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arXiv 1107.3670math.PRmath-phmath.MP
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连续经典多体系统中团簇尺寸分布的大偏差

Large deviations for cluster size distributions in a continuous classical many-body system

Sabine Jansen, Wolfgang König, Bernd Metzger

备注：
Published in at http://dx.doi.org/10.1214/14-AAP1014 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：
Annals of Applied Probability 2015, Vol. 25, 930-973
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摘要：

统计物理中一个有趣的问题是当粒子对相互作用由稳定的Lennard-Jones型势给出时经典粒子在液滴或团簇中的凝聚。我们研究这个问题的两个方面。我们首先推导出一个大偏差原则的团簇尺寸分布的任何逆温度$β\in (0,\infty)$和粒子密度$ρ\in(0,ρ_{\mathrm{cp}})$的热力学极限。这里$ρ_{\mathrm{cp}}>0$是紧密堆积密度。虽然在一般情况下的速率函数是一个抽象的对象，我们的第二个主要结果是$Γ$ -收敛的速率函数向一个明确的限制速率函数在低温稀限$β\to \infty$，$ρ\downarrow0$，使$-β^{-1}\logρ\toν$的一些$ν\in(0,\infty)$。在最近的工作中出现了极限速率函数和它的极小值，其中温度和粒子密度与粒子数耦合。在这里考虑的解耦限制，我们证明，只有一个集群大小是占主导地位的，取决于参数$ν$。在对势的附加假设下，$Γ$ -沿曲线的收敛可以被加强到一致的边界，在低温、低密度矩形中有效。

英文摘要：

An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $β\in (0,\infty)$ and particle density $ρ\in(0,ρ_{\mathrm{cp}})$ in the thermodynamic limit. Here $ρ_{\mathrm{cp}}>0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $Γ$-convergence of the rate function toward an explicit limiting rate function in the low-temperature dilute limit $β\to \infty$, $ρ\downarrow0$ such that $-β^{-1}\logρ\toν$ for some $ν\in(0,\infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the decoupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $ν$. Under additional assumptions on the potential, the $Γ$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.

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