Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor Page: 46
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For this class of trajectories we have
(Ap(N))2 in2 [x(0)]K2N2 < K2N2, (5.5)
so for some individual trajectories the upper bound to (Ap(N))2 is K2N2. References [16, 56]
mention that the momentum grows quadratic in time only for those trajectories in accelerator modes.
Since only a few points from 0 to 21r can satisfy Eq. (5.3), the average momentum diffusion is
expected somewhat less than quadratic. Our numerical results that follow are consistent with this
5.3 Calculation of Momentum Diffusion
To obtain the momentum diffusion in Eq. (5.2) for a given K we solve Eq. (4.2) numerically for
initial values x(0) and p(0) = 0 and obtain the momentum difference Ap= p(N) - p(0) at each
value of time N from 0 to 100 kicks. To compute the average value of the momentum squared to
get the momentum diffusion/(Ap)2), we have used 1000 values of x(0) evenly spaced between 0
Figure 5.1 shows our calculated linear diffusion rate DL divided by K2/2 =(Ap)2)0 /N, the
diffusion rate in the absence of correlation effects [16, 28, 37], as a function of the control parameter
K. The linear diffusion rate DL was calculated by a least squares fit of DLN to the numerical data
for (Ap)2)for N from 0 to 100. Figure 5.1 covers the range of control parameter K between 0 to
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Zheng, Yindong. Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor, dissertation, August 2005; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc4824/m1/63/: accessed July 22, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .