(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a linear chain of N atoms. Each atom can be in 3 states (A,B,C) but an atom is state A cannot be next to an atom in state C. Find the entropy per atom as N approaches infinity.

Accomplish this by defining the 3-vector [tex] \vec{v}^{j} [/tex] to be the number of allowed configurations of the j-atom chain ending in type A, B, C. Then show that [tex] \vec{v}^{j} = \textbf{M}\vec{v}^{j-1}[/tex]. Then [tex] \vec{v}^{j} = \textbf{M}^{j-1}\vec{v}^{1}[/tex]. Show that in the limit of large N, the entropy per atom is dominated by the largest eigenvalue ofM, and is given by [tex] k ln(1 + \sqrt{2})[/tex].

3. The attempt at a solution

For the first j-atom chains, it is evident that

[tex] \vec{v}^{1} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} [/tex], [tex] \vec{v}^{2} = \begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix} [/tex], [tex] \vec{v}^{3} = \begin{bmatrix} 5 \\ 7 \\ 5 \end{bmatrix} [/tex]

which implies that

[tex] \textbf{M} = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{bmatrix} [/tex]

Right now I am having trouble with the first part: show that [tex] \vec{v}^{j} = \textbf{M}\vec{v}^{j-1}[/tex]. It is easy to show for specific cases using the vectors I have determined above, but I am confused on how to generalize this relation.

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# Homework Help: Linear algebra application: entropy

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