On the Mayer Series of Two-Dimensional Yukawa Gas at Inverse Temperature in the Interval of Collapse

We prove a theorem on the minimal specific energy for a \(\pm 1\) charged particles system, interacting through a class of pair potential v, that may be stated as follows: suppose v may be represented by a scale mixtures of d-dimensional Euclid’s hat. If the number of particles n is even, then their interacting energy \(U_{n}\) divided by n is minimized by a constant B at the configurations with total charge zero and all particles collapsed to a point; if n is odd, then the ratio \(U_{n}/(n-1)\) is minimized by a constant \(\bar{B}=B\) at the configurations with total charge \(\pm 1\) and all particles collapsed to a point. The theorem is then used to investigate the convergence of the Mayer series for a gas of \(\pm 1\) charged particles interacting through the two-dimensional Yukawa pair potential v for inverse temperatures in the collapse interval \([4\pi ,8\pi )\). The convergence is proved in the present paper up to the second threshold \(6\pi \) using the decomposition of the Yukawa potential into scales of modified Bessel functions (standard) and into scale mixtures of Euclid’s hat. Moreover, assuming that (i) neutral subclusters of size smaller than an odd number \(k>1\) do not collapse inside a cluster of size larger than k for \(\beta \) in the threshold interval \([8\pi (k-2)/(k-1),8\pi k/(k+1))\) and (ii) they satisfy a technical condition, then the Mayer series, discarding the first even coefficients of order smaller than k, converges.