广义非线性西格玛模型与约束随机动力学的联系
On the connection of the generalized nonlinear sigma model with constrained stochastic dynamics
摘要：
在连续极限下的自由连接链的动力学由场论描述，它与非线性西格玛模型非常相似。这个场论的生成泛函$Ψ[J]$包含非完整约束，这些约束是通过在表示$Ψ[J]$的路径积分中插入δ函数的适当乘积而施加的。同样的程序通常应用在统计力学中，以便在连接的聚合物系统上施加拓扑条件。这种方法的缺点是，显然失去了与控制链扩散的随机过程的联系。这项工作的主要目标是重新建立这种联系。为此目的，它是在这里示出的生成功能$Ψ[J]$符合的生成功能的关联函数的约束朗之万方程的解决方案。在离散的情况下，这个朗之万方程描述了预期的布朗运动的珠子连接在一起的链接的固定长度。
英文摘要：
The dynamics of a freely jointed chain in the continuous limit is described by a field theory which closely resembles the nonlinear sigma model. The generating functional $Ψ[J]$ of this field theory contains nonholonomic constraints, which are imposed by inserting in the path integral expressing $Ψ[J]$ a suitable product of delta functions. The same procedure is commonly applied in statistical mechanics in order to enforce topological conditions on a system of linked polymers. The disadvantage of this method is that the contact with the stochastic process governing the diffusion of the chain is apparently lost. The main goal of this work is to reestablish this contact. To this purpose, it is shown here that the generating functional $Ψ[J]$ coincides with the generating functional of the correlation functions of the solutions of a constrained Langevin equation. In the discrete case, this Langevin equation describes as expected the Brownian motion of beads connected together by links of fixed length.
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