Mayer级数的Cauchy优次法收敛性及其在塌陷区Yukawa气体中的应用
Convergence of the Mayer Series via Cauchy Majorant Method with Application to the Yukawa Gas in the Region of Collapse
摘要：
本文对压力的Mayer级数满足一阶非线性微分方程构造了优函数$Φ(t,z)$，并利用特征线法求解。利用特征交点的包络给出了Mayer级数的收敛域$| z| <(eτ) ^{-1}$.对于非负电位，我们推导出一个明确的解决方案的朗伯$W$ -功能，这是有关的指数生成函数$T$的根树作为$T(x)=-W(-x)$。对于稳定的电势，该解由非负电势解优化。在这种情况下有很多选择，我们将这种自由度与拉格朗日乘数结合起来来研究坍缩区域中的汤川气体。本文给出了Benfatto，Gallavotti和Nicoló猜想成立的一个充分条件.对于任何$β\in \lbrack 4π, 8π)$，迈耶系列的领先条款的扩展省略（有多少取决于$β$）被证明是收敛提供了一个改进的稳定性条件。数值计算表明，这一条件是满足的，如果很少的粒子参与。
英文摘要：
We construct majorant functions $Φ(t,z)$ 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 $| z| <(eτ) ^{-1}$ 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 Nicoló. For any $β\in \lbrack 4π, 8π)$, the Mayer series with the leading terms of the expansion omitted (how many depending on $β$) 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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