On the Mayer series of two-dimensional Yukawa gas at inverse temperature in the interval of collapse

TL;DR

This paper proves a theorem on the minimal energy configurations of a two-dimensional Yukawa gas, resolving issues with the convergence of the Mayer series in the collapse interval, and extends the convergence up to the second threshold.

Contribution

It introduces a new theorem on minimal energy configurations for Yukawa interactions and applies it to prove convergence of the Mayer series up to 6π, with insights to extend to 8π.

Findings

Proved minimal energy configurations for charged particles with Yukawa potential.

Established convergence of the Mayer series up to the second collapse threshold 6π.

Provided mechanisms to extend convergence results up to 8π.

Abstract

A Theorem on the minimal specific energy for a system with \pm 1 charged particles interacting through the Yukawa pair potential v is proved which may stated as follows. Let v be represented by scale mixtures of d-dimensional Euclid's hat (cutoff at short scale distances) with d\geq 2. For any even number of particles n, the interacting energy U_{n} divided by n, attains an n-independent minimum at a configuration with zero net charge and particle positions collapsed altogether to a point. For any odd number of particles n, the ratio U_{n}/(n-1) attains its minimum value, the same of the previous cases, at the configuration with \pm 1 net charge and particle positions collapsed to a point. This Theorem is then used to resolve an obstructive remark of an unpublished paper (Remark 7.5 of \cite{Guidi-Marchetti}) which, whether the standard decomposition of the Yukawa potential into scales were adopted, would impede a direct proof of the convergence of the Mayer series of the two-dimensional Yukawa gas for the inverse temperature in the whole interval [4\pi ,8\pi ) of collapse. In the present paper, it is proven convergence up to the second threshold 6\pi and its given explanations on the mechanism that allow it to be extend up to 8\pi . The paper distinguishes the matters concerning stability from those related to convergence of the Mayer series. In respect to the latter the paper dedicates to the Cauchy majorante method applied to the density function of Yukawa gas in the interval of collapses. It also dedicates to the proof of the main Theorem and estimates of the modified Bessel functions of second kind involved in both representations of two-dimensional Yukawa potential: standard and scale mixture of the Euclid's hat function.

Equations

$$\sum _{1\le i,j\le n}{\bar{z}}_i{z}_{j}f\left(x_i-x_j\right)\ge 0,\forall z_1,\dots ,z_n\in \mathbb{C}.$$

$$\frac{4}{\pi s^2}{\chi }_{s/2}\ast {\chi }_{s/2}(x)\equiv h(\left|x\right|/s)$$

$${\int }_0^{\infty }\phi \left(\left|x\right|/s\right)d\nu (s)$$

$$v(x)=\frac{1}{2\pi }{\int }_{{\mathbb{R}}^2}{e}^{i\xi \cdot x}\frac{1}{{\xi }^2+1}d\xi =\frac{1}{2\pi }{K}_0(\left|x\right|)$$

$$v(x)={\int }_0^{\infty }h(\left|x\right|/s)g(s)ds$$

$$h(w)=\frac{2}{\pi }\left(\arccos w-w\sqrt{1-w^2}\right),\text{if}0<w\le 1$$

$$g(s)=\frac{-s}{4\pi }{\int }_s^{\infty }{K}_0^{\prime \prime \prime }(r)\frac{r}{\sqrt{r^2-s^2}}dr.$$

$$v(x)={\int }_0^1\dot{v}(s;x)ds$$

$$v(x)=\frac{1}{2\pi }{\int }_0^1\tilde{h}(\left|x\right|/s)\frac{ds}{s}$$

$$\varphi =\sum _{j\ge 0}{\zeta }_j$$

$$\mathbb{E}{\zeta }_j(x){\zeta }_j(y)={\int }_{I_j}g(s)h\left(\left|x-y\right|/s\right)ds\equiv v_{I_j}(x-y).$$

$${\mathcal{Z}}_0\left(\varphi \right)$$

$${\mathcal{V}}_0(\varphi )$$

$$v(t,x)={\int }_{t_0}^{t}h(\left|x\right|/s)g(s)ds$$

$$\frac{\partial \mathcal{Z}}{\partial t}=\frac{1}{2}{\Delta }_{\dot{v}}\mathcal{Z},\underset{t\searrow t_0}{\lim }\mathcal{Z}(t,\varphi )={\mathcal{Z}}_0(\varphi )$$

$${\Delta }_{\dot{v}}\mathcal{Z}={\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}dxdy\dot{v}(t,x-y)\frac{{\delta }^2\mathcal{Z}}{\delta \varphi (x)\delta \varphi (y)}.$$

$$\frac{\partial \mathcal{V}}{\partial t}=\frac{1}{2}{\Delta }_{\dot{v}}\mathcal{V}+\frac{1}{2}{\left(\nabla \mathcal{V},\nabla \mathcal{V}\right)}_{\dot{v}},\underset{t\searrow t_0}{\lim }\mathcal{V}(t,\varphi )={\mathcal{V}}_0(\varphi )$$

$${\left(\nabla \mathcal{V},\nabla \mathcal{V}\right)}_v={\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}dxdy\dot{v}(t,x-y)\frac{\delta \mathcal{V}}{\delta \varphi (x)}\frac{\delta \mathcal{V}}{\delta \varphi (y)}.$$

$$\mathcal{V}\left(t,\varphi \right)=\sum _{n\ge 1}\sum _{\alpha :|\alpha |=n}\frac{1}{\alpha !}\frac{{\partial }^{\alpha }\mathcal{V}}{\partial {\varphi }^{\alpha }}(t,0){\varphi }^{\alpha }$$

$$\mathcal{V}\left(t,\varphi \right)=\sum _{n\ge 1}\frac{1}{n!}\int d^n\varrho {\psi }_n^c(t,{\zeta }_1,\dots ,{\zeta }_n):\exp \left(i\sqrt{\beta }{\sum }_{j=1}^n{\sigma }_j\varphi (x_j)\right):$$

$${\Delta }_{\dot{v}}\mathcal{V}\left(t,\varphi \right)$$

$$U_n({\zeta }_1,\dots ,{\zeta }_n;v)=\sum _{1\le i<j\le n}{\sigma }_{i}v\left(x_i-x_j\right){\sigma }_j.$$

$$U_n({\zeta }_1,\dots ,{\zeta }_n;v)\ge -nB$$

$$U_n({\zeta }_1,\dots ,{\zeta }_n;v)\ge -\frac{1}{2}v(0)\sum _{j=1}^n{\sigma }_j^2$$

$$e$$

$$e_n$$

$$\bar{e}=\underset{n\ge 2}{\inf }\underset{\begin{array}{c}\left({\zeta }_1,\dots ,{\zeta }_n\right)\\ \mathrm{non}-\mathrm{neutral}\end{array}}{\inf }\frac{1}{n-1}{U}_n({\zeta }_1,\dots ,{\zeta }_n;h)$$

$$e\left(v\right)={\int }_{t_0}^{t}e\left(h(\cdot /s)\right)g(s)ds=e\left(h\right)\cdot {\int }_{t_0}^{t}g(s)ds.$$

$$U_n({\zeta }_1,\dots ,{\zeta }_n;h)\ge \frac{1}{2}\left(\left|{\sum }_{j=1}^n{\sigma }_j\right|-{\sum }_{j=1}^n{\sigma }_j^2\right)$$

$$\mu (a)=\underset{n\ge 2}{\sup }\underset{\begin{array}{c}\left(x_1,\dots ,x_n\right)\in {\mathbb{R}}^n:\\ |x_i-x_j|>a\end{array}}{\sup }\sum _{i=1}^n\max \left(-\varphi \left(\left|x_i\right|\right),0\right)$$

$${\psi }_n^c({\zeta }_1,\dots ,{\zeta }_n;v)=\sum _{G\mathrm{connected}}\prod _{⟨ij⟩\in E(G)}\left(\exp \left(-\beta {\sigma }_i{\sigma }_{j}v\left(\left|x_i-x_j\right|\right)\right)-1\right),$$

Full text

MSC: 60J45, 42A82, 97K10, 82B21, 82B28

Keywords: two-dimensional Yukawa potential, minimal specific energy, scale mixtures of Euclid’s hat, stability, Mayer series, collapse interval

1 Introduction and background of tools and methods

The present paper investigates a system of particles with $\pm 1$ charges living in a two dimensional Euclidian space and interacting through the Yukawa pair potential $v(x)=\left(-\Delta+1\right)^{-1}(0,x)$. Because of Yukawa and Coulomb potentials look like the same at short distances, the two-dimensional Yukawa gas inherits the same instabilities of the corresponding Coulomb system when the inverse temperatures $\beta$ belongs to the interval $[4\pi,8\pi)$, in which a sequence of collapses of neutral cluster of size $2n$ occurs at the thresholds $\beta_{2n}=8\pi\left(1-1/2n\right)$, $n\in\mathbb{N}$. It remains an open problem for this system to establish convergence of the Mayer series in powers of activity $z$, with the first even terms removed from the series, how many depending on $\beta\in[4\pi,8\pi)$. It is our purpose to revisit this long standing problem.

Benfatto[Be] and collaborators from the Italian school (see references therein) initiate a program using iterated Mayer series for pressure (and correlation functions) together with ideas from the work of Gopfert-Mack [GoMa] and Imbrie [I]. Brydges and Kennedy [BK] have also considered the Mayer expansion of the two-dimensional Yukawa gas in the context of the Hamilton-Jacobi equation. We adopt in present investigation their continuum scaling renormalization method, adding to that approach a new ingredient. The novelty is related with the (short–range) decomposition of the Yukawa potential into scales. Instead of the standard decomposition $v(x)=\displaystyle\int_{-\infty}^{0}\left(d\left(-\Delta+e^{-s}\right)^{-1}(0,x)/ds\right)ds$ (or the discrete version of it) adopted in the previous work, we use the scale mixture $v(x)=\displaystyle\int_{0}^{1}g(s)h(\left|x\right|/s)ds$ of Euclid’s hat $h(r)$. Using a concept introduced by Basuev [Ba1], we first prove a theorem that the minimal specific energy $e(v)$ and the constrained (to non zero net charge) modified minimal specific energy $\bar{e}(v)$ are equal.

Our main theorem on specific energy when applied in the investigation of the two-dimensional Yukawa gas resolve an obstructive limitation that has been posed in an umpublished paper by Guidi and one of the authors (see Conjecture 2.3 and Remark 7.5 of [GM]) towards a direct proof of the convergence of the Mayer series on the entire interval $[4\pi,8\pi)$ whether the standard decomposition of the Yukawa potential were adopted. The limitation value for the $3$–particles interacting energy given by a numerical evaluation in [GM] is proven in Proposition 2.2 and Remark 2.3.

In the present paper, the convergence of the Mayer series is proven up to the second threshold $\beta\in[4\pi,6\pi)$ and it is provided a full explanation of the mechanisms that allow it to be extend up to $8\pi$. We distinguished the issues concerning stability from those related to convergence of the Mayer series. In respect to the latter, the paper dedicates in Section 3 to the Cauchy majorante method applied to the density function of Yukawa gas in the region of collapse. Regarding the former, we considered only the simplest case of collapse prevention of neutral pair of charges due the presence of other charges in the configuration.

We shall now review the tools and methods employed in present investigation. We refer to the references for detail.

Decomposition of radial positive functions of positive type.

Positive definite functions have arisen in many areas of (pure and applied) mathematics and physics (see [S] for an historical survey). A continuous function $f$ defined in $\mathbb{R}^{d}$ is called positive definite (abbreviated as p. d.) if the $n\times n$ real matrix $\left[f\left(x_{i}-x_{j}\right)\right]_{1\leq i,j\leq n}$ is positive definite for $n\in\mathbb{N}$ arbitrary elements $x_{1}$, …, $x_{n}$ of $\mathbb{R}^{d}$:

[TABLE]

The celebrate work of Bochner (see e.g [B]) characterizes these functions as follows: $f$ (with $f(0)=1$) is positive definite if, and only if, is a Fourier-Stieltjes transform $\check{\mu}(x)=\displaystyle\int_{\mathbb{R}^{d}}e^{ix\cdot\xi}d\mu(\xi)$ of a probability Borel measure $\mu$ on $\mathbb{R}^{d}$. Although powerful, Bochner’s theorem may be difficult to use in practice: how do we know that a given $f$ satisfies (1.1)? Even when explicit computation of Fourier transform is available, how do we represent $f$ into suitable scale mixture of elementary functions?

