Stationary state

See xournal 223.
See this in the context of Quantum Mechanics, for the moment.

On the other hand, in general linear evolution equations (finite or infinite dimensional) I like to think of stationary states as special superpositions of states which behaves particularly well with respect to time evolution.
For example, consider two species of fishes $F_{1}$ and $F_{2}$ , whose populations is given respectively by $x (t), y (t)$ . Assume that the dynamics is given by

(\begin{matrix} \dot{x} \\ \dot{y} \end{matrix}) = L \cdot (\begin{matrix} x \\ y \end{matrix}) .

An eigenvector $V$ of $L$ ( $L V = λ V$ ) is something like a stationary state. For me, every eigenvector $V_{1}, V_{2}$ with eigenvalues $λ_{1}, λ_{2}$ , is something like a new "being" made of both fish species. This way we have two kind of "cluster of species" that behave in a fairly simple way with respect to time: it undergoes an exponential growth. That is, the vector $(\begin{matrix} 1 \\ 0 \end{matrix})$ represents the specie $F_{1}$ and $(\begin{matrix} 0 \\ 1 \end{matrix})$ represents $F_{2}$ ; and the vector $Y (t) = (\begin{matrix} x (t) \\ y (t) \end{matrix}) = x (t) (\begin{matrix} 1 \\ 0 \end{matrix}) + y (t) (\begin{matrix} 0 \\ 1 \end{matrix})$ is the complete system. But we can now think that we have two new species represented by the vectors $V_{1}$ and $V_{2}$ in such a way that the system is expressed as

Y (t) = {(\begin{matrix} α_{1} (t) \\ α_{2} (t) \end{matrix})}_{B} = α_{1} (t) V_{1} + α_{2} (t) V_{2} .

where the $B$ is to mean "another basis". What we gain from it is that

\dot{Y} (t) = {(\begin{matrix} {\dot{α}}_{1} (t) \\ {\dot{α}}_{2} (t) \end{matrix})}_{B}

and

L Y (t) = α_{1} (t) L V_{1} + α_{2} (t) L V_{2} = α_{1} (t) λ_{1} V_{1} + α_{2} (t) λ_{2} V_{2}

and then

α_{i} (t) = e^{t λ_{i}}

So the population of these two new species have a fairly simple growth. And moreover, there is a linear coordinate change to obtain $x (t), y (t)$ from $α_{1} (t), α_{2} (t)$ :

(\begin{matrix} x (t) \\ y (t) \end{matrix}) = (V_{1}, V_{2}) (\begin{matrix} α_{1} (t) \\ α_{2} (t) \end{matrix})