We investigated the transverse beam dynamics in a thermal wave model by using a functional method. It can describe the beam optical elements separately with a kernel for a component. The method can be applied to general quadrupole magnets beyond a thin lens approximation as well as drift spaces. We found that the model can successfully describe the PARMILA simulation result through an FODO lattice structure for the Gaussian input beam without space charge effects. The thermal wave model is an efficient way to study the beam dynamics of relativistic charged particles. The Schrö-dinger-type equation in the model governs the time evolution of the beam wave function whose squared magnitude is proportional to the particle number densities [1]. The model has successfully explained the filamentation of a particle beam and the self-pinching equilibrium in collisionless plasma [2]. It was also used to estimate the luminosity in a linear collider where a spherical aberration was present [3]. The model can also provide some insight into a halo formation by introducing a Gaussian slit [4].Transverse beam dynamics in a one spatial dimension is another application area of the thermal wave model. In Ref.[5], the authors investigated the beam wave function through a quadrupole magnet with sextupole and octupole perturbations followed by a long drift space under a thin lens approximation. There is also a Letter on the phase space behavior of particle beams in the transverse directions where the Wigner and Husimi functions are used as the phase space distribution functions [6].In this work, we investigate the transverse beam dynamics in a two-dimensional trace (x − x or y − y ) space in the thermal wave model by using the functional integral method [7]. Because the method can be extended to general lattice structures including quadrupole magnets beyond a thin lens approximation limit and it can treat the beam optical elements individually, it is possible to systematically analyze a beam motion in a realistic environment such as an FODO lattice. We found that the model can successfully explain the PARMILA [8] simulation results with Gaussian input beams in a two-dimensional trace space under the condition that the space charge effects are negligible. We note that this method can explain a low energy particle behavior as well as the relativistic motion of the charged particles if the important interactions are related to the external linear optical elements such as the quadrupole magnets and the random motion described by a beam emittance.In the thermal wave model, the time evolution of the beam wave function for the relativistic charged particles can be described by the Schrödinger-type equation in the transverse directions. Because the beam dynamics are usually described in 0375-9601/$ -see front matter
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