Fractional Beam Dynamics of Anomalous Diffusion An approach towards resonance enhanced transport

1. Fractional Beam Dynamicsof Anomalous DiffusionAn approach towards resonance enhanced transport Tanaji Sen APC Seminar

2. Basic Questions • Beam loss occurs and halos develop near resonances. Is particle growth diffusive and does the regular diffusion equation suffice? • What is the right statistical mechanical model to describe the beam distribution over long times? • Can we use the model to describe halo formation, emittance growth and beam lifetimes near resonances? APC Seminar

3. Outline • Beam dynamics near beam-beam driven low order synchro-betatron (SBR) resonances • Emittance growth, beam profiles • Detailed look at phase space dynamics • Introduce continuous time random walk (CTRW) model • Jump size and waiting time distributions in the model. • Derivation of a fractional diffusion equation (FDE) APC Seminar

4. Crossing angle with beam-beam interactions at two IPs. Crossing angles in hor, and vert. planes (LHC parameters) Choose SBRs of low order betatron resonances to see effect in short times Resonances are 2(3νx - 2νs) = 2 4νx - 2νs = 1 Beam-beam kick depends on longitudinal position – couples transverse and longitudinal Synchro-betatron resonances APC Seminar

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6. Phase space : resonance 2(3νx - 2νs) = 2 x0 = 0.2σ x0 = 1.5σ x0 = 7σ x0 = 3σ APC Seminar

7. Phase space: Resonance 4νx - 2νs = 1 x0 = 1σ x0 = 3σ x0 = 7σ x0 = 5σ APC Seminar

8. Emittance growth Horizontal Vertical • Growth is larger for resonance 2(3νx - 2νs) = 2 • Significantly smaller growth in vertical plane • Good fits to power law behaviour in time APC Seminar

9. Beam profiles • Resonance I: 2(3νx - 2νs) = 2 • Initial profile is Gaussian • Horizontal profile grows tails beyond 8σ • Vertical profile is not changed much • Resonance II : 4νx - 2 νs = 1 • Horizontal profile grows tails to 7σ • Growth of tails is slower here • Vertical profile is not much changed APC Seminar

10. Central Limit Theorem & Generalization The distribution of a sum of a sequence of random, identically distributed and independent variable with finite mean and second moment tends to a Gaussian distribution in the limit that the number in the sequence approaches infinity. Generalizing the CLT by dropping the requirement of a finite second moment leads to the family of Levy stable distributions APC Seminar

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12. Heavy (tailed) Levy Available as a built-in function in Mathematica 8 APC Seminar

13. Levy stable distribution – does it fit? • Resonance II: 4νx - 2 νs=1 • Index α = 1.3 • Wider than a Lorentzian • Falls off as 1/|x|2.3 • Resonance I: 2(3νx - 2 νs)=2 • Index α = 0.95 • Narrower than a Lorentzian • Falls off as 1/|x|1.95 APC Seminar

14. Measured and simulated proton beam halo in LEDA PRSTAB 5, 124201 (2002) APC Seminar

15. Inside the beam: how amplitudes grow 4νx-2 νs=1 2(3νx-2 νs)=2 • Initially 4000 particles at each amplitude • No growth below 1σ • Large variation at 1σ • Particles at 1.5 σ and above move to large amplitude • Same initial distribution • No growth below 1.5 σ • Large variations at 2σ • Particles above 2.5 σ move to • large amplitude APC Seminar

16. How does the variance grow? Variance at different amplitudes fitted by < ΔJ2> ~ tp p < 1 : sub-diffusion, p > 1 : super-diffusion Resonance I: 2(3νx - 2 νs)=2 No diffusion at x < 1 σ Bounded chaos at x ~ 1 σ Power law growth at x > 1.5 σ Resonance II: 4νx - 2 νs=1 No diffusion at x < 2 σ Bounded chaos at x ~ 2 σ Power law growth at x > 2.5 σ APC Seminar

17. Diffusion types: power index p Resonance: 2(3νx - 2 νs)=2 Resonance: 4νx - 2 νs=1 Super diffusive Super-diffusive Sub-diffusive Sub-diffusive No diffusion No diffusion • At small amplitudes, no diffusion • Followed by a narrow region (depends on y, as) with super-diffusion • Then, a broad range of amplitudes with sub-diffusion • At large amplitude (≥ 7σ), no diffusion APC Seminar

