Fractional Dynamics of Open Quantum Systems QFTHEP 2010 Vasily E. Tarasov Skobeltsyn Institute of...

Fractional Dynamics of Open Quantum Systems

QFTHEP 2010

Vasily E. Tarasov

Skobeltsyn Institute of Nuclear Physics,Moscow State University, Moscow tarasov@theory.sinp.msu.ru

Fractional dynamics 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

Fractional 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

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Mathematics Books 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).

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 Reviews R. 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.6 /42QFTHEP 2010

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

Fractional Relaxation-Oscillation Effects;

Fractional Diffusion-Wave Effects;

Viscoelastic Materials;

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 Equations 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.

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