Recently (see [G, HS, JMR] and references therein), investigations towards extending Bochner’s theorem seek for concrete examples and easy checkable criteria of p. d. function. A particularly interesting subclass of p. d. functions, denoted in [JMR] by $\Omega_{d}^{+}$, is provided by radial continuous functions: $f(x)=\varphi(\left|x\right|)$ for some positive continuous function $\varphi$ of $\mathbb{R}_{+}$. A simple example of these functions that vanishes out of a ball $B_{s}$ of radius $s>0$ centered at origin is given by the Euclid’s hat ($d=2$)

[TABLE]

where $\chi_{r}(x)=\chi_{B_{r}}(x)$ is the characteristic function of $B_{r}$. In [JMR] Jaming, Matolcsi and Révész have identified certain compactly supported functions, alike this one, as extrema rays of the cone $\Omega_{d}^{+}$, playing the same role as the family $\left\{e^{i\xi\cdot x}\right\}$ for the Bochner’s theorem. So, if $\varphi$ is an extremum ray of $\Omega_{d}^{+}$ then, by Choquet representation,

[TABLE]

is an element of $\Omega_{d}^{+}$ for a suitable positive measure $\nu$ supported on the family of scales $\left\{\varphi\left(\left|x\right|/s\right)\right\}$ of $\varphi$. An open problem is to find all extrema of $\Omega_{d}^{+}$ (see [JMR]).

Geiting [G] and Hainzl-Seiringer [HS] give, on the other hand, complete characterizations of the subclass $H_{d}\subset\Omega_{d}^{+}$ that are formed by scaling mixtures of $d$–dimensional Euclid’s hat, extending Polya’s criterion on $\mathbb{R}^{d}$, for $d\geq 2$. Hainzl-Seiringer’s representation however suffices to make our point in the present work. Let us start with the two-dimensional Yukawa potential, which is an element of $\Omega_{2}^{+}$ given by the Green’s function $v(1/\sqrt{\kappa},x)=\left(-\Delta+\kappa\right)^{-1}(0,x)$ (the resolvent kernel of the Laplacian operator $\Delta=\partial^{2}/\partial x_{1}^{2}+\partial^{2}/\partial x_{2}^{2}$). Applying Fourier transform yields (with $v(x)\equiv v(1,x)$ and $v(1/\sqrt{\kappa},x)=v(1,\sqrt{\kappa}x)$. See e.g. Sec. 7.2 of [GJ])

[TABLE]

where $K_{0}$ is the modified Bessel function of second kind of order [math]. Hainzl-Seiringer’s formula for this function reads

[TABLE]

where ($h(0)=1$)

[TABLE]

$h(w)=0$ if $w>1$ and

[TABLE]

The mixture density function $g(s)$ for the Yukawa potential in $d=1$ and $3$ dimensions and the Coulomb potential in $d$–dimensions have closed forms (see Examples 1 and 2 of [HS]). For the Yukawa function in $2$–dimensions, however, $g(s)$ can only be written in term of Meijer $G$ –functions (see [BS] for an introduction): $g(s)=\sqrt{\pi}G_{13}^{30}\left(s^{2}/4\left|\genfrac{}{}{0.0pt}{1}{1/2}{0,1,3}\right.\right)/(2\pi s)$.

We observe that $h(w)$ is a convex function of $w\in\mathbb{R}_{+}$ and a mixture of the Euclid’s hat (1.4) preserves convexity. This useful property, as we shall see, distinguishes (1.4) from another common decomposition of (1.3) into scales (see Fig. 4): with $v(s,x)=\left(-\Delta+1/s^{2}\right)^{-1}(0,x)=\left(-\Delta+1\right)^{-1}(0,x/s)=K_{0}(\left|x\right|/s)$, we write

[TABLE]

by the fundamental theorem of calculus. Substituting the derivative $\dot{v}(s;x)$ with respect to $s$, yields

[TABLE]

where $\tilde{h}(w)=-wK_{0}^{\prime}(w)=wK_{1}(w)$, with $K_{1}$ the modified Bessel function of second kind of order $1$, like $h$ given by (1.5), decreases monotonously to [math] and satisfies $h_{1}(0)=1$ but changes from concave to convex as $w\in\mathbb{R}_{+}$ varies. The mixture density $g(s)$ for both decompositions of $v$, (1.4) and (1.7), behaves in the neighborhood of $s=0$ as $\left(2\pi s\right)^{-1}$ implying that $v(x)$ behaves as the Coulomb potential $(-1/2\pi)\log\left|x\right|$ at short distances.

Gaussian Processes and renormalization group.

Positive definite functions plays an important role on renormalization group (RG) methods in statistical physics. Brydges and collaborators [BGM] (see also [BT]) coined a term “finite range decomposition” to the mixture of different scales (1.2) for some compactly supported radial extremal functions $\varphi$. They used a probabilistic argument as follows: breaking up the range of integration into disjoint union of intervals $I_{j}=[L^{-j},L^{-j+1})$, $j\geq 1$ for $L>1$ and $I_{0}=[1,\infty)$, (1.4) may be seen as the “finite range” decomposition

[TABLE]

of a Gaussian process $\phi$ of mean $\mathbb{E}\phi(x)=0$ and covariance $\mathbb{E}\phi(x)\phi(y)=v\left(x-y\right)$ into a family of independent Gaussian processes $\left\{\zeta_{j}\right\}$ of mean $\mathbb{E}\zeta_{j}=0$ and covariance

[TABLE]

Since the covariance of a sum of independent Gaussian random variable is the sum of their covariances, $v=\displaystyle\sum_{j\geq 0}v_{I_{j}}$ equals (1.4). The authors of [BGM, BT] were also capable of applying suitable finite range decomposition to a large class of positive definite functions on $\mathbb{R}^{d}$ and $\mathbb{Z}^{d}$ that comprises integral kernels (Green’s functions) of certain elliptic operators, their corresponding finite differences and fractional powers.

When a statistical system is represented by the expectation $\mathbb{E}\mathcal{Z}$ of a functional $\mathcal{Z}\left(\phi\right)$, the decomposition (1.8) of the Gaussian field $\phi$ can be used to integrate out each $\zeta_{j}$ at a time. Let $\mathbb{E}^{(j)}$ denote the expectation with respect the Gaussian field $\zeta_{j}$. The renormalization group is a method of calculating the expectation $\mathbb{E}\mathcal{Z}$ through the sequence of maps $\mathcal{Z}_{j}\longmapsto\mathcal{Z}_{j+1}=\mathbb{E}^{(j+1)}\mathcal{Z}_{j}$ starting from $\mathcal{Z}_{0}=\mathcal{Z}$. The limit $\lim_{j\rightarrow\infty}\mathcal{Z}_{j}=\mathbb{E}\mathcal{Z}$, supposing it exists, is obtained provided $\mathcal{Z}_{j}\longmapsto\mathcal{Z}_{j+1}$ is amenable to be analyzed as a dynamical system depending on parameters in the initial condition. For instance, in the decomposition (1.8) of $\phi$ into finite range fields $\zeta_{j}$ corresponding to the Yukawa potential (1.4), the limit $j\rightarrow\infty$ drives the statistical system into the short scaling limit $s\rightarrow 0$ for which the potential diverges logarithmically. For an infinitely many-particle system with $\pm 1$ charges, the existence of $\lim_{j\rightarrow\infty}\mathcal{Z}_{j}$ expresses the thermodynamical stability of the system. We shall come back to this issue below.

Hamilton–Jacobi equation and majorant method.

Under the Kac–Siegert transformation, [FS, BM] the grand partition function for the two–dimensional Yukawa gas of particles with $\pm 1$ charges can be written as the expectation $\mathbb{E}\mathcal{Z}_{0}$ (with respect to the Gaussian field $\phi$) of

[TABLE]

where the parameters $\beta$ and $z$ are, respectively, the inverse temperature and activity and $:\cdot:_{v}$ indicates Wick ordering with respect to the potential $v$. In the present work, we shall adopt the continuum scale decomposition (1.3) instead of (1.8). The induced RG dynamics is thus generated by a Hamilton-Jacobi equation as proposed in [BK] by Brydges and Kennedy. Let us expand these ideas in some detail. A scale–dependent–interaction $v:\mathbb{R}_{+}\times\mathbb{R}^{2}\longrightarrow\mathbb{R}$ is introduced replacing (1.4) by a mixture supported in a finite interval $\left[t_{0},t\right]$ of scales

[TABLE]

where $t_{0}>0$ is a cutoff of the short scale distances. Since $g$ and $h$ are continuous, we have $\lim_{t\searrow t_{0}}v(t,x)\equiv 0$. The renormalization group is now given by a convolution mapping $\left(t,\phi\right)\longmapsto\mathcal{Z}(t,\phi)=\mathbb{E}^{(t)}\mathcal{Z}_{0}\mathcal{(\phi+\cdot)}$ with initial data $\mathcal{Z}(t_{0},\phi)=\mathcal{Z}_{0}\mathcal{(\phi)}$, where $\mathbb{E}^{(t)}$ denotes the expectation with respect the Gaussian field $\zeta$ with covariance $v\left(t,x-y\right)$. Formally, $\mathcal{Z}(t,\phi)$ satisfies the initial value problem of a “heat equation”

[TABLE]

where $\dot{v}(t,x):=\partial v/\partial t(t,x)=g(t)h(\left|x\right|/t)$, by the fundamental theorem of calculus, is the weighted Euclid’s hat scaled by $t$ and $\Delta_{\dot{v}}$ is the “Laplacian” operator

[TABLE]

Writing $\mathcal{Z}(t,\phi)=\exp\left(\mathcal{V}(t,\phi)\right)$, the heat equation turns into a nonlinear equation for $\mathcal{V}$:

[TABLE]

where $\Delta_{\dot{v}}$ acts as in (1.11) and

[TABLE]

In [BK], the authors considered the random field $\phi$ on $\mathbb{Z}^{d}$ instead, for which the functional derivative $\displaystyle\int dx~{}v(x)~{}\delta/\delta\phi(x)~{}\mathcal{V}(\phi)=\lim_{\varepsilon\rightarrow 0}\left(\mathcal{V}(\phi+\varepsilon v)-\mathcal{V}(\phi)\right)/\varepsilon$ becomes partial derivative $\partial/\partial\phi_{x}$ with respect to the variable $\phi_{x}\in\mathbb{R}$ at site $x$. Because of translation invariance, (1.11) and (1.13) diverges even if $\mathbb{R}^{2}$ is replaced by $\mathbb{Z}^{2}$ but this can be solved by fixing one point $x$ of $\mathbb{Z}^{2}$. Inserting the Taylor expansion (multi-index formula):

[TABLE]

into an integral equation equivalent to (1.12), a system of equations for derivatives of $\mathcal{V}$ (by collecting order by order terms), together with an appropriate norm, is used to majorize $\mathcal{V}\left(t,\phi\right)$ by the solution $\nu(t,\varphi)$ of a first order PDE equation in two independent real variables $\left(t,\varphi\right)$, $\varphi$ playing the role of chemical potential. The local existence and uniqueness of the initial value problem (1.12) are then proved in ref. [BK] (see Theorem 2.2 and Proposition 2.6 therein) for a domain in plane $\left(t,z\right)$ with $z=e^{\varphi}$ ($\beta$ may be included as well). Quoting the authors, these results are “the precise version of the Mayer expansion” for the pressure or correlations functions of statistical systems.

Brydges and Kennedy have also provided an equivalent system of ordinary differential equations for the Ursell functions (Lemma 3.3 of [BK]) which replaces (1.12) defined on $\mathbb{Z}^{d}$ and can be used for systems of point particles in $\mathbb{R}^{d}$. If $\left(\Omega,\mathcal{B},d\varrho(\zeta)\right)$ denotes the finite measure space on $\left\{-1,1\right\}\times\mathbb{R}^{2}$ corresponding to the possible states of a single particle (we united $\sigma$ and $x$ into $\zeta=\left(\sigma,x\right)$), the solution of (1.12) may be represented formally as

[TABLE]

where the Ursell functions $\psi_{n}^{c}(t,\zeta_{1},\ldots,\zeta_{n})$ are translational invariant and invariant under the action of the symmetric group $\mathbb{S}_{n}$ of permutations of the index set $\left\{1,\ldots,n\right\}$.111 Applying the functional calculus on (1.14) we obtain formally from (1.12) the system of equations (see eq. (3.5)) satisfied by the $\psi_{n}^{c}$’s. For instance, the Laplacian of $\mathcal{V}$ gives

$\displaystyle\Delta_{\dot{v}}\mathcal{V}\left(t,\phi\right)$ $\displaystyle=$ $\displaystyle\lim_{\varepsilon,\eta\rightarrow 0}\frac{1}{\varepsilon\eta}\left(\mathcal{V}\left(t,\phi+\varepsilon\dot{v}+\eta\dot{v}\right)-\mathcal{V}\left(t,\phi+\varepsilon\dot{v}\right)-\mathcal{V}\left(t,\phi+\eta\dot{v}\right)+\mathcal{V}\left(t,\phi\right)\right)$

$\displaystyle=$ $\displaystyle\sum_{n\geq 1}\frac{1}{n!}\int d^{n}\varrho\frac{-\beta}{2}\sum_{i\neq j}\sigma_{i}\sigma_{j}\dot{v}(t,x_{i}-x_{j})\psi_{n}^{c}(t,\zeta_{1},\ldots,\zeta_{n}):\!\exp\left(i\sqrt{\beta}\sum\nolimits_{j=1}^{n}\sigma_{j}\phi(x_{j})\right)\!:\ .$

. To make mathematical sense of the above equations (1.12) and (1.14) one can check, at the very end, whether the solution of the system of ODE’s for $\psi_{n}^{c}$’s agrees with the statements of [L1, L2] on point processes of infinitely many particles (consult [R] for the definition of $n$–point correlation and cluster functions and Theorem 5.4 of [Gi] for a hybrid approach combining methods employed for Poisson point process with correlation functions satisfied by the (sine-Gordon) representation (1.9) of the Yukawa gas). The present work will take the system of equations satisfied by the $\psi_{n}^{c}(t,\zeta_{1},\ldots,\zeta_{n})$ (see (3.5) and (3.6) below), together with the scale decomposition (1.10) for the Yukawa potential, as the starting point for our analysis.

