We study the dynamics of a single semiflexible filament coupled to a Hookean spring at its boundary. The spring produces a fluctuating tensile force on the filament, the value of which depends on the filament's instantaneous end-to-end length. The spring thereby introduces a nonlinearity, which mixes the undulatory normal modes of the filament and changes their dynamics. We study these dynamics using the Martin–Siggia–Rose–Janssen–De Dominicis formalism, and compute the time-dependent correlation functions of transverse undulations and of the filament's end-to-end distance. The relaxational dynamics of the modes below a characteristic wavelength $\sqrt{\kappa /{\tau }_R}$, set by the filament's bending modulus $\kappa$ and spring-renormalized tension ${\tau }_R$, are changed by the boundary spring. This occurs near the crossover frequency between tension- and bending-dominated modes of the system. The boundary spring can be used to represent the linear elastic compliance of the rest of the filament network to which the filament is cross linked. As a result, we predict that this nonlinear effect will be observable in the dynamical correlations of constituent filaments of networks and in the networks' collective shear response. The system's dynamic shear modulus is predicted to exhibit the well-known crossover with increasing frequency from ${\omega }^{1/2}$ to ${\omega }^{3/4}$, but the inclusion of the network's compliance in the analysis of the individual filament dynamics shifts this transition to a higher frequency.

中文翻译：

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网络中半柔丝的波动波动动力学

我们研究了单个半柔性细丝在其边界处耦合到Hookean弹簧的动力学。弹簧在细丝上产生波动的拉力，该拉力的值取决于细丝的瞬时端到端长度。弹簧因此引入了非线性，该非线性混合了细丝的起伏的正常模式并改变了它们的动力学。我们使用Martin–Siggia–Rose–Janssen–De Dominicis形式主义研究了这些动力学，并计算了横向波动和灯丝端到端距离的时间相关函数。低于特征波长的模式的弛豫动力学$\sqrt{\kappa /{\tau }_{\mathrm{[R}}}$，由灯丝的弯曲模量设定 $\kappa$ 和弹簧归一化张力 ${\tau }_{\mathrm{[R}}$，由边界弹簧改变。这发生在系统的张力模式和弯曲模式之间的交叉频率附近。边界弹簧可以用来表示长丝交联的其余长丝网络的线性弹性柔度。结果，我们预测这种非线性效应将在网络组成细丝的动力学相关性以及网络的集体剪切响应中可见。预测该系统的动态剪切模量会随着频率的增加而表现出众所周知的交叉。${\omega }^{\mathrm{1个}/2}$ 至 ${\omega }^{3/4}$，但是在分析单个灯丝动力学时考虑到网络的顺应性，则使这种过渡转变为更高的频率。