Fractional Relaxation-Oscillation* /42QFTHEP 2010

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  • 2. Fractional Diffusion-Wave Effects

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  • 3. Viscoelastic Materials* /42QFTHEP 2010

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  • 4. Dielectric Media: Universal Responce* /42QFTHEP 2010

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  • Universal Response - Jonscher laws* /42QFTHEP 2010

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  • * A.K. Jonscher, Universal Relaxation Law (Chelsea Dielectrics Pr, 1996);* T.V. Ramakrishnan, M.R. Lakshmi, (Eds.), Non-Debye Relaxation in Condensed Matter (World Scientific, 1984).* /42QFTHEP 2010

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  • Fractional equations of Jonscher laws * /42QFTHEP 2010

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  • Universal electromagnetic waves* /42QFTHEP 2010

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  • Markovian dynamics for quantum observables* /42QFTHEP 2010

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  • Fractional non-Markovian quantum dynamics* /42QFTHEP 2010

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  • Semigroup property ?* /42QFTHEP 2010

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  • The dynamical maps with non-integer cannot form a semigroup. This property means that we have a non-Markovian evolution of quantum systems.

    The dynamical maps describe quantum dynamics of open systems with memory.The memory effect means that the present state evolution depends on all past states.

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  • Example: Fractional open oscillator* /42QFTHEP 2010

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  • Exactly solvable model.

    Step 1* /42QFTHEP 2010

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  • Step 2* /42QFTHEP 2010

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  • Step 3* /42QFTHEP 2010

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  • Step 4* /42QFTHEP 2010

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  • Step 5* /42QFTHEP 2010

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  • Solutions:* /42QFTHEP 2010

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  • For alpha = 1*/42QFTHEP 2010

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  • ConclusionsEquations of the solutions describe non-Markovian evolution of quantum coordinate and momentum of open quantum systems.

    This fractional non-Markovian quantum dynamics cannot be described by a semigroup. It can be described only as a quantum dynamical groupoid.

    The long-term memory of fractional open quantum oscillator leads to dissipation with power-law decay.

    Tarasov V.E. Quantum Mechanics of Non-Hamiltonian and Dissipative Systems(Elsevier, 2008) 540p.

    Tarasov V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics ofParticles, Fields and Media, (Springer, 2010) 516p.

    Final page *QFTHEP 2010

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