Stability condition and minimal specific energy.

Stability of the interaction $v$ is a condition under which there exist the thermodynamic functions describing an infinitely large statistical system. Let $U_{n}$ be the total energy potential of the classical charged system of $n$ point particles at positions $x_{1}$, …, $x_{n}$ of $\mathbb{R}^{2}$, with respective charges $\sigma_{1}$, …, $\sigma_{n}\in\left\{-1,1\right\}$, interacting through a pair Yukawa potential:

[TABLE]

An interacting potential $v$ satisfies the stability condition if there exists $B>0$ such that

[TABLE]

holds for all $(\zeta_{1},\ldots,\zeta_{n})$ on the configurations space $\displaystyle\bigcup_{n}$ $\left(\left\{-1,1\right\}\times\mathbb{R}^{2}\right)^{n}$ (otherwise the specific energy $U_{n}/n$ would not be bounded from below).

The standard stability theorem for charged system due to Fisher and Ruelle [FR] (see Theorem I and eq. (III.7) therein) assures that: if $\hat{v}(\xi)=\left(1/2\pi\right)\int_{\mathbb{R}^{2}}v(x)e^{-i\xi\cdot x}dx\geq 0$ and $v(0)=\left(1/2\pi\right)\int_{\mathbb{R}^{2}}\hat{v}(\xi)d\xi<\infty$, then

[TABLE]

and, since $\sigma_{j}^{2}=1$, (1.16) is satisfied with $B=v(0)/2$. The proof of (1.17) follows from the “if” direction of Bochner’s theorem. For this, note that adding $1/2$ of each $i=j$ diagonal terms to (1.15) (i.e., (1.17) with the right hand side passed to the left), the quadratic form has to be positive as $v$ is positive definite. It follows from (1.3) that $\hat{v}(\xi)=\left(2\pi\left(\xi^{2}+1\right)\right)^{-1}\geq 0$ is a positive density but $v(x)$, which yields the stability constant $B$, grows unboundedly at $x=0$. As the self–energy $v(0)$ diverges logarithmically, the decomposition (1.7) or (1.4) has to be used instead. The thermodynamic functions are defined when the scales smaller than an $s_{0}>0$ are removed from the decomposition of $v$, but one has to prove that they remain well defined after the cutoff is removed.

Let $v$ be the scale mixtures of Euclid’s hat (1.10), cutoff on the short scales. We introduce the minimal specific energy $e=e(h)$ of $h$ at the scale $s=1$

[TABLE]

and the modified minimal specific energy $\bar{e}=\bar{e}(h)$,

[TABLE]

where the infimum is now taken over all non–neutral configurations $\left(\zeta_{1},\ldots,\zeta_{n}\right)$: $\left(x_{1},\ldots,x_{n}\right)\in\mathbb{R}^{2n}$ and $\left(\sigma_{1},\ldots,\sigma_{n}\right)\in\left\{-1,1\right\}^{n}$ such that $\sum_{j=1}^{n}\sigma_{j}\neq 0$. It is clear that minimal specific energy (modified or not) of the scaled Euclid’s hat $h(\cdot/s)$ with $s\neq 1$ have all the same value by homogeneity of the infimum and we have

[TABLE]

In the present paper we determine both specific energies $e$ and $\bar{e}$ and characterize the configuration that they are attained for $h$ and, consequently, for (1.10) by (1.20). From definitions (1.16) and (1.18), we have $-e(h)\leq h(0)/2=1/2$. We show that this is in fact an equality and, moreover, $e=\bar{e}=-1/2$. More precisely, we have proven in Sec. 2 an improvement of (1.17)

[TABLE]

from which the equality of specific energies follows at once. Observe that the inequality (1.21) turns out to be an equality for certain configurations.

We should mention a short note written by Basuev [Ba1] on the minimal specific energy for classical one–specie system of particles in $\mathbb{R}^{3}$, interacting through a radial two-body potential $\phi(\left|x-y\right|)$. The conclusions of this investigation may be stated as follows. Suppose that $\phi$ satisfies the two conditions that defines what now-a-days is called Basuev potentials:[LPY] there is $a>0$ such that: i. $\phi\left(\left|x\right|\right)\geq\phi\left(a\right)>0$, for all $\left|x\right|\leq a$; and ii. $\phi\left(a\right)>2\mu(a)$ where

[TABLE]

is finite (the supremum is taken over all configurations whose distance between any pair exceeds $a$). Then $\phi$ and the potential $\phi^{a}$, given by $\phi^{a}\left(\left|x\right|\right)=\phi\left(\left|x\right|\right)$ if $\left|x\right|>a$ and $\phi(\left|x\right|)=\phi(a)$ if $\left|x\right|\leq a$, are stable with stability constant $B=\mu(a)/2$, and their minimal specific energy are equal: $e\left(\phi\right)=e\left(\phi^{a}\right)$ and $\bar{e}\left(\phi\right)=\bar{e}\left(\phi^{a}\right)$. Colloquially, it says that an increase of the positive part of the potential does not reduce the binding energy of the system. Basuev class includes potentials of Lenard-Jones type introduced by Fisher (see e.g. [FR], [RT] for an overview and [LPY] for a proof of this statement).

Basuev [Ba1] has in addition shown that $e\left(\phi\right)\leq\bar{e}\left(\phi\right)\leq 13e\left(\phi\right)/12$ for potentials $\phi$ such that $\min_{x\in\mathbb{R}^{3}}\phi(\left|x\right|)=-\lambda<0$ and has stated that $e\left(\phi\right)=\bar{e}\left(\phi\right)$ holds for the majority of stable potentials which is useful in applications. Our result on the equality $e(h)=\bar{e}(h)$ differs, however, in many respects. Typical Basuev potentials are bounded from below by a negative constant $-\lambda$, repulsive at short, attractive and integrable at large distances. Equality in this situation occurs when the infimum in $n$ of (1.18) is attained at $\infty$.222 Because the minimal specific energy may be written as $\bar{e}=\inf_{n\geq 2}\frac{n}{n-1}e_{n}$. The Yukawa potential $v$ on the other hand, as scale mixtures of Euclid’s hat $h(\left|x\right|/s)$ weighted by $g(s)$ where $h$, $g$ and consequently $v$ are all positive functions, repeals (attracts) two particles with the same (opposite) charges in its entire support. The infimum in (1.18) is attained for neutral configurations, when $n_{+}=m\in\mathbb{N}$ positive and $n_{-}=m$ negative charges collapse into one point while the infimum in the modified specific energy $\bar{e}(v)$ is attained when $\left|n_{+}-n_{-}\right|=1$.

One versus iterated Mayer expansion.

The Ursell functions can be written by the well known formula introduced by Mayer (see [UF]):

[TABLE]

where the sum runs over all connected linear graphs $G$ with vertices in the index set $\left\{1,\ldots,n\right\}$ and $E(G)$ denotes the set of edges of $G$. As far as the estimation of pressure and correlation functions are concerned, equation (1.22) is not useful due the cardinality of its sum. To reduce the sum over connected Mayer graphs to labeled trees, Penrose [P] has exploited cancellations occurring on the formula under proper re-summation and proved that the Mayer series converge provided the potential $v$ is stable, integrable at large distances and has, in addition, a hard core condition which recently has shown [PY] to be unnecessary (see also [BM] for an overview and extensions). The cardinality of labeled trees of order $n$ is $n^{n-2}$ by the famous Cayley theorem, which makes the tree graph identities suitable for the estimation of thermodynamical functions. Among the proposed tree graph formulas now available we indicate the one in Theorem 3.1 of [BK] as the most adequate to our purposes of representing the Ursell functions $\psi_{n}^{c}(t,\zeta_{1},\ldots,\zeta_{n})$ defined by (1.22) with the scale–dependent–interaction (1.10) in the place of $v$. Such Ursell functions satisfy the system of ordinary differential equations (3.5).

One particular tree graph identity due to Basuev [Ba2] is however worth mentioning in the context of the present work. The representation of Basuev works for radial potentials in $\mathbb{R}^{d}$ of the form $\phi=\phi^{a}+\delta$ (see definition of Basuev potentials above), where $\phi^{a}(r)=\phi(r)$ for $r=\left|x\right|>a$, $\phi^{a}(r)=\phi(a)$ for $r\leq a$, is stable and $\delta(r)=\phi(r)-\phi^{a}(r)>0$ for $r\leq a$ and $\delta(r)=0$ for $r>a$, which may include hard–core: $\delta(r)=\infty$, $r\leq a$. To estimate the Ursell functions efficiently, Basuev uses the modified stability condition

[TABLE]

instead of (1.16), where in the majority of cases important for applications $\bar{B}$ is equal or closed to $B$. It might appear that a slight improvement on the stability bound would not affect the radius of convergence of Mayer series. It turns out, however, that the estimate of the Ursell functions through the Basuev tree graph identity works so well when (1.23) is applied (see particularly equations (15) and (16) of [Ba2]) that expressive improvements on the convergence are reported (at low temperatures) in Basuev paper, as well as in [LPY].

Let us now explain how the estimate on the Ursell functions gets improved by (1.23) in our case. It is known that the Mayer series for the pressure of a two-dimensional Yukawa gas [BK, Be, GM]

[TABLE]

converges if $\left|z\right|<(4\pi-\beta)/(4\pi e\beta)$, the radius of convergence being positive provided $\beta<4\pi$. Since the Yukawa potential (1.3) diverges logarithmically as $\left|x\right|\rightarrow 0$, the proof of such statement requires the use of Brydges–Kennedy approach or iterated Mayer expansion (no one-scale tree expansion formula would be able to deal with this issue). The problem at our hand is to extend the stability of $2$–dimensional Yukawa gas to the inverse temperature in the range $4\pi\leq\beta<8\pi$, passing through the sequence of thresholds $\beta_{2r}=8\pi\left(1-1/2r\right)$, $r\in\mathbb{N}$. Here, $\beta_{2r}$ is the inverse temperature in which a clusters with $r$ positive and $r$ negative charges collapse altogether at once, heuristically given by an argument of entropy–energy (there are $r^{2}$ and $r(r-1)$ distinct pairings of opposite, respectively, same charges):

[TABLE]

By the second mean value theorem and $g(s)\asymp 1/(2\pi s)$ as $s\rightarrow 0$, the balance expressed by (1.25) is in favor of entropy $S(\delta)=\delta^{2(2r-1)}$ if $\beta<\beta_{2r}$ while the energy contribution $e^{-\beta E(\delta)}\simeq\delta^{-\beta r/(2\pi)}$ dominates if $\beta>\beta_{2r}$ so we have

[TABLE]

for some constant $c>0$.

Avoiding the collapse of neutral clusters: a conjecture

As a consequence of the alluded collapses, the leading even coefficients $b_{2j}$, $j=1,\ldots,n$, of the Mayer series (1.24) diverges for $\beta_{2n}\leq\beta<\beta_{2(n+1)}$ when the short scale cutoff $t_{0}$, introduced in (1.10) (or in (1.7)) to make the system conditionally stable, is removed. A conjecture stated as an open problem in [Be] is as follows:

Conjecture 1.1

If the leading $n$ even coefficients $b_{2j}$’s are removed from the Mayer series (1.24), the radius of convergence of the corresponding series remains positive for any $\beta\in[\beta_{2n},\beta_{2(n+1)})$ and, consequently, for any $\beta<\beta_{2(n+1)}$.

Brydges-Kennedy [BK] have proved convergence of (1.24) with $O\left(z^{2}\right)$ term omitted for $4\pi\leq\beta<16\pi/3$ and have explained how it would be extended up to the second threshold $6\pi$. It turns out that the claimed improvement on the estimate of the three–particle energy from $U_{3}(\xi_{1},\xi_{2},\xi_{3};\dot{v})\geq-3\dot{v}(t,0)/2$ to $U_{3}(\xi_{1},\xi_{2},\xi_{3};\dot{v})\geq-\dot{v}(t,0)$ does not hold uniformly on $\left(\left\{-1,1\right\}\times\mathbb{R}^{2}\right)^{3}$ at each scale for the decomposition (1.7) used by the authors. It has been shown by numerical calculation in [GM] that the factor $3$ (the number $n$ of charged particles involved) in the lower bound of $U_{3}$ may be improved to $2.14..$., which is enough to extend the convergence of Mayer series to any $\beta\in[4\pi,6\pi)$ but insufficient to establish the conjecture beyond a certain threshold (about $\beta_{15}=112\pi/15$) up to $8\pi$. For the latter, it is indeed necessary to improve the factor from $3$ to $2$ ($n=3$ to $n-1=2$). Both statements are proved in the present work. In addition, we have proved that, if the decomposition (1.4) for the Yukawa potential is used instead, then by (1.21) $3$ can be replaced by $2$ in the stability bound for $U_{3}$ and for every odd $n$ (1.16) can be substituted by (1.23) with $B=\bar{B}=v(0)/2$.

