{
  "annotation_id": "265ae043-eae4-4fbd-8a05-5a038e754ba7",
  "date_created": "2026-03-02T18:01:21.686000Z",
  "date_modified": "2026-03-02T18:01:21.686000Z",
  "file_hash": "07d02289e4167343b3e386dcbf54988a381b02ce40899f2d91c1e847459c64bc",
  "private": false,
  "record": {
    "abstract": "It is well-known that the Liouville equation of statistical mechanics is\nrestricted to systems where the total number of particles (N) is fixed. In this\npaper, we show how the Liouville equation can be extended to systems where the\nnumber of particles can vary, such as in open systems or in systems where\nparticles can be annihilated or created. A general conservation equation for an\narbitrary dynamical variable is derived from the extended Liouville equation\nfollowing Irving and Kirkwood\u0027s2 technique. From the general conservation\nequation, the particle number conservation equation is obtained that includes\ngeneral terms for the annihilation or creation of particles. It is also shown\nthat the grand canonical ensemble distribution function is a particular\nstationary solution of the extended Liouville equation, as required. In\ngeneral, the extended Liouville equation can be used to study nonequilibrium\nsystems where the total number of particles can vary.",
    "arxiv_id": "physics/9809039",
    "authors": [
      "Michael H. Peters"
    ],
    "categories": [
      "physics.chem-ph",
      "cond-mat.stat-mech"
    ],
    "title": "An extended liouville equation for variable particle number systems",
    "url": "https://arxiv.org/abs/physics/9809039"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "ede09759-de3b-43ac-9bc5-04d358302168",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}