Prove that if $H:R^n\rightarrow R^n$ is any Dirichlet operator, and $\psi\in R^n$ is any vector, then
Prove that if $H$ is an infinitesimal stochastic operator and $O$ is an observable, then if $\psi(t)$ obeys the master equation, then
Suppose $U$ is a stochastic operator and $O$ is an observable. Show that $O$ commutes with $U$ iff the expected values of $O$ and its square don’t change when we evolve our state one time step using $U$. Hence or otherwise, show that
if and only if
and
for all stochastic states $\psi(t)$.
Let $T$ be a nonnegative $n\times n$ matrix. Show that $T$ is irreducible if and only if for all $i,j\geq 0$, $(T^m)_{ij}$ greater than $0$ for some natural number $m$.
We defined in class the usual concept of irreducibility for nonnegative matrices, but also defined a concept of irreducibility for Dirichlet operators. Luckily there’s no conflict: Dirichlet operators aren’t nonnegative matrices, but if we add a big multiple of the identity to a Dirichlet operator it becomes a nonnegative matrix, and then: Show that a Dirichlet operator $H$ is irreducible if and only if the nonnegative operator $H+cI$ (where $c$ is any sufficiently large constant) is irreducible.
Irreducibility is also related to the nonexistence of interesting conserved quantities. In class we saw a version of Noether’s Theorem for stochastic mechanics. Remember that an observable $O$ in stochastic mechanics assigns a number $O_i$ to each configuration $i=1...n$. We can make a diagonal matrix with $O_i$ as its diagonal entries, and by abuse of language we call this $O$ as well. Then we say $O$ is a conserved quantity for the Hamiltonian $H$ if the commutator $[O,H]=OH-HO$ vanishes.
In fact this works more generally.
Here we will consider the SI, and SIR model acting on a lattice of three spin-1 particles, each interacting with the other two. Consider the basis $|I\rangle:=|0\rangle$, $|S\rangle:=|1\rangle$ and $|R\rangle:=|2\rangle$ to denote a person as being either infected, susceptible or recovered. We will consider the following transitions
with rate $\alpha$ and
with rate $\beta$
Write down the pseudo Hamiltonian modeling the process.
What are the conserved quantities for $\beta = 0$ (SI) and positive $\alpha$?
What are the conserved quantities for $\beta$, $\alpha$ positive (SIR)?
Consider the initial state $|S\rangle|S\rangle|S\rangle$. With $\alpha =\beta =1$ what is the probability of finding two or more infected particles at time $t$? (repeat this for the SI case, with $\beta = 0$). Plot these expressions using Matlab etc.