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A classic reference on cluster expansions in mathematical physics (specially statistical mechanics) is these lecture notes by professor Brydges for a les Houches course in 1984 on the mentioned topic. However, these lectures took place over 30 years ago and I wonder if the material is outdated by now. Is it still the best place to study cluster expansions in statistical mechanics? How did the results and/or techniques used in his lectures changed by now? Are there more modern references on this subject?

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  • $\begingroup$ The number of terms in the Mayer expansion grows so copiously that for many realistic computations is almost useless. I haven't seen applications of the technique beyond typical classroom examples. For examples, the 4th order has 64 terms, not all of them inequivalent, of course, but the ultimate number of integrals you have to compute is still very large. If the number of particles is large one usually uses a (mean-)field theory and the perturbation techniques therein. $\endgroup$
    – Daniel Castro
    Commented Jan 11, 2021 at 8:37
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    $\begingroup$ @DanielCastro: The purpose of these cluster expansions is not so much doing computations but proving theorems and in particular producing convergent expansions for quantities of interest like pressure, correlations etc. in models of statistical mechanics and quantum field theory. So in this regard they are very far from "useless". $\endgroup$
    – Abdelmalek Abdesselam
    Commented Jan 11, 2021 at 18:35

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An impressive result of the method of cluster expansion is to derive the van der Waals equation of state for gases. Indeed, two-body interactions give the corrections to the ideal gas law. So it can be used for practical calculations!

But as Abdelmalek has remarked, the method is often used in mathematical physics to prove results. Connections with combinatorics have also been pointed recently, including the related viral expansion.

Below are some recent references; the list is not exhaustive! Notice that the method is well explained in the recent book of Friedli and Velenik (Statistical mechanics of lattice systems. A concrete mathematical introduction. Cambridge University Press, Cambridge, 2018. xix+622 pp. ISBN: 978-1-107-18482-4).

Nguyen, Tong Xuan; Fernández, Roberto Convergence of cluster and virial expansions for repulsive classical gases. J. Stat. Phys. 179 (2020), no. 2, 448–484.

Jansen, Sabine; Tsagkarogiannis, Dimitrios Cluster expansions with renormalized activities and applications to colloids. Ann. Henri Poincaré 21 (2020), no. 1, 45–79.

Jansen, S. Cluster expansions for Gibbs point processes. Adv. in Appl. Probab. 51 (2019), no. 4, 1129–1178.

Pulvirenti, Elena; Tsagkarogiannis, Dimitrios Finite volume corrections and decay of correlations in the canonical ensemble. J. Stat. Phys. 159 (2015), no. 5, 1017–1039.

Jansen, Sabine; Tate, Stephen J.; Tsagkarogiannis, Dimitrios; Ueltschi, Daniel Multispecies virial expansions. Comm. Math. Phys. 330 (2014), no. 2, 801–817.

Morais, Thiago; Procacci, Aldo Continuous particles in the canonical ensemble as an abstract polymer gas. J. Stat. Phys. 151 (2013), no. 5, 830–849.

Faris, William G. Combinatorics and cluster expansions. Probab. Surv. 7 (2010), 157–206.

Poghosyan, Suren; Ueltschi, Daniel Abstract cluster expansion with applications to statistical mechanical systems. J. Math. Phys. 50 (2009), no. 5, 053509, 17 pp.

Faris, William G. A connected graph identity and convergence of cluster expansions. J. Math. Phys. 49 (2008), no. 11, 113302, 14 pp.

Fernández, Roberto; Procacci, Aldo Cluster expansion for abstract polymer models. New bounds from an old approach. Comm. Math. Phys. 274 (2007), no. 1, 123–140.

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Let me add to Daniel's list a couple links to some hopefully pedagogical notes I wrote for a course I taught a while ago.

For two hopefully instructive examples of applications to seemingly unrelated problems in many-body quantum mechanics and nonrelativistic QFT, see

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