18. Schizophrenic phase space Stickiness of islands, unstable fixed points and persistence of KAM tori are expected to lead to sub-diffusion. APC Seminar

19. Continuous Time Random Walk Model • Continuous Time Random Walk (CTRW) model: time between steps is random – E.W. Montroll and G. Weiss (1965) • Waiting time –particle can wait a random time interval before making a random jump. Described by a distribution w(t) : w(t)dt = probability that particle waits for a time between t and t+dt before jumping • Jump distribution – the length of a jump is a random variable with distribution ψ(J,ΔJ). ψ(J,ΔJ)d(ΔJ) is the probability that particle makes a jump from J to J+ ΔJ APC Seminar

20. Time series Time between jumps in amplitude appears to be random. Length of jump (short or long) appears to be random. Ingredients of a CTRW model. APC Seminar

21. Jump distribution in x Jump size distributions in x 2(3νx - 2νs) = 2 x0 = 3σ x0 = 7σ x0 = 1.5σ x0 = 0.2σ Periodic function distribution: 4νx - 2νs = 1 x0 =1σ x0 = 5σ x0 = 7σ x0 = 3σ APC Seminar

22. Divide and rule phase space APC Seminar

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24. Why the standard diffusion equation may not be applicable • Particles can have short or long waiting periods before making sizable jumps. • Sizes of jumps vary and can be comparable to rms beam size • Diffusion coefficients are not constant in time. Diffusion type in most regions is sub-diffusive • Beam profiles develop long non-Gaussian tails – can these be solutions to the diffusion equation? APC Seminar

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28. Waiting for - the exponential? • Resonance • 2(3νx - 2 νs)=2 • Tails are too long to be fit by exponential No diffusion APC Seminar

29. Waiting time: power law? • Resonance : 2(3νx – 2 νs) = 2 • Power law fits data to some degree • Slope varies in the range -2.5 < α < -2.1 APC Seminar

30. Waiting time: power law? • Resonance : 4νx - 2 νs = 1 • At small amplitudes, no jumps in action • Slope varies in the range: -3 < α < -1.2 No diffusion APC Seminar

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32. Transporting the density APC Seminar

33. Validation of the model • Measurement of beam profile. A fit to a Levy stable distribution would be suggestive. These are solutions of some fractional diffusion equations. • Direct measurement of the waiting time w(t:J) ? • Solve fractional diffusion equation for the density. Find • Calculate loss rate and compare with beam loss rate • Calculate the moments, i.e. emittance • Application to space charge driven resonances ? APC Seminar

34. Cliff Notes Summary • Beam profiles near these resonances are described by Levy stable distributions. • Phase space is divided into zones: no diffusion, thin layer of super-diffusion, sub-diffusion, no diffusion. No evidence of regular diffusion. • Bounded chaos in super-diffusive zone. This zone lies below the resonance islands. • In zones of sub-diffusion, particles migrate to large amplitudes. • Fractions appear in the emittance growth rate, Levy distribution index, the power laws of the variance, the waiting time • A fractional diffusion equation in action derived, based on a CTRW model. • Waiting time is not exponential, so dynamics is non-Markovian. • Fractional diffusion equation reduces to an ODE for power law waiting times. Could be numerically solved to compare with the SBR dynamics. • Application to beam lifetime and emittance growth rates awaits. APC Seminar

35. A brief history of anomalous diffusion Beam physics • “A mechanism of anomalous diffusion in particle beams” by D. Jeon et al, Phys. Rev. Lett, 80, 2314 (1998) • “The applicability of diffusion phenomenology to particle losses in hadron colliders” by A. Gerasimov, Fermilab-Pub-92/185 Other fields • ‘Observation of Anomalous Diffusion and Fractional Self-Similarity in One Dimension’, Y. Sagi et al, PRL, 108, 093002 (2012) • ‘Nondiffusive transport in plasma turbulence: a fractional diffusion equation approach’ D. del-Castillo-Negrete et al, PRL, 94,065003 (2005) • Search for “anomalous diffusion” until April 2012 Phys. Rev. Lett: 114, Phys. Rev. E: 270 Phys. Rev. ST – AB: 0 APC Seminar