Equilibration Diffusion and Halo Suggested experiments with proton

Equilibration, Diffusion and Halo – Suggested experiments with proton beams in IOTA & & D. L. Bruhwiler, S. D. Webb, A. Valishev, R. Kishek&, % and S. Nagaitsev & % Workshop on Scientific Opportunities in IOTA 28, 29 April 2015 – Fermilab Special thanks to Eric Stern, Jim Amundsen and the Synergia development team at Fermilab. This work is supported by the US DOE Office of Science, Office of High Energy Physics under SBIR award DE-SC 0011340.

Overview • We have developed a physical picture of the dynamics for 2 D nonlinear integrable & bounded chaotic systems – rapid equilibration due to nonlinear decoherence – slower diffusion of the invariant(s) over intermediate time scales • two invariants for nonlinear-integrable dynamics • one invariant for bounded-chaotic dynamics • This physical model needs to be explored experimentally – experimental verification of Hamiltonian diffusion would be very exciting IOTA Workshop – 28, 29 April 2015 – Fermilab #2

Rapid Equilibration due to Nonlinear Decoherence • Previous work with Tech-X Corp. – simulations of a toy problem with Py. ORBIT code • More recent work by Radia. Soft – simulations of IOTA with Octupoles , using Synergia IOTA Workshop – 28, 29 April 2015 – Fermilab #3

For a linear lattice, core mismatch oscillations quickly drive test-particles into the halo IOTA Workshop – 28, 29 April 2015 – Fermilab #4

For integrable nonlinear magnetic fields (D&N 2010), nonlinear decoherence suppresses halo formation IOTA Workshop – 28, 29 April 2015 – Fermilab #5

Synergia – vertical offset (linear) • 1 mm vertical offset applied – vertical oscillations observed – no coupling to horizontal • linear optics is applied – SC adds frequency content • this can drive parametric resonances zero current 10% current IOTA Workshop – 28, 29 April 2015 – Fermilab #6

Synergia – vertical offset (w/ octupoles) • 1 mm vertical offset applied – vertical oscillations suppressed • nonlinear decoherence – same behavior with SC • helps to suppress parametric resonances zero current IOTA Workshop – 28, 29 April 2015 – Fermilab 10% current #7

Diffusion of the Invariants on Intermediate Time Scale • Nonlinear space charge forces break integrability – Vlasov quasi-equilibria are evolving over time – show movie – invariants of the motion show signs of diffusion rapid equilibration IOTA Workshop – 28, 29 April 2015 – Fermilab #8

Calculating the Diffusion Coefficient (Part 1) we use a transfer map approach integrable Hamiltonian plus perturbation due to nonlinear space charge forces Green’s function representation of the nonlinear space charge forces fundamental assumption: change in the invariant during a single turn (or a few turns) is a Markov process, which means there is no dependence on earlier states, which in turn imples strong phase space mixing. W(J-d. J, n, d. J) is the probability that a trajectory with invariant value J-d. J at time ‘n’ will be kicked to value J at time ‘n+1’ Analytical form may be possible in some limits. In general, numerics will be required. IOTA Workshop – 28, 29 April 2015 – Fermilab #9

Calculating the Diffusion Coefficient (Part 2) Taylor series expansion of the Markov process equation yields a proto-diffusion equation: time derivative of f(J) dynamical friction (often vanishes) diffusion phase-averaged form of the 1 st-order kick in d. J (may have to be obtained numerically): phase-averaged form of the 2 nd-order kick in d. J (may have to be obtained numerically): IOTA Workshop – 28, 29 April 2015 – Fermilab # 10

Summary & Future Work • Diffusion of invariants is predicted over 100 s of turns – this can in principle be observed in IOTA • Proposed experiments – kick beam and measure rate of return to <x>=<y>=0 • very fast (just a few turns) – measure 2 nd moments accurately over 100 s of turns • need to understand 2 nd moments of phase space • numerical work is required to infer distribution in H, I • theoretical work should help • Experimental measurement of diffusion in a Hamiltonian system could be a Nature paper IOTA Workshop – 28, 29 April 2015 – Fermilab # 11