The purpose of the present paper is also to provide a majorant candidate for the pressure of the Yukawa gas at $\beta_{2n}\leq\beta<\beta_{2(n+1)}$, uniformly in the cutoff $t_{0}$, when it is extracted from the even leading Mayer coefficients $b_{2j}$, $j=1,\ldots,n$, their divergent part. Such majorant has been proposed in [GM] but our presentation is neater than the original paper making it more transparent. We adapt to the potential decomposition (1.4) all ingredients and the construction used in that reference through the scale decomposition (1.7).

The majorant construction is based on the idea already present in the early works by Imbrie [I] and [Br], according to which the Mayer series (1.24) (after some combinatorics together with the stability estimate) is dominated by an expansion in powers of $ez\left\|\beta v\right\|_{1}e^{B}$, where $\left\|\beta v\right\|_{1}$ is the $L^{1}$–norm of $\beta v(x)$ and $B=\beta v(0)/2$. A one–step Mayer expansion is not suitable to potentials that $B$ is large in the range that $\left\|\beta v\right\|_{1}$ contributes little, as typically occurs for the two–dimensional Yukawa potential $v$ (see [I, GoMa, BK] for other applications). When $v$ is decomposed into a continuum of scales (see (1.7), or alternatively (1.10)), the Mayer series becomes, roughly speaking, an expansion in powers of $ez\tau(t_{0},t)$, where

[TABLE]

solves a linear equation (3.19) satisfied by the majorant $C_{2}$ of two times the second Mayer coefficient: $2\left|b_{2}\right|\leq C_{2}$ (see (3.14) and (3.16)). It has been shown that the Mayer expansion (1.24) converges provided $\beta\in[0,4\pi)$ and $e\left|z\right|\tau(t_{0},t)<1$ uniformly in $t_{0}>0$ (see Theorem 4.1 together with pgs. 41-42 of [BK] and Proposition 3.4, Remarks 3.5 and 3.6 below). Inside the first threshold, the domain of convergence is replaced by $\left(\beta,z\right)\in\mathbb{R}_{+}\times\mathbb{C}$ such that $\beta\in$ $[4\pi,16\pi/3)$ and $e\left|z\right|\displaystyle\int_{t_{0}}^{t}\left\|\beta\dot{v}(s,\cdot)\right\|_{1}e^{(3\beta/2)\int_{s}^{t}\dot{v}(\tau,0)d\tau}ds<1$ and we shall see that our candidate to majorant series converges provided $\beta\in[\beta_{k},\beta_{k+1})$ and the factor $3\beta/2$ in the exponent of this domain is replaced by $(k+1)\beta/k$ for any $k>1$. If $C_{n}$, $1<n\leq k$ , denote the first $k-1$ majorant coefficients: $n\left|b_{n}\right|\leq C_{n}$, we observe by (3.27) that $(k+1)(n-1)B/k$ multiplies the linear term of the equation satisfied by $C_{n}$ (after the divergent part of the even $n\leq k$ coefficients have been extracted through a Lagrage multiplier $L_{k}$). In particular, for $n=2$, $\tau_{k}(t_{0},t)$ given by (3.26) generalizes (1.26) and solves the linear equation (3.28) for $C_{2}$. Since the modified stability condition (1.23) applies for every $n>1$ odd, the coefficient that multiplies the linear term of the equation for $C_{n}$, which is given by $(n-1)B<(k+1)(n-1)B/k$, implies that the same equation satisfied by $C_{n}$ with $n$ even holds for $n$ odd. To understand why the modified stability bound (1.23) is so crucial, we observe that anything large than $(n-1)B$ would prevent the convergence of the majorant series in the whole interval of collapse $[4\pi,8\pi)$. Recall that, when the standard scale decomposition is used, the interacting energy $U_{n}$ with $n=3$ is bounded below by a factor $-2.14B$, instead of $-2B$, preventing the $nb_{n}$ to be dominated by $C_{n}$ for $\beta>\beta_{k}$ with $k$ verifying the inequality $2.14>2(k+1)/k$, i. e. $k>15$.

Outlines of the present work

The present paper is organized as follows. Section 2 is dedicated to the proof of the main Theorem 2.8 and Corollary 2.10 together with estimates on the modified Bessel functions of second kind envolved in both representations of two-dimensional Yukawa potential: standard (Proposition 2.2) and scale mixtures of Euclid’s hat (Proposition 2.11).

The main Theorem is then used to resolve an obstructive remark of an umpublished paper (Remark 7.5 of [GM]) which, whether the standard decomposition of the Yukawa potential into scales were adopted, would impede a direct proof of the convergence of the Mayer series of the two-dimensional Yukawa gas for the inverse temperature up to $8\pi$. We dedicate Section 3 to the Cauchy majorant method applied to the density function of Yukawa gas on the whole interval $[4\pi,8\pi)$ of collapses. For this system it is proven that the Mayer series converge up to the second threshold $\beta\in[4\pi,6\pi)$ and its given explanations on the mechanism that allows it to be extended up to $8\pi$. The paper distinguishes the stability issues from those matters related to convergence of the Mayer series. In respect to the former its is proven at the end of Section 3 that dipoles in the presence of other charges are prevented to collapse.

2 Minimal specific energy: main theorem and estimates involving

modified Bessel functions

We prove in this section our main theorem (1.21) and the implications of it on the minimal specific energies $e(h)$ and $\bar{e}(h)$ for the Euclid’s hat $h$ in $\mathbb{R}^{2}$.

Three particles minimal specific energy

To begin with, let $U_{n}$ be the $n$–particle total energy (1.15) and let

[TABLE]

and

[TABLE]

be the $n$–particles minimal, and constrained minimal, specific energies. As the particles of our system have either $+1$ or $-1$ charges, these two quantities are related to each other when $n$ is an odd number as $e_{n}(v)=\bar{e}_{n}(v)(n-1)/n$. Let us first consider the case $n=3$ and let $v(x)$ be given by the two–dimensional Yukawa potential (1.3) under the standard decomposition into scales (1.7), cut-off at short distances $s\leq t_{0}$: $v(x)=\displaystyle\int_{t_{0}}^{1}\tilde{h}(\left|x\right|/s)/(2\pi s)~{}ds$. Since, by (1.20),

[TABLE]

it is enough to consider the minimal specific energy of $3$–particles $\bar{e}_{3}(\tilde{h})$ for $\tilde{h}(w)=wK_{1}(w)$, where $K_{1}$ is the modified Bessel function of second kind of order $1$.

We shall need among other properties some general features of $\tilde{h}(w)$.

Proposition 2.1

$w\longmapsto\tilde{h}(w)=wK_{1}(w)$* is a regular function at every point $w\in\left(0,\infty\right)$. The function $\tilde{h}(w)$ strictly decreases from its maximum $\tilde{h}(0)=1$, decays to [math] at $\infty$ exponentially fast and changes its concavity: $\tilde{h}^{\prime\prime}(w)<0$ for $w<w_{0}$ and $\tilde{h}^{\prime\prime}(w)>0$ for $w>w_{0}$ at $1/2<w_{0}<\left(1+\sqrt{17}\right)/8$ whose numerical value is $w_{0}=0.5950$(…).*

Proof. Regularity and positivity of $K_{\nu}(x)$ for every $\nu\in\mathbb{R}$ and $x>0$ are known facts (see e.g. Appendix A of [Ga]). It follows from the equation

[TABLE]

with $n=1$ together with $\lim_{w\rightarrow 0}wK_{0}(w)=0$ and $\lim_{w\rightarrow 0}wK_{1}(w)=1$ (see [Ga] and Lemma 2.2 of [YC]) that

[TABLE]

for $w>0$, proving the strictly decreasing property of $\tilde{h}$ and $\tilde{h}(0)=1$. The inequalities for $x>0$:

[TABLE]

find in ref. [YC], imply the exponential decaying of $\tilde{h}(w)$ and together with

[TABLE]

(2.4) yield

[TABLE]

provided $w<1/2$ and

[TABLE]

provided $w>(2w+1/2)(1-w)=3w/2+1/2-2w^{2}$ or, equivalently, $w>\left(1+\sqrt{17}\right)/8=0.640\,39$. The unique solution of $wK_{1}(w)/K_{0}(w)-1=0$, whose numerical value is $w_{0}=0.5950$(…), satisfies $1/2<w_{0}<0.640\,39$ (see proof of Lemma 2.5). This concludes the proof.

$\Box$

Because the particles interact via a pair potential, it is easy to see that the minimum potential energy is due to a system in which two of the three particles have equal signs and the third has charge with the opposite sign. The potential energy (1.15) with $n=3$ and $\sigma_{1}=\sigma_{3}=-\sigma_{2}$ is then given by

[TABLE]

To simplify the expression, we write $r_{1}=\left|x_{1}-x_{2}\right|$, $r_{2}=\left|x_{2}-x_{3}\right|$ and $r_{3}=\left|x_{1}-x_{3}\right|$ can be written, as the particles are located at the vertices of a triangle, by the law of cosine, as

[TABLE]

Since $\tilde{h}(w)$ is a strictly decreasing function, the minimal specific energy of $3$–particles (2.1) thus reads

[TABLE]

The next proposition shows that this quantity does not reach from below the value $-1/2=\left(-\sum_{i=1}^{3}\sigma_{i}^{2}+\left|\sum_{i=1}^{3}\sigma_{i}\right|\right)/(2\cdot(3-1))$ that one would expected for a convex function $h$.

Proposition 2.2

[TABLE]

Remark 2.3

As the numerical evaluations used in the proofs are sharp up to high decimal order, we may claim that $\bar{e}_{3}=-0.530$(…), which is certainly less than $-1/2$ ($-0.527$(…)$>\bar{e}_{3}>-0.535$), according to the precision of the machine used to calculate it.

Proof. To prove (2.6), it is enough by (2.5) to show that

[TABLE]

holds for all $x$, $y\geq 0$. Defining $f(x)=\tilde{h}(x)-1.07$, equation (2.7) is equivalent to show superadditivity of $f(x)$:

[TABLE]

But this is implied by the following

Lemma 2.4

Let $q(x)=f(x)/x$ be defined for $x>0$. If $q(x)$ is monotone increasing, then $f(x)$ is superadditive.

Proof of Lemma 2.4. Suppose that $g(x)$ is monotone increasing function. Then $q(x+y)\geq q(x)$, $q(x+y)>q(y)$ and it follows that

[TABLE]

which proves the lemma.

$\Box$

It remains thus to prove that $q(x)=(\tilde{h}(x)-1.07)/x=K_{1}(x)-1.07/x$ is monotone increasing. From (2.3) with $n=1$, we deduce

[TABLE]

which implies that

[TABLE]

for $x>0$ provided

[TABLE]

This inequality, however, holds in view of the following:

Lemma 2.5

The function $x\longmapsto p(x)=xK_{1}(x)+x^{2}K_{0}(x)$ defined in $\mathbb{R}_{+}$ has a global maximum at $x_{0}$, $1/2<x_{0}<\left(1+\sqrt{17}\right)/8$. It strictly increases with $p(0)=1$ as $x$ varies from [math] to $1/2$ and strictly decreases to [math], exponentially fast, as $x$ varies from $\left(1+\sqrt{17}\right)/8$ to $\infty$. The second derivative $p^{\prime\prime}(x)$ of $p(x)$ is negative in the interval $1/2\leq x_{0}\leq\left(1+\sqrt{17}\right)/8$. Numerically, $x_{0}=0.5950$ (…) and its (global) maximum values $p(x_{0})=1.061$(…)$~{}<1.07$.

Proof of Lemma 2.5. Using (2.3) with $n=1$ together with $K_{0}^{\prime}(x)=-K_{1}(x)$, as in the proof of Proposition 2.1, we have

[TABLE]

provided $x>\left(1+\sqrt{17}\right)/8$ and

[TABLE]

provided $x<1/2$. These prove that $p(x)$ increases in $\left(0,1/2\right)$ and decreases in $\left(\left(1+\sqrt{17}\right)/8,\infty\right)$, exponentially fast in view of (2.4).

$p(x)$ attains its maximum value at the same point at which $\tilde{h}(w)$ changes its concavity. The maximum $x_{0}$ of $p(x)$ solves $K_{0}(x)-xK_{1}(x)=0$ and satisfies $1/2<x_{0}<$ $\left(1+\sqrt{17}\right)/8\approx\allowbreak 0.64$, as stated above and showed in Proposition 2.1. To prove that $x_{0}$ is the global maximum, it suffices to show that the second derivative of $p(x)$, which may be calculated exactly as before,

[TABLE]

takes negative values for $x\in\left[1/2,\left(1+\sqrt{17}\right)/8\right]$. By equation (2.4) and positivity of $K_{0}(x)$, this is implied by

[TABLE]

Denoting the function on the right hand side by $l(x)=x+x/(2x+1/2)-(1+x^{2})/2$, we need to show that $l(x)>0$ for $x\in\left[1/2,\left(1+\sqrt{17}\right)/8\right]$. But $l(1/2)=5/24\approx 0.20$ and $l(\left(1+\sqrt{17}\right)/8)=\left(23+\sqrt{17}\right)/64\approx 0.29$ are both positive and the second derivative of $l(x)$,

[TABLE]

for all $x>0$, proving therefore the statement.

