# Local Dynamics in an Infinite Harmonic Chain

## Abstract

**:**

## 1. Introduction

## 2. Method of Recurrence Relations

#### 2.1. Kubo Scalar Product

#### 2.2. Basis Vectors

#### 2.3. Basis Functions

#### 2.4. Continued Fractions

## 3. Local Dynamics in a Harmonic Chain

#### Langevin Dynamics

## 4. Dispersion Relation for Harmonic Chain

- (a)
- $N=2,\phantom{\rule{4.0pt}{0ex}}\sigma =(2,2)$${P}_{2}={z}^{2}+2,$${q}_{2}=1$
- (b)
- $N=4,\phantom{\rule{4.0pt}{0ex}}\sigma =(2,1,1,2)$${P}_{4}={z}^{4}+4{z}^{2}+2$${q}_{4}={z}^{2}+2$
- (c)
- $N=6;\phantom{\rule{4.0pt}{0ex}}\sigma =(2,1,1,1,1,2)$${P}_{6}={z}^{6}+6{z}^{4}+9{z}^{2}+2$${q}_{6}={z}^{4}+4{z}^{2}+3$
- (d)
- $N=8;\phantom{\rule{4.0pt}{0ex}}\sigma =(2,1,1,1,1,1,1,2)$${P}_{8}={z}^{8}+8{z}^{6}+20{z}^{4}+16{z}^{2}+2$${q}_{8}={z}^{6}+6{z}^{4}+10{z}^{2}+4$
- (e)
- $N=10;\phantom{\rule{4.0pt}{0ex}}\sigma =(2,1,1,1,1,1,1,1,1,2)$${P}_{10}={z}^{10}+10{z}^{8}+35{z}^{6}+50{z}^{4}+25{z}^{4}+2$${q}_{10}={z}^{8}+8{z}^{6}+21{z}^{4}+20{z}^{2}+5$
- (f)
- $N=12;\phantom{\rule{4.0pt}{0ex}}\sigma =(2,1,1,1,1,1,1,1,1,1,1,2)$${P}_{12}={z}^{12}+12{z}^{10}+54{z}^{8}+112{z}^{6}+105{z}^{4}+36{z}^{2}+2$${q}_{12}={z}^{10}+10{z}^{8}+36{z}^{6}+56{z}^{4}+35{z}^{2}+6$

#### 4.1. Zeros of ${q}_{N}$

#### 4.2. ${a}_{0}(t)$ for Finite N

#### 4.3. ${a}_{0}(t)$ When $N\to \infty $

#### 4.4. ${\tilde{a}}_{0}(z)={\Psi}_{N}(z)$ When N$\to \infty $

## 5. Ergodicity of Dynamical Variable $A={p}_{0}$

#### 5.1. Infinite Harmonic Chain

#### 5.2. Infinite Harmonic Chain with One End Attached to a Wall

## 6. Harmonic Chain and Logistic Map

## 7. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

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Local Dynamics in an Infinite Harmonic Chain. *Symmetry* **2016**, *8*, 22.
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Lee MH.
Local Dynamics in an Infinite Harmonic Chain. *Symmetry*. 2016; 8(4):22.
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2016. "Local Dynamics in an Infinite Harmonic Chain" *Symmetry* 8, no. 4: 22.
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