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Fractional Dynamics of Open Quantum Systems

Vasily E. Tarasov

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

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 letter to L'Hospital

30 September 1695

, when the derivatives of order 1/2 has been described by Leibniz in a  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.

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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

 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

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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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