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Convergence of the Mayer Series via Cauchy Majorant Method with Application to the Yukawa Gas in the Region of Collapse
Mathematical Physics · Submitted 14 Oct 2003
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For any $\beta \in \lbrack 4\pi, 8\pi)$, the Mayer series with the leading terms of the expansion omitted (how many depending on $\beta $) is shown to be convergent provided an improved stability condition holds.
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We construct majorant functions $\Phi (t,z)$ for the Mayer series of pressure satisfying a nonlinear differential equation of first order which can be solved by the method of characteristics. The domain $| z| <(e\tau) ^{-1}$ of convergence of Mayer series is given by the envelop of characteristic intersections. For non negative potentials we derive an explicit solution in terms of the Lambert $W$ --function which is related to the exponential generating function $T$ of rooted trees as $T(x)=-W(-x)$. For stable potentials the solution is majorized by a non negative potential solution. There are many choices in this case and we combine this freedom together with a Lagrange multiplier to examine the Yukawa gas in the region of collapse. We give, in this paper, a sufficient condition to establish a conjecture of Benfatto, Gallavotti and Nicol\'{o}. For any $\beta \in \lbrack 4\pi, 8\pi)$, the Mayer series with the leading terms of the expansion omitted (how many depending on $\beta $) is shown to be convergent provided an improved stability condition holds. Numerical calculations presented indicate this condition is satisfied if few particles are involved.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We construct majorant functions $\Phi (t,z)$ for the Mayer series of pressure satisfying a nonlinear differential equation of first order which can be solved by the method of characteristics. | method 0.81 | 0.95 | 0.08 | asserted 1.00 |
| s1 | The domain $| z| <(e\tau) ^{-1}$ of convergence of Mayer series is given by the envelop of characteristic intersections. | result 0.96 | 0.83 | 0.17 | asserted 1.00 |
| s2 | For non negative potentials we derive an explicit solution in terms of the Lambert $W$ --function which is related to the exponential generating function $T$ of rooted trees as $T(x)=-W(-x)$. | result 0.55 | 0.82 | 0.08 | asserted 1.00 |
| s3 | For stable potentials the solution is majorized by a non negative potential solution. | result 0.87 | 0.83 | 0.06 | asserted 1.00 |
| s4 | There are many choices in this case and we combine this freedom together with a Lagrange multiplier to examine the Yukawa gas in the region of collapse. | method 0.72 | 0.90 | 0.04 | asserted 0.98 |
| s5 | We give, in this paper, a sufficient condition to establish a conjecture of Benfatto, Gallavotti and Nicol\'{o}. | contribution 0.70 | 0.88 | 0.30 | asserted 0.94 |
| s6 | For any $\beta \in \lbrack 4\pi, 8\pi)$, the Mayer series with the leading terms of the expansion omitted (how many depending on $\beta $) is shown to be convergent provided an improved stability condition holds. | result 0.97 | 0.92 | 0.58 | asserted 1.00 |
| s7 | Numerical calculations presented indicate this condition is satisfied if few particles are involved. | result 0.94 | 0.93 | 0.08 | hedged possibility 0.99 |
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Original abstract
We construct majorant functions $\Phi (t,z)$ for the Mayer series of pressure satisfying a nonlinear differential equation of first order which can be solved by the method of characteristics. The domain $| z| <(e\tau) ^{-1}$ of convergence of Mayer series is given by the envelop of characteristic intersections. For non negative potentials we derive an explicit solution in terms of the Lambert $W$ --function which is related to the exponential generating function $T$ of rooted trees as $T(x)=-W(-x)$. For stable potentials the solution is majorized by a non negative potential solution. There are many choices in this case and we combine this freedom together with a Lagrange multiplier to examine the Yukawa gas in the region of collapse. We give, in this paper, a sufficient condition to establish a conjecture of Benfatto, Gallavotti and Nicol\'{o}. For any $\beta \in \lbrack 4\pi, 8\pi)$, the Mayer series with the leading terms of the expansion omitted (how many depending on $\beta $) is shown to be convergent provided an improved stability condition holds. Numerical calculations presented indicate this condition is satisfied if few particles are involved.
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How the literature uses it
2 arXiv papers cite this in 20 sentences
Papers citing it · 17 sentences without a confident label
Contests it 1 paper
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However, the numerical evaluation (see Remark 7.5 of [ref] ) of the total energy $U_2n-1(ξ;βv̇)$ for the standard scaling decomposition ([ref]) when $n=2$ and $3$ have indicated that it would fail for sufficient large $k$ and Proposition [ref] now proves that $U_3(ξ;βv̇)$ does not satisfy the improved stability for $k>15$.
On the Mayer series of two-dimensional Yukawa gas at inverse temperature in the interval of collapse · Summary and open question · contest 0.85
Uses its method or tool 1 paper
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Writing $ω=η(t)z$, we have [equation] and the system of equations for $( Q̃_k) _k≥1$, going backward through the steps ([ref])-([ref]), is equivalent to the following PDE (compare with ([ref])) [equation] with $Q̃(0,ω)=1$, whose solution can be explicitly written in terms of the Lambert $W$–function (see Subsection 5.1 of [ref] ) [equation] (i.e., $W(x)$ is the principal branch of the inverse of $f(W)=We^W$, regular at origin [ref] ).
On the Virial Series for a Gas of Particles with Uniformly Repulsive Pairwise Interaction · Convergence of Power Series: Miscellaneous Methods · 0.91
Uses its data or values 1 paper
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Numerical calculations performed in [ref] indicate that $e̅_2k+1( h̃)$ remains for $k>1$ strictly smaller than $-1/2$.
On the Mayer series of two-dimensional Yukawa gas at inverse temperature in the interval of collapse · Minimal specific energy: main theorem and estimates involving modified Bessel functions · 0.83
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