QFTHEP 2010
Fractional Dynamics of
Open Quantum Systems
Vasily E. Tarasov
Skobeltsyn Institute of Nuclear Physics,
Moscow State University, Moscow
tarasov@theory.sinp.msu.ru
Fractional dynamics
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Fractional dynamics is a field of study in physics and
mechanics, studying the behavior of physical systems
that are described by using
integrations 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 calculus
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Fractional calculus is a theory of integrals and derivatives of
any arbitrary real (or complex) order.
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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
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The fractional differentiation and fractional integration go
back to many great mathematicians such as
Leibniz, Liouville, Riemann, Abel, Riesz, Weyl.
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B. Ross, "A brief history and exposition of the fundamental theory of fractional
calculus", Lecture Notes in Mathematics, Vol.457. (1975) 1-36.
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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
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Mathematics Books
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The 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).
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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).
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I. Podlubny, Fractional Differential Equations (Academic Press, 1999).
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A.M. Nahushev, Fractional Calculus and Its Application (Fizmatlit, 2003) in Russian.
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Special Journals
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"Journal of Fractional Calculus";
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"Fractional Calculus and Applied Analysis";
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"Fractional Dynamic Systems";
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"Communications in Fractional Calculus".
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Physics Books and Reviews
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R. Metzler, J. Klafter, "The random walk's guide to anomalous diffusion: a
fractional dynamics approach" Physics Reports, 339 (2000) 1-77.
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G.M. Zaslavsky, "Chaos, fractional kinetics, and anomalous transport"
Physics Reports, 371 (2002) 461-580.
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R. Hilfer (Ed.), Applications of Fractional Calculus in Physics (World
Scientific, 2000).
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A.C.J. Luo, V.S. Afraimovich (Eds.), Long-range Interaction, Stochasticity
and Fractional Dynamics (Springer, 2010) .
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F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An
Introduction to Mathematical Models (World Scientific, 2010).
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V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to
Dynamics of Particles, Fields and Media (Springer, 2010).
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V.V. Uchaikin, Method of Fractional Derivatives (Artishok, 2008) in Russian.
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1. Cauchy's differentiation formula
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2. Finite difference
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Grunwald (1867), Letnikov (1868)
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3. Fourier Transform of Laplacian
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Riesz integral (1936)
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4. Fourier transform of derivative
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Liouville integral and derivative
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Liouville integrals, derivatives (1832)
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5. Caputo derivative (1967)
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Riemann-Liouville and Caputo
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Physical Applications
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Fractional Relaxation-Oscillation Effects;
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Fractional Diffusion-Wave Effects;
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Viscoelastic Materials;
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Dielectric Media: Universal Responce.
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1. Fractional Relaxation-Oscillation
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2. Fractional Diffusion-Wave Effects
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3. Viscoelastic Materials
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4. Dielectric Media: Universal Responce
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Universal Response - Jonscher laws
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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).
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Fractional equations of Jonscher laws
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Universal electromagnetic waves
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Markovian dynamics for
quantum observables
Alicki R., Lendi K., Quantum Dynamical Semigroups and Applications (Springer, 1987)
Attal S., Joye A., Pillet C.A., Open Quantum Systems: The Markovian Approach (Springer, 2006)
Tarasov V.E., Quantum Mechanics of Non-Hamiltonian and Dissipative Systems (Elsevier, 2008)
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Fractional non-Markovian
quantum dynamics
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Semigroup property ?
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The dynamical maps with non-integer α 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
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Exactly solvable model.
Step 1
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Step 2
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Step 3
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Step 4
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Step 5
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Solutions:
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For alpha = 1
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Conclusions
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Equations of the solutions describe non-Markovian evolution of
quantum coordinate and momentum of open quantum systems.
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This fractional non-Markovian quantum dynamics cannot be
described by a semigroup. It can be described only as a
quantum dynamical groupoid.
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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 of
Particles, Fields and Media, (Springer, 2010) 516p.
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