We have thus proven that $x_{0}$ is a global maximum of $p(x)$, concluding the proof of Lemma 2.5.

$\Box$

Returning to the proof of Proposition 2.2, the lower bound stated in (2.6) follow from the superadditivity of $f(x)=\tilde{h}(x)-1.07$, which is proven in Lemmas 2.4 and 2.5. The numerical estimate for the specific energy $\bar{e}_{3}$ stated in Remark 2.3 is obtained when $1.07$ is replaced by the maximum values $p(x_{0})=1.061$(…), given in Lemma 2.5, since at this point $f(x_{0})=\tilde{h}(x_{0})-1.061$(…) satisfies (2.8) as an equality and consequently, by (2.7), $\tilde{h}(2x_{0})-\tilde{h}(x_{0})-\tilde{h}(x_{0})=-1.061(\ldots)$.

By definition (2.5), taking $r_{1}=r_{2}=1/2$ in the expression inside the minimum, we have an upper bound

[TABLE]

$\Box$

Remark 2.6

It does not seem easy to extend the superadditivity method used to estimated the (restricted) minimum specific energy of $3$–particles to $(2k+1)$–particles with $k>1$. As we shall see in the next section, the result on the minimal specific energy $\bar{e}_{3}\left(\tilde{h}\right)$ prevents that the third Mayer coefficient be defined uniformly in the cutoff $t_{0}$ in the entire collapse interval $\left[4\pi,8\pi\right]$, although it is enough for concluding convergence of the Mayer series up to the second threshold $[4\pi,6\pi)$. Numerical calculations performed in [GM] indicate that $\bar{e}_{2k+1}\left(\tilde{h}\right)$ remains for $k>1$ strictly smaller than $-1/2$. We should mention that if $\tilde{h}(w)$ were convex, the minimal of (2.5) would be attained at $r_{1}=r_{2}=0$, obtaining the expected value $\bar{e}_{3}=-1/2$ as it is exactly the case when we use decomposition (1.4) of the Yukawa potential (1.3) instead of (1.7). Since the method based on superadditivity cannot be easily extended to $k>1$, another method will be employed to obtain $\bar{e}_{2k+1}(h)=-1/2$ with $h$ the Euclid’s hat function (1.5).

The main theorem

We shall now turn to the representation of Yukawa potential (1.3) given by $v(x)=v_{(0,\infty)}(x)=K_{0}(\left|x\right|)/(2\pi)$ where (see (1.4)):

[TABLE]

is a scale mixtures of Euclid’s hat. Here, for $x\in\mathbb{R}^{2}$ and $s\in\mathbb{R}_{+}$,

[TABLE]

is the self convolution of indicator function $\chi_{[0,s/2]}(x):=\theta(s/2-\left|x\right|)$ of the $2$–dimensional ball (disc) $B_{r}\equiv B_{r}(0)$ of radius $r=s/2$ centered at origin and $g(s)$ is the scale mixtures density given by Hainzl–Seiringer: [HS]

[TABLE]

We observe that (2.11) differs from the $g(s)$ in equation (11) of [HS] by a pre–factor $\pi\left(s/2\right)^{2}$ that we have used in (2.10) in order to normalize $h$ at origin: $h(0)=1$. This normalization is suitable when the radial function $\varphi(\left|x\right|)=v(x)$ is the characteristic function of a spherically symmetric probability distribution in $\mathbb{R}^{d}$ or the covariance of a stationary and isotropic random field on $d$–dimensional Euclidean space. The latter is the point of view of the present paper, while the former were the focus of Gneiting paper [G], for which the classes $H_{d}$ of radial positive definite functions generated by scale mixtures of $d$–dimensional Euclid’s hat $h_{d}(\left|x\right|)$ played an important role in the proof of an analogue of Pólya’s criterion for $d>1$. We observe however that the scale mixture used in [G] is of the form $\varphi(t)=\displaystyle\int_{0}^{\infty}h_{d}(rt)dG(r)$, where $G(r)$ is a probability distribution function in $\left(0,\infty\right)$ with $G(0+)=c\in\left[0,1\right]$. In order to compare with our $g(s)$ given by (2.11) (by (2.9) $s=1/r$), which behaves as $s$ goes to [math] as $1/(2\pi s)$, in the case that $dG(r)$ is absolutely continuous and finite positive measure in $\mathbb{R}_{+}$, we write $dG(r)=f(r)dr=-f(1/s)ds/s^{2}=-\tilde{f}(s)ds$. We see that $\tilde{f}(s)\sim 1/s$ would lead to a nonintegrable mixture density $f(r)\sim 1/r$ at infinity and, consequently, $\varphi(t)$ with such a density would not belong to the class $H_{2}$ considered in that paper.

Equation (2.11) can be written in terms of a Meijer $G$–functions that is regular at $s=0$ as

[TABLE]

as one can check using Mathematica program together with the shift property: $t^{2}G_{13}^{30}\left(t\left|\genfrac{}{}{0.0pt}{1}{-3/2}{-2,-1,0}\right.\!\right)=G_{13}^{30}\left(t\left|\genfrac{}{}{0.0pt}{1}{1/2}{0,1,2}\right.\!\right)$.

We begin by describing the general features of $h(w)$. We shall state and prove our main theorem afterwards and return to the asymptotic properties of (2.12) required for the next section.

Proposition 2.7

$w\longmapsto h(w)$* defined by (2.10) is regular at every point $w\in\left(0,1\right)$, convex and non increasing function in $\left(0,\infty\right)$. Moreover, it can be written as*

[TABLE]

$h(w)=0$* if $w>1$ so, writing $\varphi(x)=h(\left|x\right|)$ we have $\varphi(0)=h(0)=1$ and $\hat{\varphi}(0)=\displaystyle\int_{\mathbb{R}^{2}}h(\left|x\right|)dx=\pi/4$ is its Fourier transform $\hat{\varphi}(\xi)$ at $\xi=0$.*

Proof. We shall deduce (2.13) from (2.10) by means of a geometric representation of the convolution integral

[TABLE]

(see e.g Sec. 2 of [G]). The product of indicator functions does not vanish if their support, the discs $B_{s/2}(x)$ and $B_{s/2}(0)$ centered at $x$ and [math], intersect and this occurs when the distance $\left|x\right|$ between their centers is less than their diameter $s$. Writing $w=\left|x\right|/s$, we have

[TABLE]

From this point of view, the convolution integral (2.14) is given by the area $A(\theta)$ of two ”caps”, of common bases, made of a sector of opening angle $\theta$ and radius $s/2$ with the triangular region inside removed (see Fig. 5):

[TABLE]

where, with $b$ the length of the caps common bases,

[TABLE]

By $\left|x\right|^{2}+b^{2}=s^{2}$, we deduce $b=s\sqrt{1-w^{2}}$. Solving equations (2.16) for $\theta$ and $\sin\theta$:

[TABLE]

together with (2.15), yields

[TABLE]

Equation (2.13) follows from (2.14), (2.15) and (2.17). The regularity of $h(w)$ in $\left(0,1\right)$ follows from this representation. Since

[TABLE]

for any $w\in\left(0,1\right)$, we conclude that $h(w)$ is such that $h(0)=1$, by definition, is strictly decreasing in $\left(0,1\right)$ monotone non increasing and convex in $\left(0,\infty\right)$, concluding the proof.

$\Box$

Before we state and prove our main theorem, we use (2.10) to write

[TABLE]

Changing the integration variables for each term of the sum to $z=y+x_{j}$ yields

[TABLE]

since the function $\chi_{[0,s/2]}(x)$ is even.

Theorem 2.8

For any integer $n\geq 2$, any configuration of $n$–particle $\left(\zeta_{1},\ldots,\zeta_{n}\right)$, $\zeta_{j}=\left(\sigma_{j},x_{j}\right)\in\left\{-1,1\right\}\times\mathbb{R}^{2}$ any $s\in\mathbb{R}_{+}$, the total energy with interacting potential $h$ satisfies

[TABLE]

Proof. Since $h(0)=1$, we add $n/2$ to the total energy in order to include the $i=j$ terms into the sum in (2.19):

[TABLE]

So, the result is proven if we show that

[TABLE]

Using the fact that $\sum_{j=1}^{n}\sigma_{j}\chi_{[0,s/2]}(z-x_{j})$ is always an integer number, we have

[TABLE]

and this, together with (2.18), implies that

[TABLE]

concluding the proof.

$\Box$

Remark 2.9

Since the proof does not set any condition on the dimension of the Euclidean space, Theorem 2.8 holds for any $d\geq 2$. In this case, $h(w)$ has to be replaced by the Euclid’s hat $h_{d}(w)$ (see Sec. 2 of [G] for the proof of the statements in Proposition 2.7 for the corresponding $d$–dimensional Euclid’s hat).

Theorem 2.8 implies the following

Corollary 2.10

The minimal specific energy $e(h)$ and the minimal constrained specific energy $\bar{e}(h)$, defined by (1.18) and (1.19), are both $-1/2$.

Proof. This result follows from the definitions (2.1) and (2.2) and the inequality (2.19). The minimal specific energy $e(h)=\inf_{n\geq 2}e_{n}(h)$ of $h$ is attained for even number of particles $n$ satisfying $\displaystyle\sum\nolimits_{j=1}^{n}\sigma_{j}=0$ and $\displaystyle\sum\nolimits_{j=1}^{n}\sigma_{j}^{2}=n$ when they collapse to a single point since, in this case, the inequality (2.19) becomes an equality. Likewise, the constrained minimal specific energy $\bar{e}(h)=\inf_{n\geq 2}\bar{e}_{n}(h)$ of $h$ is attained for odd number of particles $n$ satisfying $\left|\displaystyle\sum\nolimits_{j=1}^{n}\sigma_{j}\right|=1$ and $\displaystyle\sum\nolimits_{j=1}^{n}\sigma_{j}^{2}=n$ when they collapse to a single point. Note that, for a calculation similar to the energy in (1.25), the potential energy (1.15) with $n=2r+1$, $\sigma_{1}=\cdots=\sigma_{r}=-\sigma_{r+1}=\cdots=-\sigma_{2r+1}$ and $x_{1}=\cdots=x_{2r+1}=x_{0}\in\mathbb{R}^{2}$, is given by ($h(0)=1$)

[TABLE]

$\Box$

As a consequence of Corollary 2.10, representing the Yukawa potential $v$ as scale mixtures of Euclid’s hat (2.9) regularized at short distances, the stability bound (1.16) can be replaced by (1.23) with

[TABLE]

Properties of the mixture function

Regarding the mixture function, we have the following

Proposition 2.11

The function $g:\left(0,\infty\right)\longrightarrow\left(0,\infty\right)$ given by (2.11) can be written as

[TABLE]

where

[TABLE]

is a regular function such that $m(0)=1$, increases monotonously in $\left(0,s_{0}\right)$, where $s_{0}=0.812$(…) and $m(s_{0})=m_{0}=1,075$(…), then decreses monotonously in $\left(s_{0},\infty\right)$ to [math], exponentially fast. Globally, it is bounded from above and from below as

[TABLE]

In the vicinity of the origin, it satisfies

[TABLE]

where $a=(1-3\gamma+\log 4-\psi(-1/2))/8=0.07726$(…), being the r.h.s. of (2.22) asymptotic to $m(s)$ at $s=0$.

Proof. We begin by showing that (2.11) multiplied by $2\pi s$ can be written as (2.20). For this, we use $K_{1}(w)=-K_{0}^{\prime}(w)$ and the representation (see [GJ], Section 7.2)

[TABLE]

from which we infer that $K_{0}$ is regular in $\left(0,\infty\right)$. We may thus differentiate (2.23) three times, replace it into (2.11), exchange the integration order and, after multiplying by $2\pi s$ it can be written as

[TABLE]

where

[TABLE]

We have changed variable $sz=\sqrt{r^{2}-s^{2}}$, so $r=s\sqrt{z^{2}+1}$ and $rdr/\sqrt{r^{2}-s^{2}}=sdz$. Replacing (2.25) back into (2.24), making one more change of variable: $s\sqrt{k^{2}+1}=y$, so that $sk=\sqrt{y^{2}-s^{2}}$ and $sdk=y/\sqrt{y^{2}-s^{2}}$, yields (2.20).

The sequence of operations bringing (2.11) into the form (2.20) will be applied some more times. Let us start by finding a lower bound for (2.20). By monotonicity of the modified Bessel functions with respect to their order (see [C]) and integration by parts, we have

[TABLE]

where

[TABLE]

Observe that the boundary term in the partial integration, $\left.y^{2}K_{0}(y)\sqrt{y^{2}-s^{2}}/2\right|_{y=s}^{\infty}$ vanishes for all $s\in\left(0,\infty\right)$ because the exponential decay of $K_{0}(y)$ and boundedness of $y^{2}K_{0}(y)$.

