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This analysis focuses on the probability distribution function, $P_i(\mathbf{r}_i-\mathbf{r}_\mathrm{cm})$, of the $i$th segment relative to the center of mass in an ideal chain. An ideal chain is modeled as a multidimensional random walk with independent steps. The distribution of each step, with a mean length of $b$, is assumed to be Gaussian: $P(\mathbf{r})\sim\mathcal{N}(0,b^2)$. Let $\mathbf{b}_i=\mathbf{r}_{i+1}-\mathbf{r}_{i}$ […]
A circulant matrix is defined as: $C=\begin{bmatrix}c_{0}&c_{n-1}&\dots &c_{2}&c_{1}\\c_{1}&c_{0}&c_{n-1}&&c_{2}\\\vdots &c_{1}&c_{0}&\ddots &\vdots \\c_{n-2}&&\ddots &\ddots &c_{n-1}\\c_{n-1}&c_{n-2}&\dots &c_{1}&c_{0}\\\end{bmatrix}$ where $C_{j, k}=c_{j-k \mod n}$. The $k$-th eigenvalue $\lambda_k$ and eigenvector $x_k$ satisfy $C\cdot x_k=\lambda_k x_k$, which can be expressed as $n$ equations: $\sum_{j=0}^{m-1}c_{m-j}x_j+\sum_{j=m}^{n-1}c_{n-j+m}x_j=\lambda_k x_m\quad m=0,1,\dots,n-1$ with $c_n=c_0$, where $x_m$ is the $m$-th component of the eigenvector $x_k$. By changing the […]
Introduction Saddle point methods are widely used to estimate integrals of the form: $I = \int \exp\left(-f(x)\right)\mathrm{d}x$ where the function $f(x)$ can be approximated by the first two terms of its Taylor series expansion around a point $x_0$: $f(x)\approx f(x_0) + f^\prime (x_0)(x-x_0) + \frac{1}{2}f^{\prime\prime}(x_0)(x-x_0)^2$ The integral is then approximated by its value at the […]
The Boltzmann superposition principle states that the strain response of a viscoelastic material is a superposition of the responses to all previous stress histories. Assuming no stress before time $t=0$, the constitutive equation can be written as: $\begin{equation} \gamma(t)=\int_0^t J(t-t^\prime)\dot{\sigma}(t^\prime)\mathrm{d}t^\prime \end{equation}$ Applying the Laplace transform to this equation yields: $\begin{align} \hat{\gamma}(s)&=\hat{J}(s)\hat{\dot{\sigma}}(s)\\ &=\hat{J}(s)\left(s\hat{\sigma}(s)-\sigma(0^{-})\right) \\ &=\hat{J}(s)\left(s\hat{G}(s)\hat{\dot{\gamma}}(s)-\sigma(0^{-})\right)\\ &=\hat{J}(s)\left(s\hat{G}(s)(s\hat{\gamma}(s)-\gamma(0^-))-\sigma(0^{-})\right)\\ […]
Persistence length, $L_p$, is a fundamental mechanical property that quantifies the bending stiffness of a polymer. It’s defined as the characteristic length scale over which the correlation of bond angles decays. This correlation is expressed as the average cosine of the angle, $\theta$, between bonds separated by a distance $s$ along the chain: $\langle\cos(\theta(s))\rangle = […]