{
  "annotation_id": "595edc7f-a367-45de-9f7b-ae21dd8d8392",
  "date_created": "2026-03-02T18:00:32.233000Z",
  "date_modified": "2026-03-02T18:00:32.233000Z",
  "file_hash": "2705dc99afac9f5ba4413119b44217acb9b3426f73441a104756bd01d10d2d31",
  "private": false,
  "record": {
    "abstract": "We have developed an efficacious algorithm for simulation of the beam-beam\ninteraction in synchrotron colliders based on the nonlinear $\\delta f$ method,\nwhere $\\delta f$ is the much smaller deviation of the beam distribution from\nthe slowly evolving main distribution $f_0$. In the presence of damping and\nquantum fluctuations of synchrotron radiation it has been shown that the slowly\nevolving part of the distribution function satisfies a Fokker-Planck equation.\nIts solution has been obtained in terms of a beam envelope function and an\namplitude of the distribution, which satisfy a coupled system of ordinary\ndifferential equations. A numerical algorithm suited for direct code\nimplementation of the evolving distributions for both $\\delta f$ and $f_0$ has\nbeen developed. Explicit expressions for the dynamical weights of\nmacro-particles for $\\delta f$ as well as an expression for the slowly changing\n$f_0$ have been obtained.",
    "arxiv_id": "physics/0010055",
    "authors": [
      "Yunhai Cai",
      "Alexander W. Chao",
      "Stephan I. Tzenov",
      "Toshi Tajima"
    ],
    "categories": [
      "physics.acc-ph",
      "physics.comp-ph"
    ],
    "title": "Nonlinear $\\delta f$ Method for Beam-Beam Simulation",
    "url": "https://arxiv.org/abs/physics/0010055"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "6646a8a5-6509-44b6-9966-766c52193f3c",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}