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Quantum techniques for stochastic mechanics (course) quiz

HWK Assignment

Conductance networks and Dirichlet operators

Prove that if H:R nR n H:R^n\rightarrow R^n is any Dirichlet operator, and ψR n\psi\in R^n is any vector, then

ψ|Hψ=12 ijH ij(ψ iψ j) 2 \langle\psi|H\psi\rangle =-\frac{1}{2}\sum_{i\neq j}H_{ij}(\psi_i - \psi_j)^2

Equations of motion for stochastic processes

Prove that if HH is an infinitesimal stochastic operator and OO is an observable, then if ψ(t)\psi(t) obeys the master equation, then

ddtOψ(t)=[O,H]ψ(t) \frac{d}{dt}\int O \psi(t) = \int [O, H] \psi(t)

Noether’s theorem for Markov chains

Suppose UU is a stochastic operator and OO is an observable. Show that OO commutes with UU iff the expected values of OO and its square don’t change when we evolve our state one time step using UU. Hence or otherwise, show that

[O,U]=0 [O, U]=0

if and only if

OUψ(t)=Oψ(t) \int O U \psi(t) = \int O \psi(t)

and

O 2Uψ(t)=O 2ψ \int O^2 U \psi(t) = \int O^2 \psi

for all stochastic states ψ(t)\psi(t).

Irreducibility and Perron’s theorem

Let TT be a nonnegative n×nn\times n matrix. Show that TT is irreducible if and only if for all i,j0i,j\geq 0, (T m) ij(T^m)_{ij} greater than 00 for some natural number mm.

Irreducibility and Perron–Frobenius theory

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 HH is irreducible if and only if the nonnegative operator H+cIH+cI (where cc is any sufficiently large constant) is irreducible.

Conserved quantities and irreducibility in stochastic process

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 OO in stochastic mechanics assigns a number O iO_i to each configuration i=1...ni=1...n. We can make a diagonal matrix with O iO_i as its diagonal entries, and by abuse of language we call this OO as well. Then we say OO is a conserved quantity for the Hamiltonian HH if the commutator [O,H]=OHHO[O,H]=OH-HO vanishes.

  • Let HH be a Dirichlet operator. Show that HH is irreducible if and only if every conserved quantity OO for HH is a constant, meaning that for some cRc\in R we have O i=cO_i=c for all ii. (Hint: examine your proof of Noether’s theorem.)

In fact this works more generally.

  • Let HH be an infinitesimal stochastic matrix. Show that H+cIH+cI is an irreducible nonnegative matrix for all sufficiently large cc if and only if every conserved quantity OO for HH is a constant.

Modeling infections as spin particles on a lattice

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:=|0|I\rangle:=|0\rangle, |S:=|1|S\rangle:=|1\rangle and |R:=|2|R\rangle:=|2\rangle to denote a person as being either infected, susceptible or recovered. We will consider the following transitions

SI,ISII SI, IS \rightarrow II

with rate α\alpha and

IR I \rightarrow R

with rate β\beta

  • Write down the pseudo Hamiltonian modeling the process.

  • What are the conserved quantities for β=0\beta = 0 (SI) and positive α\alpha?

  • What are the conserved quantities for β\beta, α\alpha positive (SIR)?

  • Consider the initial state |S|S|S|S\rangle|S\rangle|S\rangle. With α=β=1\alpha =\beta =1 what is the probability of finding two or more infected particles at time tt? (repeat this for the SI case, with β=0\beta = 0). Plot these expressions using Matlab etc.

  • Link to course homepage