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Cluster and virial expansions for the multi-species Tonks gas. (English) Zbl 1341.82015

The author deals with one-dimensional systems of non-overlapping rods on a continuous segment \([0,L]\) or a discrete interval \(\{0,1,\dots,L-1\}\). There are countably many types \(k\) of rods, coming each with a length \(l_k \geq 0\) and an activity \(z_k\). The main results of this paper are necessary and sufficient convergence criteria for the expansion of the pressure in terms of the activities \(z_k\) and the densities \(\rho_k\). The author provides explicit formulas for the pressure-activity expansion that are interesting from a combinatoric point of view. In the continuous case, the author finds that the activity expansion is (up to signs) the multivariate exponential generating function for labeled rooted colored trees. This generalizes the well-known relation between non-overlapping rods and labeled rooted trees. Noting that the key tool is the fixed point equation for the pressure, the author provides a physical explanation in terms of van der Waals mixtures and a probabilistic explanation in terms of renewal processes.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C15 Coloring of graphs and hypergraphs
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