- #1
buffordboy23
- 548
- 2
Homework Statement
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 of M, and is given by [tex] k ln(1 + \sqrt{2})[/tex].
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.