fractional dynamics of open quantum systems qfthep 2010 vasily e. tarasov skobeltsyn institute of...
TRANSCRIPT
Fractional Dynamics of Open Quantum Systems
QFTHEP 2010
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.
2/42QFTHEP 2010
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
3/42QFTHEP 2010
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 /42QFTHEP 2010
Special Journals
"Journal of Fractional Calculus";
"Fractional Calculus and Applied Analysis";
"Fractional Dynamic Systems";
"Communications in Fractional Calculus".
5 /42QFTHEP 2010
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
7 /42QFTHEP 2010
2. Finite difference
8 /42QFTHEP 2010
Grunwald (1867), Letnikov (1868)
9 /42QFTHEP 2010
3. Fourier Transform of Laplacian
10 /42QFTHEP 2010
Riesz integral (1936)
11/ /42QFTHEP 2010
4. Fourier transform of derivative
12 /42QFTHEP 2010
Liouville integral and derivative
13 /42QFTHEP 2010
Liouville integrals, derivatives (1832)
14 /42QFTHEP 2010
5. Caputo derivative (1967)
15 /42QFTHEP 2010
Riemann-Liouville and Caputo
16 /42QFTHEP 2010
Physical Applications
Fractional Relaxation-Oscillation Effects;
Fractional Diffusion-Wave Effects;
Viscoelastic Materials;
Dielectric Media: Universal Responce.
17 /42QFTHEP 2010
1. Fractional Relaxation-Oscillation
18 /42QFTHEP 2010
2. Fractional Diffusion-Wave Effects
19 /42QFTHEP 2010
3. Viscoelastic Materials
20 /42QFTHEP 2010
4. Dielectric Media: Universal Responce
21 /42QFTHEP 2010
Universal Response - Jonscher laws
22 /42QFTHEP 2010
* 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 /42QFTHEP 2010
Fractional equations of Jonscher laws
24 /42QFTHEP 2010
Universal electromagnetic waves
25 /42QFTHEP 2010
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)
26 /42QFTHEP 2010
27 /42QFTHEP 2010
Fractional non-Markovian quantum dynamics
28 /42QFTHEP 2010
29 /42QFTHEP 2010
30 /42QFTHEP 2010
Semigroup property ?
31 /42QFTHEP 2010
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 /42QFTHEP 2010
Example: Fractional open oscillator
33 /42QFTHEP 2010
34 /42QFTHEP 2010
Exactly solvable model.
Step 1
35 /42QFTHEP 2010
Step 2
36 /42QFTHEP 2010
Step 3
37 /42QFTHEP 2010
Step 4
38 /42QFTHEP 2010
Step 5
39 /42QFTHEP 2010
Solutions:
40 /42QFTHEP 2010
For alpha = 1
41/42QFTHEP 2010
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.
Final page 42QFTHEP 2010