# Anomalous Diffusion Near Resonances Page: 2 of 3

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Proceedings of IPAC'10, Kyoto, Japan

1e-14

- 1-11-

2. x

Ic-5b

1c-12

0 100 200 300 400 500 600 700 800 900 1000

Turns [x 10001

Figure 3: Variance in the actions over time at a tune corre-

sponding to the resonance 3q, - 2q= 1. Also shown are

the (barely visible) monomial fits to the data.

09

08

Ic 1-13A

1e5s

0.2 U

01 1e-17

-01

0 05 I US 2 15 3 15 4 45 S

Initial amplitude [6i

Figure 4: The exponent a, (red, left vertical scale) and the

coefficient AJ O (blue, right vertical scale) as a function

of the initial amplitude.

tudes. The increase in the vertical variance is smaller by a

few orders of magnitude but the growth rate is steeper. We

model the growth in the variance of the actions (J, Jy) as:

(AJZ ) =AJT o , (AJ ) =A Ji- of For normal dif-

fusive behaviour, both the exponents (a,, ay) 1 while

anomalous sub-diffusive behaviour is characterized by ex-

ponents < 1 and super-diffusive behaviour has exponents

> 1. Both exponents are < 1 at all amplitudes, but in many

regions ay is about twice a,, yet (AJ ) grows much more

slowly because the constant coefficient AJ2 > is about five

orders of magnitude smaller than AJT o. Figure 4 shows

the exponent ax and the constant coefficient (AJI O) as

a function of the initial amplitude. At amplitudes below

lQ and above 4a, ax ~ 0 implying no diffusion . The

diffusion is fastest at 1.7Q, the location of the peak in the

coefficient AJT o.

We can define diffusion coefficients in the usual man-

ner, e.g. Dxx Var(Jx)/N, Dxy Covar(Jx, Jy)/N,

where N is the total number of turns. Fig 5 shows that

the "diffusion coefficients" Dxx calculated at a few am-

plitudes with x y for the different resonance tunes.

We do not show here the dependence of the coefficients

Dx,x, Dx,y, Dyy on both (Jx, Jy). There is a sharp rise in

these coefficients at ~ 2.0a for the (3,0,-1) resonance and

at ~ 1.5a for the (3,0,-2) resonance with a larger jump for

the latter. In the following we will focus on the (3,0,-2) res-

onance although the qualitative conclusions are applicable

to the other resonances. We can (without a priori justifica-

tion) use the diffusion coefficients in the normal diffusion496

1e-1

1e-1 -

1e-12 -

1e-13

1e-I4

le-l6

3v -2

0.5 1 15 2 25 3 35 4 45 5

Radial amplitude [6i

Figure 5: The horizontal "diffusion" coefficients as a func-

tion of radial amplitude at tunes corresponding to the four

SBR tunes.

in action equation for the density distibution,ap(J , J) = -V[D2;Vs]p(J , Jy)

(1)

Here (i, j) run over (x, y) and Dj are the diffusion coef-

ficients defined above. Numerically solving this diffusion

equation with the diffusion coefficients found above leads

to predictions that disagree spectacularly with the direct

particle tracking. For example, the solution to this diffu-

sion equation shows that about 20% of particles are lost

at a 6Q aperture within a few seconds while direct track-

ing shows an insignificant loss over this period. This is not

surprising given that the action does not grow as rapidly

as assumed by the diffusion equation. Clearly we need a

different transport equation to model the diffusion process.

CTRW MODEL FOR ANOMALOUS

DIFFUSION

The fact that the motion is sub-diffusive is to be expected

since the persistence of the KAM tori both below and above

the resonance islands will slow growth. Particles can cir-

culate around resonance islands for long periods of time

which can also lead to subdiffusion. We need to identify

the most plausible model for subdiffusion applicable to our

problem.

At the chosen tune of interest, the resonance islands in

horizontal phase space lie at around 2Q. Motion in their

vicinity but at slightly smaller amplitudes can be quite

complicated. Single trajectories starting from amplitudes

around 1.5a explore much of phase space: regions well

below the islands, regions around the islands as well as

regions outside the islands. The motion jumps between

these regions with different amplitudes, and the time spent

in each region appears to be random. These are the usual

ingredients needed for the continuous time random walk

(CTRW) model [2] of anomalous diffusion where the time

at which a step occurs is also taken to be a random variable.

The CTRW model leads to a fractional diffusion equation

where the order of the time derivative is fractional.

The step size distribution is one of the quantities that

characterize a CTRW model. Figure 6 shows distributions

in horizontal step sizes for initial amplitudes of 0.la and

1.55a respectively. The left plot in this figure is typical for

01 Circular Colliders

A01 Hadron CollidersMOPECO17

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Sen, Tanaji. Anomalous Diffusion Near Resonances, article, May 1, 2010; Batavia, Illinois. (digital.library.unt.edu/ark:/67531/metadc1014391/m1/2/: accessed June 25, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.