1. 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

2. 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. 2/42 QFTHEP 2010

3. 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 3/42 QFTHEP 2010

4. 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. 4 /42 QFTHEP 2010

5. Special Journals "Journal of Fractional Calculus"; "Fractional Calculus and Applied Analysis"; "Fractional Dynamic Systems"; "Communications in Fractional Calculus". 5 /42 QFTHEP 2010

6. 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 /42 QFTHEP 2010

7. 1. Cauchy's differentiation formula 7 /42 QFTHEP 2010

8. 2. Finite difference 8 /42 QFTHEP 2010

9. Grunwald (1867), Letnikov (1868) 9 /42 QFTHEP 2010

10. 3. Fourier Transform of Laplacian 10 /42 QFTHEP 2010

11. Riesz integral (1936) 11/ /42 QFTHEP 2010

12. 4. Fourier transform of derivative 12 /42 QFTHEP 2010

13. Liouville integral and derivative 13 /42 QFTHEP 2010

14. Liouville integrals, derivatives (1832) 14 /42 QFTHEP 2010

15. 5. Caputo derivative (1967) 15 /42 QFTHEP 2010

16. Riemann-Liouville and Caputo 16 /42 QFTHEP 2010

17. Physical Applications Fractional Relaxation-Oscillation Effects; Fractional Diffusion-Wave Effects; Viscoelastic Materials; Dielectric Media: Universal Responce. 17 /42 QFTHEP 2010

18. 1. Fractional Relaxation-Oscillation 18 /42 QFTHEP 2010

19. 2. Fractional Diffusion-Wave Effects 19 /42 QFTHEP 2010

20. 3. Viscoelastic Materials 20 /42 QFTHEP 2010

21. 4. Dielectric Media: Universal Responce 21 /42 QFTHEP 2010

22. Universal Response - Jonscher laws 22 /42 QFTHEP 2010

23. * 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). 23 /42 QFTHEP 2010

24. Fractional equations of Jonscher laws 24 /42 QFTHEP 2010

25. Universal electromagnetic waves 25 /42 QFTHEP 2010

26. Markovian dynamics for quantum observables 26 /42 QFTHEP 2010

27. 27 /42 QFTHEP 2010

28. Fractional non-Markovian quantum dynamics 28 /42 QFTHEP 2010

29. 29 /42 QFTHEP 2010

30. 30 /42 QFTHEP 2010

31. Semigroup property ? 31 /42 QFTHEP 2010

32. 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. 32 /42 QFTHEP 2010

33. Example: Fractional open oscillator 33 /42 QFTHEP 2010

34. 34 /42 QFTHEP 2010

35. Exactly solvable model. Step 1 35 /42 QFTHEP 2010

36. Step 2 36 /42 QFTHEP 2010

37. Step 3 37 /42 QFTHEP 2010

38. Step 4 38 /42 QFTHEP 2010

39. Step 5 39 /42 QFTHEP 2010

40. Solutions: 40 /42 QFTHEP 2010

41. For alpha = 1 41/42 QFTHEP 2010

42. 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 of Particles, Fields and Media, (Springer, 2010) 516p. Final page 42 QFTHEP 2010