A polymer-chain network is a collection of interconnected polymer-chains, made themselves of the repetition of a single pattern called a monomer. Our first main result establishes that, for a class of models for polymer-chain networks, the thermodynamic limit in the canonical ensemble yields a hyperelastic model in continuum mechanics. In particular, the discrete Helmholtz free energy of the network converges to the infimum of a continuum integral functional (of an energy density depending only on the local deformation gradient) and the discrete Gibbs measure converges (in the sense of a large deviation principle) to a measure supported on minimizers of the integral functional. Our second main result establishes the small temperature limit of the obtained continuum model (provided the discrete Hamiltonian is itself independent of the temperature), and shows that it coincides with the $$\Gamma $$ Γ -limit of the discrete Hamiltonian, thus showing that thermodynamic and small temperature limits commute. We eventually apply these general results to a standard model of polymer physics from which we derive nonlinear elasticity. We moreover show that taking the $$\Gamma $$ Γ -limit of the Hamiltonian is a good approximation of the thermodynamic limit at finite temperature in the regime of large number of monomers per polymer-chain (which turns out to play the role of an effective inverse temperature in the analysis).

中文翻译：

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从统计高分子物理学到非线性弹性

聚合物链网络是相互连接的聚合物链的集合，它们由称为单体的单一模式的重复构成。我们的第一个主要结果表明，对于聚合物链网络的一类模型，规范系综中的热力学极限会产生连续介质力学中的超弹性模型。特别是，网络的离散亥姆霍兹自由能收敛到连续积分泛函（能量密度仅取决于局部变形梯度）的下界，而离散吉布斯测度收敛到（在大偏差原理的意义上）支持积分泛函的最小值的度量。我们的第二个主要结果建立了所获得的连续模型的小温度限制（假设离散哈密顿量本身与温度无关），并表明它与离散哈密顿量的 $$\Gamma $$ Γ -limit 重合，从而表明热力学和小温度极限相互交换。我们最终将这些一般结果应用于聚合物物理的标准模型，从中我们可以得出非线性弹性。此外，我们还表明，在每个聚合物链中有大量单体的情况下，采用哈密顿量的 $$\Gamma $$ Γ -limit 是有限温度下热力学极限的一个很好的近似值（结果证明分析中的有效逆温度）。