Revisiting the microfield formulation using Continuous. Time Random. Walk

Transcription

1 Revisiting the microfield formulation using Continuous Time Random Walk Theory D. Boland, A. Bouzaher, H. Capes, F. Catoire, M. Christova a, L. Godbert-Mouret, M. Koubiti, S. Mekkaoui, Y. Marandet, J. Rosato, R. Stamm Physique des Interactions Ioniques et Moléculaires Université de Provence et CNRS, Marseille, France a Department of Applied Physics, Technical University-Sofia

2 Outline Microfield Model Method The Continuous Time Random Walk Theory and its application to Stark Broadening The Kangaroo Hypothesis Time Memory effect Application to Ly-a References

3 Semi-classical theory for line profile d ih U t, + dt V ( t' ) = ( H + V( t) B) U( t, t' ) r r ( t) = D.( t) H Hamiltonian operator for the unperturbed atom V(t) time-depending perturbing potential due to the motions of ions B operator of electron impacts Assumptions: dipole approximation for the perturbing potential perturbers cause only radiative transitions perturbers are classical uncorrelated particals moving in straight lines

4 Microfield Model Method d dt P Γ ( t' ) = M ( t) U t, r ( ) r r r () t = () t.( ) ( ) U( t, t' ) Linear stochastic equation of the evolution operator (t) fluctuating microfield Probability distribution and covariance of the microfield crucial role for Stark profiles U. Frisch and A. Brissaud, J. Quant. Spectrosc. Radiat. Transfer 11, 1753 (1971) C. Stehlé, Astron. Astrophys. Suppl. Ser. 14, 59 (1994) B. Talin et al., Phys. Rev. A 51, 4917 (1995)

5 This work -To revise the Stochastic Processes used to model the plasma microfield, namely the Kangaroo Process, developped by Frisch and Brissaud (1971), at the light of the Continuous Time Random Walk (CTRW) of Montroll, Weiss (1965) -Use the CTRW for calculating the ion dynamics effect on the Lyman alpha line -xplore possible improvements of the stochastic process. Montroll and J. Weiss, J. of Mathematical Physics 6, 167 (1965)

6 Continuous Time Random Walk (CTRW) CTRW: the time interval between two jumps in the stochastic variable obeys a given probalility law (Waiting Time Distribution) Kangaroo Process is a particular case of CTRW process using the Kangaroo Hypothesis and a Poisson Waiting Time Distribution lliott Montroll ( ) Consider (t) as a stochastic vector (t) {, } i t i 1 i+1 i i+2 t i-1 t i t i+1 t i+2

7 Continuous Time Random Walk (CTRW) Key Quantity: probability density ψ(t -t )(i 1, i) i i -1 + Stationary process Statistically independent jumps d 3 dtψ t (, ') = 1 d 3 ' ψ t (', ) = ϕ (t) Waiting Time Distribution Φ (t) = 1 dt' ϕ t O (t') Probability that no jumps occur during t

8 Solution of the Schrödinger equation for a stepwise constant process du(t,t )/dt= i(h D.(t))U(t,t ) U(t,t ) = U (t tm)u (tm tm 1)... U (t1,t ) m+ 1 m 1 Single field history of an atom during t U (t k k t ) = exp i(h d. k 1 )(t k k t ) k 1 Time evolution for an atom in a static field k

9 Averaging of the evolution operator over electric fields 1) Calculate the atomic time evolution operator for all possible field time histories with the initial value ' and final value at time t T t (,') ' ' ' T t = + + t τ t τ 1 τ 2 t t dτ d 1 T t (,') =δ(-')φ Ε (t)u (t) + T t-τ (, 1 ) U ' (τ) Ψ τ ( 1,')

10 Averaging of the evolution operator over electric fields 2) Average over all initial and final electric fields lectric field Distribution Hooper T(t) = d 3 d 3 ' P(')T t (, )

11 New part: Generalization of Kangaroo Process Kangaroo Hypothesis ψ (,') = q() '(t) t ϕ q() - the probability density to find an electric field after one jump Waiting Time Distribution φ (t) any function

12 New part: Generalization of Kangaroo Process xact solution for the atomic evolution operator T ~ (s) = d 3 P ( )Y ~ (s) + d 3 q( )Y ~ (s) [ q( )X ~ I d (s)] d P ( )X ~ (s) X ~ Y ~ (s) (s) = = dtϕ dtφ (t) exp s + (t) exp s + 1 ih 1 ih (H (H r r D. + B) r r D. + B) t t

13 Standard Kangaroo process ψ (,') = q() '(t) t ϕ ϕ (t) = ν()exp ν()t In diagonal representation 1 r r = s + (H D. + B) ih p X ~ (s) ~ ~ = ϕ (p) Y ~ (s) = Φ (p) The Frisch-Brissaud s result for the atomic evolution operator T(t) is recovered if the Waiting Time Distribution φ (t) is a Poisson s one U. Frisch and A. Brissaud, J. Quant. Spectrosc. Radiat. Transfer 11, 1753 (1971)

14 Main difference from Frisch-Brissaud s solution t t 3 ( / t + ih + id.)t t(, 1) = K(t τ)t τ(, 1)dτ+ q() dτ d 'K' (t τ τ 1 )T (', ) Memory kernel K ~ (s) = pϕ ~ (p) /(1 ϕ ~ (p)) 1 r r = s + (H D. + B) ih p Poisson WTD K (t) = ν(ε)δ(t) K ~ ( s) = ν = const Arbitrary WTD K (t) non-local in time K i (t-t i-1 ) i+1 jumps with different electric fields overlap in time i+2 i t i-1 t i t i+1

15 Application to Ly-a line Low density (near impact) q()=p() ϕ(t) = ν Γ(a) ( νt) a-1 exp νt WTD Poisson (a = 1) Gamma (a = 2) ffect of the electric field dynamics

16 Application to Ly-a line High density (near static) Poisson WTD a = 1 Gamma WTD a = 2 Line profile with Gamma Waiting Time Distribution is close to the static profile

17 Conclusions -TheContinuous Time Random Walk allows to generalize the Frisch Brissaud result to arbitrary Waiting Time Distribution - The CTRW solution retains memory effects - Preliminary applications to Ly-a show a significant influence of the choice of the Waiting Time Distribution Perspectives Stark Broadening -Comparison with simulation results -Use of Levy WTD for turbulent electric fields Collisional Radiative Model with fluctuating plasmas parameters Neutral Transport in turbulent plasmas

18 References U. Frisch and A. Brissaud, J. Quant. Spectrosc. Radiat. Transfer 11, 1753 (1971) C. Stehlé, Astron. Astrophys. Suppl. Ser. 14, 59 (1994) B. Talin et al., Phys. Rev. A 51, 4917 (1995).. Montroll and J. Weiss, J. of Mathematical Physics 6, 167 (1965) Acknowledgements This work is part of a collaboration (LRC DSM 99-14) between the Laboratory of Physique des Interactions Ioniques et Moléculaires and the Fédération de recherche en fusion par confinement magnétique. It has also been supported by the Agence Nationale de la Recherche (ANR-7-BLAN-187-1), project PHOTONITR.