Continuos particles in the Canonical Ensemble as an abstract polymer gas

We revisit the expansion recently proposed by Pulvirenti and Tsagkarogiannis for a system of $N$ continuous particles in the canonical ensemble. Under the sole assumption that the particles interact via a tempered and stable pair potential and are subjected to the usual free boundary conditions, we show the analyticity of the Helmholtz free energy at low densities and, using the Penrose tree graph identity, we establish a lower bound for the convergence radius which happens to be identical to the lower bound of the convergence radius of the virial series in the grand canonical ensemble established by Lebowitz and Penrose in 1964. We also show that the (Helmholtz) free energy can be written as a series in power of the density whose $k$ order coefficient coincides, modulo terms $o(N)/N$, with the $k$-order virial coefficient divided by $k+1$, according to its expression in terms of the $m$-order (with $m\le k+1$) simply connected cluster integrals first given by Mayer in 1942. We finally give an upper bound for the $k$-order virial coefficient which slightly improves, at high temperatures, the bound obtained by Lebowitz and Penrose.