Lemma 2.12

Let $I:[0,\infty)\longrightarrow\mathbb{R}$ be defined by

[TABLE]

The integral can be written as

[TABLE]

and from these we conclude that $I(s)=\pi e^{-s}/2$.

Proof. Since the integral (2.29) converge uniformly in $[s_{0},K)$, for any $s_{0}>0$ and $K<\infty$, the integral (2.30) is minus the derivative of the integral (2.29):

[TABLE]

Observe that $\left.K_{1}(y)\sqrt{y^{2}-s^{2}}\right|_{y=s}=0$ for the same reason as before. Since $ae^{-s}$ solves (2.31) for any $a\in\mathbb{R}$, the proof will be completed once we establish that (2.29) implies (2.30) and show $I(0)=\pi/2$. We begin with the latter.

Repeating the operations bringing (2.11) into the form (2.20), it follows from (2.23) and $K_{0}^{\prime}(y)=-K_{1}(y)$ that

[TABLE]

Now, we develop (2.29) as

[TABLE]

In the second equality we exchange the integration order, then we integrate by parts; we change variable $y=s\sqrt{r^{2}+1}$ in the fourth equality, use (2.23) together with $K_{0}^{\prime}(w)=-K_{1}(w)$ and in the last equality we change again the variable $y=s\sqrt{k^{2}+1}$. This concludes the proof of the lemma.

$\Box$

Returning to the proof of Proposition 2.11, we now deduce an equation for $J(s)$ and $L(s)$ in terms of $I(s)$. Differentiating (2.28) with respect to $s$, gives

[TABLE]

by Lemma 2.12. Analogously, differentiating (2.27) with respect to $s$, together with (2.3), gives

[TABLE]

We need also initial condition to both equations. Performing as in (2.32),

[TABLE]

and

[TABLE]

Integrating (2.33) together with (2.35), yields

[TABLE]

Analogously, integrating (2.34) together with (2.37) and (2.36), yields

[TABLE]

Equations (2.37) and (2.38) replaced into (2.26) gives the lower bound (2.21).

An upper bound is obtained similarly. By monotonicity of the modified Bessel functions with respect to their order (see [C]) and integration by parts, we have

[TABLE]

by (2.3), where $L(s)$ is given by (2.27). Equation (2.39) together with (2.38) gives the upper bound (2.21).

The asymptotic behavior (2.22) of $m(s)$ follows from the mean value theorem

[TABLE]

for some $\tilde{s}=\tilde{s}(s)\in[0,s]$ depending on $s$. The value $m(0)$ may be calculated as $I(0)$ in (2.32), using the representation (2.23) for $K_{1}(y)=-K_{0}^{\prime}(y)$ and exchange the integration order:

[TABLE]

To calculate the derivative of $m(s)$ we apply integration by parts twice, before and after the derivative with respect to $s$:

[TABLE]

by $\left(y\cdot\left(yK_{1}\right)\right)^{\prime}=yK_{1}+y\left(yK_{1}\right)^{\prime}$ together with (2.3); by $K_{1}+yK_{1}^{\prime}=\left(yK_{1}\right)^{\prime}=-yK_{0}$ we have $-K_{1}^{\prime}=K_{0}+K_{1}/y$ and

[TABLE]

where

[TABLE]

in view of the inequality $\sqrt{y^{2}-s^{2}}/y\leq 1$ for $s\leq y<\infty$, $-K_{0}^{\prime}(y)=K_{1}(y)>0$ and the fundamental theorem of calculus. Both boundary terms yielded from the partial integrations vanish. We observe that

[TABLE]

where $\psi(z)=\Gamma^{\prime}(z)/\Gamma(z)$ is the digamma function and so, $K_{0}(s)=-\log(s/2)-\gamma+O(s^{2})$ where $\gamma=-\psi(1)$ is the Euler-Mascheroni constant. The other term of (2.41) can analogously be bounded by

[TABLE]

provided $s\in\left[0,\bar{s}\right]$ where $\bar{s}\geq 3/2$. For this, we used that $\sqrt{y^{2}-s^{2}}/y\leq 1$ and by (2.4),

[TABLE]

if $x\leq 3/2$. Consequently, for $s\leq 3/2$ the integral of the l.h.s. of (2.44) over $[0,s)$ is positive and, using the representation (2.23) for $K_{0}(y)$ and for $K_{1}(y)=-K_{0}^{\prime}(y)$ and exchange the integration order, we have

[TABLE]

by integration by parts. To obtain (2.22) and conclude the proof of Proposition 2.11, we need to optimize the choice of $\tilde{s}(s)$ in (2.40). So far, by (2.40), (2.41), (2.42) and (2.43) we have

[TABLE]

by Proposition 2.1 and this upper bound is asymptotic as $s$ tends to [math]: $m(s)=1+O\left(s^{2}\right)$ the $s^{2}$ order term in the upper bound is $(1-2\gamma-2\log(s/2))/8=0.1539$(…)$~{}-(\log s)/4$. The best upper bound up to $O\left(s^{2}\right)$ term is, however, stated in Proposition 2.11, given by the asymptotic expansion of (2.12), calculated algebraically by the software Mathematica.

$\Box$

3 Majorant of the density function

Set up and ingredients

Let $\left(\Omega,\mathcal{B},\varrho\right)$ denote the (translational invariant) $\sigma$–finite measure space on $\left\{-1,1\right\}\times\mathbb{R}^{2}$; the set $\Omega$ corresponds to the possible configurations of a single particle (we united $\sigma$ and $x$ into $\zeta=\left(\sigma,x\right)$) and $\displaystyle\int d\varrho(\zeta)\cdot~{}=1/2\displaystyle\sum_{\sigma\in\{-1,1\}}\displaystyle\int_{\mathbb{R}^{2}}d^{2}x~{}\cdot~{}$denotes the integration with respect to $\rho$. Let

[TABLE]

be the pressure of the Yukawa gas in the infinite volume limit, where $v$ is the Yukawa potential regularized at short distances $s\leq t_{0}$, given by the scale decomposition (1.10). We observe that, as $v$ decays exponentially fast at infinity and has its singularity at origin removed, the finite volume pressure $p_{\Lambda_{i}}$, defined for any increasing sequence $\left(\Lambda_{i}\right)_{i\geq 1}$ of squares with $\lim_{i}\Lambda_{i}=\mathbb{R}^{2}$,333Given by (3.1) with the integral over the $n$–particle configurations restricted to $\Lambda_{i}$ divided by $\left|\Lambda_{i}\right|$: $\beta p_{\Lambda_{i}}=\displaystyle\sum_{n\geq 1}\dfrac{z^{n}}{n!}\dfrac{1}{\left|\Lambda_{i}\right|}\displaystyle\int_{\Lambda_{i}^{n}}d\varrho(\zeta_{1})\cdots d\varrho(\zeta_{n})\psi_{n}^{c}$ converges by standard methods (see e.g. [R]) and translational invariance of $v$ to the expression (3.1).

The density function $\rho(\beta,z)=z\partial p/\partial z(\beta,z)$ is another thermodynamical function which will be convenient to write as an Mayer series (1.24) in power of the activity $z$:

[TABLE]

where $b_{1}=1$ and, for $n>1$,

[TABLE]

is the so called $n$–th Mayer coefficient in the infinite volume limit. Note that $\beta\rho(\beta,z)/z=1$ is the equation of state of an ideal gas and due the interaction of the charged particles through the Yukawa pair potential, the series (3.2) provides corrections to all order about it expressed in terms of the Ursell (cluster) functions $\psi_{n}^{c}$. A formal power series in $z$

[TABLE]

is a majorant of $\beta\rho(\beta,z)/z$ if the $C_{n}^{\ast}$ are nonnegative and

[TABLE]

holds for all $n\in\mathbb{N}$. It follows that, if the $\Theta^{\ast}(z)$ series converges on the open disc $D(r):=\left\{z\in\mathbb{C}:\left|z\right|<r\right\}$ for some $r>0$, then $\rho(\beta,z)$ is holomorphic function of $z$ on the same disc. The largest $r$ provides an lower bound on the radius of convergence of the Mayer series (3.2) and (1.24).

For the problem at our hand, the most efficient method of constructing majorants combines (multi)scale decomposition of $v$ together with some basic ingredients. Beginning with the latter, the following elementary lemmas are useful.

Lemma 3.1

If $a$, $b$, $c$ and $d$ are positive numbers such that $a-c$ and $b-d$ are positive, then $ab-cd$ is also positive.

Proof. Writing

[TABLE]

concluding the proof.

$\Box$

Lemma 3.2

Let $a=\left(a_{n}\right)_{n\geq 1}$, $b=\left(b_{n}\right)_{n\geq 1}$, $\tilde{a}=\left(\tilde{a}_{n}\right)_{n\geq 1}$ and $\tilde{b}=\left(\tilde{b}_{n}\right)_{n\geq 1}$ be positive numerical sequences ($a$, $b$, $\tilde{a}$ and $\tilde{b}>0$) such that $\tilde{a}-a$ and $\tilde{b}-b$ are both positive sequences (i.e., $\tilde{a}_{n}-a_{n}>0$ and $\tilde{b}_{n}-b_{n}>0$ hold for all $n\geq 1$). Let the convolution product $e=c\ast d$ and the pointwise product $f=c\cdot d$ of two sequences $c=\left(c_{n}\right)_{n\geq 1}$ and $d=\left(d_{n}\right)_{n\geq 1}$ be defined by the sequences $e=\left(e_{n}\right)_{n\geq 1}$ and $f=\left(f_{n}\right)_{n\geq 1}$ where $e_{1}=0$ and

[TABLE]

and

[TABLE]

Then, (i) $\tilde{a}\cdot\tilde{b}-a\cdot b>0$; (ii) $\tilde{a}\ast\tilde{b}-a\ast b>0$; in particular (iii) $\tilde{a}\ast\tilde{a}-a\ast a>0$ and $\tilde{b}\ast\tilde{b}-b\ast b>0$ hold.

Proof. The conclusions (i), (ii) and (iii) follow immediately from Lemma 3.1: for (i) each element of the sequence is of the form (3.3); for (ii) and (iii) each element of the sequence is a sum of terms of the form (3.3).

$\Box$

Remark 3.3

The statements of Lemmas 3.1 and 3.2 hold true if the assumption of positivity is replaced by nonnegativity.

The scale decomposition (1.10) becomes effective when the Ursell function in (3.1) is defined by a scaling limit

[TABLE]

where $\psi_{n}^{c}(t,\zeta_{1},\ldots,\zeta_{n};\beta v(t,\cdot))\equiv\psi^{c}(t,\zeta_{\left\{1,\ldots,n\right\}})$ is the unique solution of the infinite system of ordinary differential equations for $f_{I}=f_{I}(t)\equiv f(t,\zeta_{I})$, where $\zeta_{I}=\left(\zeta_{i_{1}},\ldots,\zeta_{i_{k}}\right)$ is the set of variables indexed by $I=\left\{i_{1},\ldots,i_{k}\right\}\subset\left\{1,\ldots,n\right\}$ and $n\in\mathbb{N}$: (see Lemma 3.3 of [BK])

[TABLE]

with (ideal gas) initial condition444By (1.10), the interaction $v(t,x)$ between particles is turned off at $t=t_{0}$.

[TABLE]

Here $\dot{v}_{ij}(t)\equiv\dot{v}(t,\zeta_{i},\zeta_{j})=\sigma_{i}\sigma_{j}g(t)h(\left|x_{i}-x_{j}\right|/t)$ so, as $\dot{v}(t,\zeta_{i},\zeta_{j})$ is a measurable and translational invariant function on the $2$–particle configuration space, $\psi^{c}(t,\zeta_{I})$ is a measurable and translational invariant function on the $k$–particle configuration space $\left(\left\{-1,1\right\}\times\mathbb{R}^{2}\right)^{k}$.

By the variation of constants formula the system of equations (3.5) is equivalent to a system of integrable equations: $f_{I}(t)=1$ if $\left|I\right|=1$ and

[TABLE]

if $\left|I\right|>1$, which will be usefull to our application.

