I am trying to calculate classical trajectories for a single positive ion and a single electron inside an RF trap. Therefore, I am dealing with a two-body system that possesses:

  • Coulomb potential with a singularity
  • time-dependent potential that is periodic in time and quadratic in space (smooth, relatively weak - compared to the Coulomb). However, asymptotically it becomes dominant.

I would like to calculate the classical trajectories that start from an arbitrary point in the phase space. The time interval I am interested in, is up to 10 periods of the trap field.

I've had some limited success with Tao's symplectic integrator for charged particle in electromagnetic fields (J. Comp. Phys. 327, 245 (2016)). However, the number of time steps necessary to get converged trajectories, is quite large and I would need to perform some systematic scan of the parameter space.

There are several ideas I have in mind but I am afraid that I am going to reinvent a wheel with them. Some of them:

  • When the particles approach each other, the time-dependent field can be perhaps neglected or treated perturbatively. So using different integrators in different regions of the phase space could help. However, it seems from the classical references that it would take away the symplecticity.
  • I could regularize the COulomb singularity, add an additional degree of freedom and make the Hamiltonian non-separable. Then I could try to solve it in the extended phase space.
  • Maybe it would be worth trying some existing simulation software (LAMMPS?). However, how is the singularity treated there?

Is there "canonically" appropriate integrator for this kind of situations please? Is any of the strategy mentioned above worth the shot? Thanks for any hint.

  • $\begingroup$ Welcome to Scicomp! When dealing with Orbits you will have quite rapid dynamics when the electron and ion are close to each other and slow and smoth dynamics when they are apart. You may gain significant performance increases when simulating with variable timesteps. Also you may alleviate some of the problems with your singularity, as time dependent schemes can be tuned to have sufficient precision in the vicinity of it. $\endgroup$
    – MPIchael
    Mar 14, 2023 at 9:30
  • 1
    $\begingroup$ With variable time steps you lose the advantage of symplectic methods, as each time step preserves a different perturbed energy. In that case it is better to employ variable-step RK 78 or higher order methods from the start $\endgroup$ Mar 14, 2023 at 10:12
  • $\begingroup$ @MPIchael Hi and thank you for your comment. Would you mind mentioning an example of such variable-time-step method please? Just to help me to start some reading. So far, I was disregarding these schemes due to the reason mentioned by Lutz Lehmann. Thank you. $\endgroup$
    – michalt
    Mar 15, 2023 at 1:26
  • $\begingroup$ @LutzLehmann Thank you for your comment. I will give it a try as it is quite simple thing to do. However, in my initial experiments with this problem, I first tried RK45 with completely disasterous results. Maybe the methods of higher order will work better. Is it worth introducing a "universal time" transformation so that d/dt=(1/r) d/ds, where r is the distance from the singulatiry? $\endgroup$
    – michalt
    Mar 15, 2023 at 1:31
  • $\begingroup$ Yes, or use the size of the ODE function, the first calculated derivatives vector in a step, to scale the step size, so that all steps have about the same length. This works nicely in some examples, removes the undersampling close to singularities, but might give too many steps in the more boring regions. $\endgroup$ Mar 15, 2023 at 6:59


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