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Dynamics Solved by the Three-Point Formula: Exact Analytical Results for Rings

Abstract

In this paper, we study in the framework of the Gaussian model, the relaxation dynamics, and diffusion process on structures which show a ring-shape geometry. In order to extend the classical connectivity matrix to include interactions between more distant nearest neighbors, we treat the second derivative with respect to position by using the three-point formula. For this new Laplacian matrix, we determine analytical solutions to the eigenvalue problem. The relaxation dynamics is described by the mechanical relaxation moduli and for diffusion we focus on the behavior of the residual concentration at the initial node. Additionally, we investigate the scaling behaviors of the mean squared radius of gyration and of the smallest eigenvalue. To calculate the residual concentration, we consider that initially the whole material is concentrated only in one node and then it spreads over the ring. We compare our results with the ones obtained from the incremental ratio method. We observe that the results of the two methods for the considered quantities are slightly different. At any intermediate time/frequency domain, the results obtained by using the incremental ratio method underestimate the ones obtained by using the three-point formula. This finding can turn important for many applications in polymer systems or in other systems where diffusive motion occurs.

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Acknowledgments

M.G. acknowledges the financial support of the CNPq. A. J. thanks to Prof. Dr. Jens Uwe Sommer and to Dr. Michael Lang for many fruitful discussions.

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Correspondence to Mircea Galiceanu.

Appendix A: Three-Point Formula Method for Derivatives

Appendix A: Three-Point Formula Method for Derivatives

The three-point formula is a computational good alternative to the incremental ratio method when a calculation of the derivatives of a function is required. To overcome the problem of the truncation error, more evaluation points should be used. To generalize the approximation formulas for derivatives, a generalization of the Lagrange polynomials is developed. Instead of using only two points (x 0 and x 1) to define a Lagrange polynomial f(x), a set of n + 1 distinct numbers x 0, x 1, x 2,..., x n is used:

$$ f(x)=\sum\limits_{k=0}^{n} f(x_{k})L_{k}(x)+\frac{(x-x_{0}) \cdot {\dots} \cdot(x-x_{n})}{(n+1)!}f^{n+1}(\xi(x)), $$
(A1)

for a generic function ψ(x) in the interval (x 0, x n ), where L k (x) is the kt h Lagrange coefficient. Differentiating (A1) leads to:

$$\begin{array}{@{}rcl@{}} f^{\prime}(x)&=&\!\sum\limits_{k=0}^{n} f(x_{k})L^{\prime}_{k} (x)+D_{x}\left[\frac{(x-x_{0})\cdot \dots \cdot(x-x_{n})}{(n+1)!}\right]\\ &&\times f^{n+1}(\xi(x))+ \frac{(x-x_{0})\cdot {\dots} \cdot(x-x_{n})}{(n+1)!}\\ &&\times D_{x}\left[f^{n+1}(\xi(x))\right], \end{array} $$
(A2)

where D x [] is the derivative with respect to x.

This generalization is referred as (n+1)-point formula. For futher details about this derivation and the treatment of the error term, see [32]. Usually, higher is the number of evaluation points greater the accuracy will be. However, truncation error issues and expensive computational time lead to the use of three or five points formulas. In the present work, we focus on the three-points formula. Its derivation begins with the definition of the three Lagrange coefficients and their derivatives:

$$ L_{0}(x)=\frac{(x-x_{1})(x-x_{2})}{(x_{0}-x_{1})(x_{0}-x_{2})}\quad L_{0}^{\prime}(x)=\frac{2x-x_{1}-x_{2}}{(x_{0}-x_{1})(x_{0}-x_{2})} $$
(A3)
$$ L_{1}(x)=\frac{(x-x_{2})(x-x_{0})}{(x_{1}-x_{2})(x_{1}-x_{0})}\quad L_{1}^{\prime}(x)=\frac{2x-x_{0}-x_{2}}{(x_{1}-x_{0})(x_{1}-x_{2})} $$
(A4)
$$ L_{2}(x)=\frac{(x-x_{1})(x-x_{0})}{(x_{2}-x_{1})(x_{2}-x_{0})}\quad L_{2}^{\prime}(x)=\frac{2x-x_{0}-x_{1}}{(x_{2}-x_{0})(x_{2}-x_{1})}. $$
(A5)

Substituting (A3), (A4), and (A5) in (A2), we obtain after some manipulations the derivative of f

$$ f^{\prime}(x)=f(x_{0})L_{0}^{\prime}(x)+f(x_{1})L_{1}^{\prime}(x)+f(x_{2})L_{2}^{\prime}(x). $$
(A6)

If we consider x i = x 0 + i h and h>0, we obtain the so-called forward derivative. We identify the point x i = x 0 + i h with node n + i, with i being any integer variable; thus node n corresponds to x 0. For simplicity, we choose h = 1. Choosing x = x 0 (or node n) and inserting the values of \(L_{0}^{\prime }\), \(L_{1}^{\prime }\), and \(L_{2}^{\prime }\) into the last equation, we obtain the equation for the forward derivative:

$$ f_{for}^{\prime}(n)= -\frac32 f(n) +2f(n+1)-\frac 12f(n+2). $$
(A7)

Similarly, the expression of the backward derivative is obtained by choosing x = x 2 and substituting in (A6), we get after some algebraic calculations

$$ f_{back}^{\prime}(n+2)= \frac12 f(n) -2f(n+1)+\frac32 f(n+2). $$
(A8)

Renoting n + 2 with n this equation can be rewritten as

$$ f_{back}^{\prime}(n)= \frac32 f(n) -2f(n-1)+\frac 12f(n-2). $$
(A9)

We can average the \(f_{for}^{\prime }(n)\) and \(f_{back}^{\prime }(n)\) in order to have the centered derivative

$$\begin{array}{@{}rcl@{}} f_{c}^{\prime}(n)&=& \frac 12 [f_{for}^{\prime}(n)+f_{back}^{\prime}(n)]= \frac 14f(n-2) -f(n-1)\\ &&+ f(n+1)-\frac 14f(n+2). \end{array} $$
(A10)

Following the same procedure, we find an equivalent expression for the second derivative, i.e.,

$$\begin{array}{@{}rcl@{}} f_{c}^{\prime\prime}(n)&=& \frac 1{16}f(n-4)-\frac12f(n-3)+f(n-2)\\ &&+\frac{1}{2}f(n-1)-\frac{17}{8}f(n)+\frac{1}{2}f(n+1)\\ &&+f(n+2)-\frac12f(n-3)+\frac 1{16}f(n+4). \end{array} $$
(A11)

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Galiceanu, M., Jurjiu, A., Volta, A. et al. Dynamics Solved by the Three-Point Formula: Exact Analytical Results for Rings. Braz J Phys 45, 719–729 (2015). https://doi.org/10.1007/s13538-015-0371-6

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Keywords

  • Diffusion equation
  • Three-point formula
  • Eigenvalue problem
  • Residual concentration
  • Viscoelastic relaxation