Majorant construction for $\beta<4\pi$

Using (3.6) two of the authors have proven in [GM] (see Theorem 2.2 and equations (4.10)-(4.12) therein) the following

Proposition 3.4

Let $\Theta=\Theta(t,z)$ be the classical solution of

[TABLE]

with $\Theta(t_{0},z)=1$ for all $z\geq 0$, where by (1.10), (1.5) and explicit calculation, $\Gamma=\Gamma(t)=\left\|\beta\dot{v}(t,\cdot)\right\|_{1}$ and $B=B(t)=\left|\beta\dot{v}(t,0)\right|/2$ are given by

[TABLE]

(the integral is exactly $2\pi t^{2}$ times $\displaystyle\int_{0}^{\infty}\dfrac{2}{\pi}\left(\arccos w-w\sqrt{1-w^{2}}\right)wdw=1/8$) and

[TABLE]

Then, the following majorant relation

[TABLE]

holds for all $\left(\beta,z\right)$ satisfying

[TABLE]

where

[TABLE]

and $W(x)$ denotes the Lambert $W$–function.[C-K]

Remark 3.5

The proof of Proposition 3.4 in [GM] uses the scale decomposition (1.7) of $v$, for which $B=\beta/(4\pi t)$ and $\Gamma=2\beta t$ can be exactly calculated (for comparison, we have set therein $\kappa(t)=1/t^{2}$ for $t\in(0,1]$). Here $v$ is given by (1.10) whose scaling function $g(t)$ agree with the scaling $1/(2\pi t)$ of (1.7) only asymptotically as $t\rightarrow 0$. Writing $\tau(t_{0},\infty)=\tau(t_{0},1)\exp\left(\beta\int_{1}^{\infty}g(s^{\prime})ds^{\prime}\right)+\tau(1,\infty)$ together with $0<g(s)\leq(1+s/5)/(2\pi s)$ if $0\leq s\leq 1$ by Proposition 2.11 (see Fig 7), for any $0<\beta<4\pi$ the limit

[TABLE]

exists and $\tau(1,\infty)$ is finite since $g(t)$ decays exponentially fast as $t\rightarrow\infty$.

Remark 3.6

The existence of $\tau=\lim_{t_{0}\rightarrow 0}\tau(t_{0},\infty)$ implies by (3.9) and (3.10) that the radius of convergence $r=\sup\left\{\left|z\right|:e\left|z\right|\tau<1,\ z\in\mathbb{C}\right\}$ of the Mayer series (3.2) remains strictly positive. This fact is already remarkable considering that $v$, given by (1.10) with $t_{0}=0$, does not satisfies the stability condition (1.16) (see also (1.17)), which is sufficient but not necessary for the density (3.2) be defined in the thermodynamic limit.

Proof of Proposition 3.4. By (3.4), (3.6) and stability (1.17), the sequence $\left(A_{n}\right)_{n\geq 1}$ of positive functions $A_{n}:[t_{0},\infty)\longrightarrow\mathbb{R}$, defined by

[TABLE]

satisfies a system of integral inequality equations

[TABLE]

with $A_{1}(t)\equiv 1$. Hence, the Mayer coefficients of the series (3.2) are majorized by

[TABLE]

Let $\Theta(t,z)$ be defined by the series

[TABLE]

where the sequence $\left(C_{n}\right)_{n\geq 1}$ of positive functions $[t_{0},\infty)\ni t\longmapsto C_{n}(t)\in\mathbb{R}_{+}$ satisfies equations (3.13) for $\left(nA_{n}\right)_{n\geq 1}$ as an equality and, consequently,

[TABLE]

It can be shown (see Sec. 4 of [GM]) that (3.15) satisfies the quasi-linear first order PDE (3.7). So, the first inequality of (3.9) holds and all one needs is to determine a domain in $\left(t_{0},\infty\right)\times\mathbb{R}_{+}$ for which the classical solution of (3.7) exists. Observe that (3.7) can be written as a system of first order differential equations for the coefficients $\left(C_{n}\right)_{n\geq 1}$. For this, by (3.15), we have

[TABLE]

Substituting these series back into the equation, yields

[TABLE]

with $C_{1}(t)\equiv 1$, $t\in[t_{0},\infty)$, and initial data $C_{n}(t_{0})=0$ for all $n\geq 2$.

The first non-trivial equation for $n=2$,

[TABLE]

with $C_{2}(t_{0})=0$, has a unique solution $\tau(t_{0},t)$ given by (3.11), which can be written as

[TABLE]

where $\Gamma_{1}(s)=\Gamma(s)/f_{1}(s)$ and

[TABLE]

is an integrating factor of (3.19). As we shall see $\tau(t_{0},t)=C_{2}(t)$ determines the radius of convergence of the series (3.15) for $\Theta$:

[TABLE]

uniformly in $t_{0}$ and $t$ for $\beta<\beta_{2}$, $0<t_{0}<t<\infty$, where $\beta_{2}=4\pi$ is the first threshold (see Remark 3.5). For this, let $\left(C_{n}^{(1)}(t)\right)_{n\geq 1}$ be a sequence of positive functions defined by

[TABLE]

Since $C_{n}^{(1)}=C_{n}/f_{1}^{n-1}$ and

[TABLE]

equation (3.18) in terms of the new $C_{n}^{(1)}$’s reads

[TABLE]

with $C_{1}^{(1)}(t)\equiv 1$ and initial data $C_{n}^{(1)}(t_{0})=0$ for all $n\geq 2$. Since the coefficient $-(n-2)B$ of the linear term is nonpositive for all $n\geq 2$, the solution of the above initial value problem (IVP) can, in turn, be majorized by another sequence $\left(\tilde{C}_{n}^{(1)}\right)_{n\geq 1}$:

[TABLE]

which solves the IVP

[TABLE]

with $\tilde{C}_{1}^{(1)}(t)\equiv 1$ and initial data $\tilde{C}_{n}^{(1)}(t_{0})=0$ for all $n\geq 2$.

Proof of (3.23). Using the notation introduced in Lemma 3.2, we write $a=\left(a_{n}\right)_{n\geq 1}$ and $b=\left(b_{n}\right)_{n\geq 1}$ with $a_{1}=b_{1}\equiv 0$, $a_{n}(t)=(n-2)B(t)$ and $b_{n}(t)=n\Gamma_{1}(t)/2$ for $n>1$. The difference sequence $\Delta=\left(\Delta_{n}\right)_{n\geq 1}$, given by $\Delta_{1}\equiv 0$ and $\Delta_{n}(t)=\tilde{C}_{n}^{(1)}(t)-C_{n}^{(1)}(t)$ for $n>1$, thus satisfies

[TABLE]

Let us assume that (3.23) holds for some $t\geq t_{0}$. Then, by Lemma 3.2 we have $\dot{\Delta}(t)\geq 0$ which, together with $\Delta(t_{0})\equiv 0$, implies that $\Delta(t)\geq 0$. Consequently, (3.23) holds for all $t\geq t_{0}$. $\Box$

It is shown in Sec. 5 of [GM] that the power series analogous to (3.21): $\tilde{\Psi}(t,w)=1+\displaystyle\sum_{n\geq 2}\tilde{C}_{n}^{(1)}w^{n-1}$ satisfies an equation given by (3.7) setting $B=0$, $\Gamma=\Gamma_{1}$ and together with $\tilde{\psi}(t_{0},z)\equiv 1$ has by the method of characteristics the classical solution

[TABLE]

provided $e\left|w\right|\tilde{\tau}_{1}(t_{0},t)<1$ holds, where $\tilde{\tau}_{1}(t_{0},t)=\displaystyle\int_{t_{0}}^{t}\Gamma_{1}(s)ds$ and $W(x)$ denotes the Lambert $W$–function defined implicitly by $We^{W}=x$, whose Taylor series about $x=0$ (see Lagrange-Bürmann theorem [D]): $W(x)=\displaystyle\sum_{n\geq 1}\left(-n\right)^{n-1}x^{n}/n!$ converges for $\left|x\right|<1/e$, including the branching point at $x=-1/e$ (see e.g. [C-K]).

Joining equations (3.14), (3.16) and (3.23) together, we conclude

[TABLE]

with $\tau_{1}(t_{0},t)$ given by (3.11), establishing the second inequality of (3.9).

$\Box$

Note that, since $f_{1}(t)>1$ for any $t_{0}$ and $t$ fixed, the radius of convergence of the majorant series $\Theta(t,z)=\Psi(t,w)$ is smaller in $z$ than in $w$ variable. However, in view of Remarks 3.5 and 3.6, it remains strictly positive when the cutoff $t_{0}$ is removed provided $\beta<\beta_{2}$ where $\beta_{2}=4\pi$ is the first threshold.

Majorant construction for $\beta$ inside the threshold

intervals $I_{n}$

The procedure of finding a majorant series for the density function can be extended for the inverse temperature $\beta$ in the thresholds interval $I_{n}=[\beta_{2n},\beta_{2(n+1)})$, $n\in\mathbb{N}$ where, for convenience, we write $\beta_{k+1}=8\pi\left(1-1/(k+1)\right)=8\pi k/(k+1)$. We shall present an scheme of avoiding neutral cluster collapse which holds for any thresholds interval when the cutoff $t_{0}$ is removed. The scheme consists of three stages. Firstly, for $\beta\in I_{(k-1)/2}$ where $k>1$ is an odd number, we remove from the system any neutral clusters or subclusters of size less than $k$. Such a removal, which prevents the increasing of the corresponding terms in the Mayer expansion when they collapse, is expressed in terms of the majorant method, by inserting Lagrange multipliers into the equation (3.7), as many as neutral clusters were removed.

The second stage addresses non neutral clusters of size smaller or equal to $k$ in the majorant equation (3.7) which has been overestimated by using the stability bound (1.16) instead of (1.23) (see equation (3.13)). This issue is fixed by replacing the coefficient $nB$ of the linear term of (3.18) by $(n-1)B$. Before we apply the second stage, we shall extract (insirting a Lagrange multiplier $L_{k}$) an exact amount from the linear term of (3.18) that allows the solution of (3.7) for $\beta<\beta_{k+1}$ be majorized by a series with positive radius of convergence, uniformly on cutoff $t_{0}$ (see Secs. 6 and 7 of [GM]). The Lagrange multiplier $L_{k}$ of order $k$ is given by the Cesàro mean of the first $k$ Taylor series of $(z\Theta)_{z}$ around $z=0$, truncated at order $0\leq j<k$, and this choice is optimal.

Proposition 3.7

For any $k\in\mathbb{N}$, let $\Theta=\Theta(t,z)$ be the classical solution of

[TABLE]

with $\Theta(t_{0},z)=1$ for all $z\geq 0$, where $\Gamma=\Gamma(t)=\left\|\beta\dot{v}(t,\cdot)\right\|_{1}$ and $B=B(t)=\left|\beta\dot{v}(t,0)\right|/2$ are given in Proposition 3.4 and

[TABLE]

is a Lagrange multiplier. Then, the following majorant relation

[TABLE]

holds for all $\left(\beta,z\right)$ satisfying

[TABLE]

where

[TABLE]

and $W(x)$ denotes the Lambert $W$–function.[C-K]

Remark 3.8

A calculation analogous to (3.12) yields that

[TABLE]

exists for $\beta<\beta_{k+1}$ and the radius of convergence of the majorant series (3.25) is strictly positive.

Proof of Proposition 3.7. We follow closely the proof of Proposition 3.4. Let $\Theta(t,z)$ be defined by the series (3.15). Observe that, by

[TABLE]

and the remaining series of (3.17), (3.24) can be written as a system of first order differential equations for $\left(C_{n}\right)_{n\geq 1}$:

[TABLE]

and (3.18) for $n>k$, with $C_{1}(t)\equiv 1$ and initial data $C_{n}(t_{0})=0$ for $n\geq 2$. The equation for $n=2$

[TABLE]

with $C_{2}(0)=0$ has a unique solution given by (3.26), which can be written as

[TABLE]

where $\Gamma_{k}(s)=\Gamma(s)/f_{k}(s)$ and

[TABLE]

is an integrating factor of (3.28).

Let $\left(C_{n}^{(k)}(t)\right)_{n\geq 1}$ be a sequence of positive functions defined by

[TABLE]

Since $C_{n}^{(k)}=C_{n}/f_{k}^{n-1}$ and

[TABLE]

the equations (3.27) for $1<n\leq k$ and (3.18) for $n>k$ in terms of the new $C_{n}^{(k)}$’s read

[TABLE]

with $C_{1}^{(k)}(t)\equiv 1$ and initial data $C_{n}^{(k)}(t_{0})=0$ for $n\geq 2$. Since the coefficient $-(n-k-1)B/k$ of the linear term of (3.30) is nonpositive for all $n\geq k+1$, the solution of the above IVP can be majorized by another sequence $\left(\tilde{C}_{n}^{(k)}\right)_{n\geq 1}$:

[TABLE]

which solves the IVP

[TABLE]

with $\tilde{C}_{1}^{(k)}(t)\equiv 1$ and initial data $\tilde{C}_{n}^{(k)}(t_{0})=0$ for $n\geq 2$. For (3.31), one may apply the same proof of (3.23) based in Lemma 3.2. It follows that (see in Sec. 7 of [GM]) the power series $\tilde{\Psi}(t,w)=1+\displaystyle\sum_{n\geq 2}\tilde{C}_{n}^{(k)}w^{n-1}$ satisfies (3.7) setting $B=0$, $\Gamma=\Gamma_{k}$ and together with $\tilde{\psi}(t_{0},z)\equiv 1$ has the classical solution

[TABLE]

provided $e\left|w\right|\tilde{\tau}_{k}(t_{0},t)<1$ holds, where $\tilde{\tau}_{k}(t_{0},t)=\displaystyle\int_{t_{0}}^{t}\Gamma_{k}(s)ds$ and $W(x)=\displaystyle\sum_{n\geq 1}\left(-n\right)^{n-1}x^{n}/n!$ denotes the Lambert $W$–function.

We thus have

[TABLE]

concluding the proof of Proposition 3.7.

