Anomalous Diffusion Near Resonances Page: 2 of 3
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Proceedings of IPAC'10, Kyoto, Japan
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.
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 diffusion
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
in action equation for the density distibution,
ap(J , J) = -V[D2;Vs]p(J , Jy)
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
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
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  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 Colliders
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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 February 16, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.