We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard–Jones type. The pressure (stress) is assumed to be small but positive and bounded away from zero, while the temperature $$\beta ^{-1}$$ β - 1 goes to zero. Our main results are: (1) As $$\beta \rightarrow \infty $$ β → ∞ at fixed positive pressure $$p>0$$ p > 0 , the Gibbs measures $$\mu _\beta $$ μ β and $$\nu _\beta $$ ν β for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals $$\overline{{\mathcal {E}}}_{\mathrm {bulk}}$$ E ¯ bulk and $$\overline{{\mathcal {E}}}_\mathrm {surf}$$ E ¯ surf . The minimizer of the surface functional corresponds to zero temperature boundary layers; (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of $$\overline{{\mathcal {E}}}_\mathrm {surf}$$ E ¯ surf ; (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts; (4) Bounds on the decay of correlations are provided, some of them uniform in $$\beta $$ β .

中文翻译：

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低温下原子链的表面能和边界层

我们分析了低温下原子链的表面能和边界层，以获得 Lennard-Jones 类型的相互作用势。压力（应力）被假定为很小但为正且远离零，而温度 $$\beta ^{-1}$$ β - 1 变为零。我们的主要结果是： (1) 当 $$\beta \rightarrow \infty $$ β → ∞ 在固定正压力 $$p>0$$ p > 0 时，Gibbs 测量 $$\mu _\beta $$ μ β 和 $$\nu _\beta $$ ν β 对于无限链和半无限链满足路径大偏差原则。速率函数是体积和表面能函数 $$\overline{{\mathcal {E}}}_{\mathrm {bulk}}$$ E ¯ bulk 和 $$\overline{{\mathcal {E}}}_ \mathrm {surf}$$ E¯ surf 。表面泛函的最小值对应于零温度边界层；(2) 对吉布斯自由能的表面修正收敛到零温表面能，借助 $$\overline{{\mathcal {E}}}_\mathrm {surf}$$ E¯ 的最小值表征冲浪; (3) 体积吉布斯测度和吉布斯自由能可以通过它们的高斯对应物来近似；(4) 提供了相关性衰减的界限，其中一些在 $$\beta $$ β 中是一致的。