dorsal/arxiv
View SchemaPartition functions and graphs: A combinatorial approach
| Authors | Allan I. Solomon, Pawel Blasiak, Gerard Duchamp, Andrzej Horzela, Karol A. Penson |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0409082 |
| URL | https://arxiv.org/abs/quant-ph/0409082 |
Abstract
Although symmetry methods and analysis are a necessary ingredient in every physicist's toolkit, rather less use has been made of combinatorial methods. One exception is in the realm of Statistical Physics, where the calculation of the partition function, for example, is essentially a combinatorial problem. In this talk we shall show that one approach is via the normal ordering of the second quantized operators appearing in the partition function. This in turn leads to a combinatorial graphical description, giving essentially Feynman-type graphs associated with the theory. We illustrate this methodology by the explicit calculation of two model examples, the free boson gas and a superfluid boson model. We show how the calculation of partition functions can be facilitated by knowledge of the combinatorics of the boson normal ordering problem; this naturally gives rise to the Bell numbers of combinatorics. The associated graphical representation of these numbers gives a perturbation expansion in terms of a sequence of graphs analogous to zero - dimensional Feynman diagrams.
{
"annotation_id": "9c9c4bb9-0b86-4aee-af83-66784e66bc55",
"date_created": "2026-03-02T18:02:10.072000Z",
"date_modified": "2026-03-02T18:02:10.072000Z",
"file_hash": "5a0f04c6421f91ced022f1689840792efc5ec5eff81399659c5fbc37dcf17c89",
"private": false,
"record": {
"abstract": "Although symmetry methods and analysis are a necessary ingredient in every\nphysicist\u0027s toolkit, rather less use has been made of combinatorial methods.\nOne exception is in the realm of Statistical Physics, where the calculation of\nthe partition function, for example, is essentially a combinatorial problem. In\nthis talk we shall show that one approach is via the normal ordering of the\nsecond quantized operators appearing in the partition function. This in turn\nleads to a combinatorial graphical description, giving essentially Feynman-type\ngraphs associated with the theory. We illustrate this methodology by the\nexplicit calculation of two model examples, the free boson gas and a superfluid\nboson model. We show how the calculation of partition functions can be\nfacilitated by knowledge of the combinatorics of the boson normal ordering\nproblem; this naturally gives rise to the Bell numbers of combinatorics. The\nassociated graphical representation of these numbers gives a perturbation\nexpansion in terms of a sequence of graphs analogous to zero - dimensional\nFeynman diagrams.",
"arxiv_id": "quant-ph/0409082",
"authors": [
"Allan I. Solomon",
"Pawel Blasiak",
"Gerard Duchamp",
"Andrzej Horzela",
"Karol A. Penson"
],
"categories": [
"quant-ph",
"math.CO"
],
"title": "Partition functions and graphs: A combinatorial approach",
"url": "https://arxiv.org/abs/quant-ph/0409082"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "2f873a3c-3f63-42a1-8fe4-04a34d7cbf57",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}