arXiv 2009.079172020-09-16math-phmath.MP

高密度边界条件下连续介质中的经典粒子

Classical particles in the continuum subjected to high density boundary conditions

Aldo Procacci, Sergio A. Yuhjtman

摘要：

考虑了一个被限制在有限区域$Λ$或$\mathbb{R}^d$中的经典粒子连续系统，在非自由边界条件下，通过超稳定的回火对势相互作用.我们证明了在任何固定的逆温度$β$和任何固定的逸度$λ$系统的压力的热力学极限不依赖于边界条件所产生的粒子外$Λ$的密度可能会增加次线性的距离从原点的速度取决于如何快速的对潜在的衰减在大的距离。特别地，如果对电势$v(x-y)$是Lennard-Jones类型，即它衰减为$C/\|x-y\|^{d+p}$其中$p>0$是$x$和$y$之间的欧几里得距离，则在存在由外部粒子产生的边界条件的情况下，保证压力的热力学极限的存在，外部粒子可以以随着距原点的距离$r$而增加的密度分布，$ρ(1+ r^q)$，其中$ρ$是任何正常数（甚至任意大于自由边界条件下计算的系统密度$ρ_0(β,λ)$）和$q\le {1\over 2}\min\{1, p\}$。

英文摘要：

We consider a continuous system of classical particles confined in a finite region $Λ$ of $\mathbb{R}^d$ interacting through a superstable and tempered pair potential in presence of non free boundary conditions. We prove that the thermodynamic limit of the pressure of the system at any fixed inverse temperature $β$ and any fixed fugacity $λ$ does not depend on boundary conditions produced by particles outside $Λ$ whose density may increase sub-linearly with the distance from the origin at a rate which depends on how fast the pair potential decays at large distances. In particular, if the pair potential $v(x-y)$ is of Lennard-Jones type, i.e. it decays as $C/\|x-y\|^{d+p}$ (with $p>0$) where $\|x-y\|$ is the Euclidean distance between $x$ and $y$, then the existence of the thermodynamic limit of the pressure is guaranteed in presence of boundary conditions generated by external particles which may be distributed with a density increasing with the distance $r$ from the origin as $ρ(1+ r^q)$, where $ρ$ is any positive constant (even arbitrarily larger than the density $ρ_0(β,λ)$ of the system evaluated with free boundary conditions) and $q\le {1\over 2}\min\{1, p\}$.

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