$\Box$

Returning to the second stage of our scheme, we show that the equation (3.7), under that operation, is replaced by

[TABLE]

For this, let the argument $n$ of the exponential in (3.13) be replaced by $n-1$. The modified coefficients $\left(C_{n}\right)_{n\geq 1}$ of the power series (3.15) satisfy then a system of integral equations

[TABLE]

with $C_{1}(t)\equiv 1$ where $\gamma(s,t)=\displaystyle\int_{s}^{t}B(s^{\prime})ds^{\prime}$. Summing equation (3.33) multiplied by $z^{n-1}$ over $n$ yields an integral equation for $\Theta$:

[TABLE]

and from this we deduce (3.32). Observe that an extra factor $e^{-\gamma(s,t)}$ inside the integration results from the stability improvement (1.23) and the derivative with respect to $t$ applied to this factor produces an additional term $B\left(\Theta-1\right)$ which has to be subtracted (due to the minus sign of the exponent) from the last term on the right hand side of (3.7): $B\left((z\Theta)_{z}-1\right)-B\left(\Theta-1\right)=Bz\Theta_{z}$.

The improved equation (3.32) leads to a significant outcome regarding the radius of convergence of the Mayer series (3.2) for $\beta$ inside each threshold interval $I_{n}=[\beta_{2n},\beta_{2(n+1)})$, $n\in\mathbb{N}$.

Proposition 3.9

Let $\Theta=\Theta(t,z)$ be the classical solution of (3.32) with $B$ and $\Gamma$ as in Proposition 3.4. Then the following majorant relation

[TABLE]

holds for all $k\in\mathbb{N}$ and $\left(\beta,z\right)$ satisfying $ez\tau_{k}(t_{0},t)<1$, where $\tau_{k}$ is given by (3.26).

Proof. Let $\Theta(t,z)$ be defined by the series (3.15) and observe that, by

[TABLE]

(3.32) can be written as a system of first order differential equations for $\left(C_{n}\right)_{n\geq 1}$:

[TABLE]

with $C_{1}(t)\equiv 1$ and initial data $C_{n}(t_{0})=0$ for $n\geq 2$. Since the coefficient $(n-1)B$ of the linear term of (3.35) is smaller than $(n-1)(k+1)B/k$ for $2\leq n\leq k$ and smaller than $nB$ for all $n\geq k+1$,555This part would not be necessary for keeping positive the radius of convergence, uniformly in $t_{0}$, at $\beta<\beta_{k+1}$. for any $k\in\mathbb{N}$, the solution of the above IVP can be majorized, in view of Lemma 3.2, by the solution of the IVP in (3.27), which by Proposition 3.7 satisfies (3.25). The proof of Proposition 3.9 is concluded.

$\Box$

Stability of a neutral pair in the presence of other particles

The third and last stage of our scheme deals with neutral subclusters of order smaller than $k$ that are part of a cluster of order larger or equal to $k+1$. So far, we have proved a weak version of the Conjecture 1.1. Let $k>1$ an odd number and suppose that all neutral clusters and subclusters of order smaller than $k$ have their singularities been removed by hand. Then, the density function (3.2), after the removal, satisfies

[TABLE]

and the majorant series has strictly positive radius of convergence uniformly in the cutoff $t_{0}$ for $\beta<\beta_{k+1}$. From the point of view of the Mayer coefficients $b_{n}$ however, for $n\leq k$ our hypotheses are better than the formulated in the conjecture – instead of removing the coefficients $b_{n}$’s entirely we remove the part of these that diverges as $t_{0}\rightarrow\infty$. The weakness of our hypotheses is that no assumptions on the coefficients $b_{n}$ for $n>k$ are made in Conjecture 1.1. The situations here is different from what we have done before. According to Proposition 3.7, when $n$ is larger than $k$, we don’t need improve the stability condition and we actually cannot for neutral clusters of even size $n$. However, no assumptions mean that neutral subclusters of size smaller than $k$ inside a cluster of order $n$ do not diverges as $t_{0}$ tends to [math] and this claim needs to be proven.

To deal with this scenario, instead of a sequence $\left(A_{n}\right)_{n\geq 1}$ defined by (3.13), we introduce a sequence $\left(\tilde{A}_{m}\right)_{m\geq 1}$ of appended at $\zeta_{0}$ analogous quantities

[TABLE]

which are independent of $\zeta_{0}=(x_{0},\sigma_{0})$ by translational invariance of variable $x_{0}$ and $\left|\sigma_{0}\right|=1$. In the next paper we shall study in particular the recursion relations satisfied by theses quantities together with their majorant equations as well as a systematic majorant approach for the correlation function. In the present paper, we shall restrict ourselves to the simplest case of $m=2$ of (3.36).

Let us consider a $n$–particle cluster containing a neutral pair subcluster. Referring to the formula (3.6), let $I$ be an index set of a cluster with $\left|I\right|=n$ and let $J$ be the index set of a pair $\left|J\right|=2$ of particles with opposite charges: $\sigma_{1}\sigma_{2}=-1$ located at $x_{1}$ and $x_{2}$, whose Ursell function at scale $s$ is simply given by

[TABLE]

where $r=\left|x_{2}-x_{1}\right|$. We refer to Sec. 6.3 of [GM] for detail. Let $\Delta=\Delta(s,\tilde{s},x_{0},x_{1},x_{2})$ be the $h$ part of (3.36) including (3.21) given by

[TABLE]

with $t_{0}\leq\tilde{s}<s$. Using the convolution form (2.14) of the Euclid’s hat $h(w)$ together with its geometric interpretation as the area of “caps” (see proof of Proposition 2.7), we shall find and upper bound for the integral over $x_{1}$ and $x_{2}$ of this quantity. For this, we write

[TABLE]

and observe that the integrand of $\Delta$ differs from [math] if, and only if, either $x_{1}$ is inside of the non null intersection $B_{s/2}(z)\cap B_{\tilde{s}/2}(\tilde{z})$ and $x_{2}$ is inside the complementary region $B_{\tilde{s}/2}(\tilde{z})\backslash(B_{s/2}(z)\cap B_{\tilde{s}/2}(\tilde{z}))\neq\emptyset$ or vice-versa. As a consequence, we have

[TABLE]

where, denoting by $\left|D\right|$ the area of a bounded domain $D\subset\mathbb{R}^{2}$, $A=\left|B_{s/2}(z)\cap B_{\tilde{s}/2}(\tilde{z})\right|$ and $B=\left|B_{\tilde{s}/2}(\tilde{z})\backslash(B_{s/2}(z)\cap B_{\tilde{s}/2}(\tilde{z}))\right|=\left(\pi\tilde{s}^{2}/4\right)-A$. Using

[TABLE]

and the fact that $A$ and $B$ are different from [math] if and only if

[TABLE]

we have

[TABLE]

and this implies that $\tilde{A}_{2}(s)$ given by (3.36) with $m=2$ and $\sigma_{1}\sigma_{2}=-1$ is bounded uniformly with respect to the cutoff $t_{0}$ provided $\beta\in[0,6\pi)$, i. e., inside the first threshold interval $I_{1}=[4\pi,6\pi)$. For this, observe that by Proposition 2.11 (see (3.12))

[TABLE]

if $\beta<6\pi$, uniformly in $t_{0}$.

The integral of (3.38) performed over $x_{1}$ and $x_{2}$ desregarding the minus sign would be proportional to $\tilde{s}^{2}s^{2}$, by (3.8). The small cluster neutrality condition leads to a rearrangement of the powers in $\tilde{s}$ and $s$ (3.39) in favor of $\tilde{s}$. Note that $sg(s)$ remains integrable by Proposition 2.11 and a new function $\Gamma(s)$ needs to be redefined accordingly. The rearrangement is not enough to prevent the collapse of the neutral pair inside the high order threshold intervals and we need to be more careful when $\left|x_{1}-x_{2}\right|/\tau$ is small. By the first mean value theorem, there exist $\tau^{\ast}\in\left[\tilde{s},s\right]$ such that

[TABLE]

For fixed $s$, let us say $s=1$, let $\Lambda=\left\{\left(r,\tilde{s}\right)\in\mathbb{R}_{+}\times\left[t_{0},1\right]:r/\tau^{\ast}(r,\tilde{s})\leq 0.2\right\}$ and note that $m(\tau^{\ast})h(r/\tau^{\ast})<3/4$ for $\left(r,\tilde{s}\right)$ in the complementary set $(\mathbb{R}_{+}\times\left[t_{0},1\right])\backslash\Lambda$, by Propositions 2.7 and 2.11. Under this condition $\tilde{A}_{2}(s)$ can be bounded by the last integral in (3.40) with the exponent $2-\beta/2\pi$ of $\tilde{s}$ replaced by $2-(3/4)\beta/2\pi=2-3\beta/8\pi$, which is finite for $\beta<8\pi$. On the other hand, one can show that the integral of $\Delta$ over $\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$ in (3.39) restricted to $\Lambda$, where $r=\left|x_{1}-x_{2}\right|$, is proportional to $\tilde{s}^{4}$ (instead of $\tilde{s}^{3}$) under the change of variables $x_{i}=\tilde{s}y_{i}$, $i=1,2$. Observe that $\tau^{\ast}(r,\tilde{s})$ tends to $\tilde{s}$ when $r$ tends to [math]. As a consequence, $\tilde{A}_{2}(s)$ can be bounded by the last integral in (3.40) with the exponent $2-\beta/2\pi$ of $\tilde{s}$ replaced by $3-\beta/2\pi$ which is finite again for $\beta<8\pi$.

4 Summary and open question

The main result of the present paper, Theorem 2.8, states that the energy $U_{n}(\xi;h)$ of a configuration $\xi=\left(x,\sigma\right)=\left(x_{1},\ldots,x_{n},\sigma_{1},\ldots,\sigma_{n}\right)$ of $n$ particles, with $\left(x_{i},\sigma_{i}\right)\in\mathbb{R}^{2}\times\{+1,-1\}$, interacting through the two–dimensional Euclid’s hat pair potential $h\left(\cdot/s\right)$ at scale $s$ satisfies (2.19). Since the inequality saturates when the $n$ particles collapses all together to a single point with net charge [math] if $n$ is even and $\pm 1$ if $n$ is odd, a corollary to this (see Corollary 2.10) is that the minimal specific energy $e(h)$ and the minimal constrained specific energy $\bar{e}(h)$, defined by (1.18) and (1.19), are both $-1/2$. The same statement holds to positive radial potentials of positive type in any dimension $d\geq 2$ provided it can be written as scale mixtures of Euclid’s hat: $v(x)=\displaystyle\int g(s)h\left(\left|x\right|/s\right)ds$, $g(s)\geq 0$ and the right hand side of (2.19) is multiplied by $\displaystyle\int g(s)ds$. Consequently, if $n$ is odd the stability bound (1.16) can be replaced by (1.23) for any potential of this class with $B=\bar{B}=\dfrac{1}{2}\displaystyle\int g(s)ds$.

We have applied the main result to the two–dimensional Yukawa gas with particles activity $z$ at the inverse temperature $\beta$ in the interval of collapse $[4\pi,8\pi)$. A Cauchy majorant, proposed in [GM] for the pressure and density function, can be written in terms of the principal branch of the $W$–Lambert function which is analytic provided its argument $-z\tau_{k}$, with $\tau_{k}=\tau_{k}(t_{0},t)$ given by (3.26), satisfies $e\left|z\right|\tau_{k}<1$, $\beta<\beta_{2n}=8\pi\left(1-1/2n\right)$ when the divergent part of the leading even Mayer coefficients up to order $2n\leq k+1$, $k>1$, are extracted. It has been assumed in addition that an improved stability condition (see Conjecture 2.3 of [GM]) holds for any odd number of particles $2n-1\leq k$. However, the numerical evaluation (see Remark 7.5 of [GM]) of the total energy $U_{2n-1}(\xi;\beta\dot{v})$ for the standard scaling decomposition (1.7) when $n=2$ and $3$ have indicated that it would fail for sufficient large $k$ and Proposition 2.2 now proves that $U_{3}(\xi;\beta\dot{v})$ does not satisfy the improved stability for $k>15$. We have in the present paper proved that when the Yukawa potential $v$ is represented as scale mixtures of Euclid’s hat it satisfies Conjecture 2.3 of [GM] for any $k>1$ and, moreover, all the estimates necessary to establish convergence of the majorant series in [GM] holds for this representation of $v$ due to Proposition 2.11. We have reestated Propositions 3.4, 3.7 and 3.9 accordingly for the reader convenience.

It is important to stress at this point that the classical solution $\Theta_{k}=\Theta_{k}(t,z)$ of (3.24) is actually a majorant for the density function (3.2) and the same statement (3.2) holds in Proposition 3.7 as long as $t_{0}>0$. One open question is whether the majorant $\Theta_{k}$ remain faithful when the cutoff $t_{0}$ is set to [math]. We answer the question afirmatively only for $k=3$ and argue that this question might be dealt using the superstability of the (two–species) Yukawa potential restricted to configurations in which a neutral subcluster is located in a small volume of linear size $t_{0}>0$ (see [Gi, RT] and references therein).