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Fractional Beam Dynamics of Anomalous Diffusion An approach towards resonance enhanced transport. Tanaji Sen. Basic Questions. Beam loss occurs and halos develop near resonances. Is particle growth diffusive and does the regular diffusion equation suffice?
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Fractional Beam Dynamicsof Anomalous DiffusionAn approach towards resonance enhanced transport Tanaji Sen APC Seminar
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
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
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
Phase space : resonance 2(3νx - 2νs) = 2 x0 = 0.2σ x0 = 1.5σ x0 = 7σ x0 = 3σ APC Seminar
Phase space: Resonance 4νx - 2νs = 1 x0 = 1σ x0 = 3σ x0 = 7σ x0 = 5σ APC Seminar
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
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
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
Heavy (tailed) Levy Available as a built-in function in Mathematica 8 APC Seminar
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
Measured and simulated proton beam halo in LEDA PRSTAB 5, 124201 (2002) APC Seminar
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
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
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
Schizophrenic phase space Stickiness of islands, unstable fixed points and persistence of KAM tori are expected to lead to sub-diffusion. APC Seminar
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
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
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
Divide and rule phase space APC Seminar
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
Waiting for - the exponential? • Resonance • 2(3νx - 2 νs)=2 • Tails are too long to be fit by exponential No diffusion APC Seminar
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
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
Transporting the density APC Seminar
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
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
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