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Blog - network theory (part 19) (Rev #2, changes)

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Network Theory Part 19

I think we need to keep telling the tale of the diatomic molecule in this episode, so I’ve moved some of it over here!

The master equation

The master equation has been reviewed previously in this series. The general form of the master equation is

ddtΨ=HΨ \frac{d}{d t} \Psi = H \Psi

and the general for of the Hamiltonian operators HH we are considering in the Petri net field theory series are given as

H= τTr(τ)(a n(τ)a m(τ)a m(τ)a m(τ)) H = \sum_{\tau\in T} r(\tau)(a^\dagger^{n(\tau)}a^{m(\tau)} - a^\dagger^{m(\tau)} a^{m(\tau)})

In our case, the Hamiltonian operator becomes

H=r 1(a 1 a 1 a 2a 2 a 2)+r 2(a 2 a 1a 1a 1 a 1 a 1a 1) H = r_1 (a_1^\dagger a_1^\dagger a_2 - a_2^\dagger a_2) + r_2 (a_2^\dagger a_1 a_1 - a_1^\dagger a_1^\dagger a_1 a_1)
=(a 2 a 1 a 1 )(a 1a 1r 2a 2r 1) = (a_2^\dagger -a_1^\dagger a_1^\dagger)(a_1 a_1 r_2 - a_2 r_1)

This equation is not an approximation, its exact. It governs all information we can know about the process, at the level of the individual species or molecules or whatever interacting.

Equilibrium states of the master equation

Here we will use the Anderson-Craciun-Kurtz theorem to work out the corresponding equilibrium states of the master equation. Brendon proved this in relation to what we consider here back in Part X.

Let

Ψ:=e z 1c 1+z 2c 2 \Psi := e^{z_1c_1 + z_2c_2}

Then

HΨ=[r 1(z 1 2c 2z 2c 2)+r 2(z 2c 2 2z 1 2c 1 2)]Ψ H\Psi = \left[ r_1(z_1^2c_2 - z_2c_2) + r_2(z_2c_2^2 - z_1^2c_1^2)\right]\Psi

and for Ψ0\Psi \neq 0

(r 1c 2r 2c 1 2)z 1 2+(r 2c 1 2r 1c 2)z 2=0 (r_1 c_2 - r_2 c_1^2) z_1^2 + (r_2 c_1^2 - r_1c_2)z_2 = 0

which vanishes for

r 1r 2=c 1 2c 2 \frac{r_1}{r_2} = \frac{c_1^2}{c_2}

Noether’s theorem

Now we will show how Noether’s theorem relates the conserved quantity 2N A+N B2N_A + N_B to a symmetry.

Conservation of particle number

The Hamiltonians that arise in Petri net field theory have a very particular general form. Not every Hamiltonian in this vast class preserves particle number (since we can have exponential growth or decay for instance). What we want to do is to find a good way to characterize those Hamiltonians that do preserve particle number. We want to understand symmetries in general. Those of you following the posts will recall the commutation relations from Part X. These are going to be relevant here too.

Just a reminder, the number operator for a single species is

N i=a i a i N_i = a_i^\dagger a_i

and the number operator for all the species is a sum over the single species

N= iN i N = \sum_i N_i

We will derive a few results at the end of the post. If you think we are telling the truth, you don’t need to check them, but they are there if you want to be bored with these sort of details. To get you into the mood…

It can be shown (using induction) that

[a,a k]=ka k1 [a, a^\dagger^k] = k a^\dagger^{k-1}
[a k,a ]=ka k1 [a^k,a^\dagger] = k a^{k-1}
  • exercise. Let [a,a ]=1[a, a^\dagger] = 1 be the base case and assume [a,a k]=ka k1[a, a^\dagger^k] = k a^\dagger^{k-1} and show that these assumptions imply the formula for k+1k+1 and hence or otherwise, prove the first commutation relation listed above by induction.

Now for some results.

  • (Lemma I). We arrive at the following commutation relations among number operators and general Hamiltonians in Petri net field theory
    (1)[N i,H]= τTr(τ)[n i(τ)m i(τ)]a n(τ)a m(τ) [N_i, H] = \sum_{\tau\in T} r(\tau)[n_i(\tau) - m_i(\tau)]a^\dagger^{n(\tau)}a^{m(\tau)}

This supporting lemma can be used to prove a range of things related to symmetries in the very type of Hamiltonians we are considering here.

  • (Theorem I — total particle conservation). The following quantity
    (2) τTr(τ)[n(τ)m(τ)]a n(τ)a m(τ)=0 \sum_{\tau\in T} r(\tau)[n(\tau) - m(\tau)]a^\dagger^{n(\tau)}a^{m(\tau)} = 0

    vanishes identically iff HH preserves particle number.

So to check if the total number of particles are conserved during evolution under some Hamiltonian, all one has to do is check Theorem I. The Hamiltonian we consider here does not conserve total particle number. However, the reversible reaction John did in Part 10 did.

  • (Exercise). In Network Theory Part 10 John considered the reversible reaction with Hamiltonian

    (3)H=(a b )(βbαa) H = (a^\dagger - b^\dagger)(\beta b - \alpha a)

    Use Theorem I to show that this reversible reaction conserves particle number.

  • (Theorem II — particle conservation symmetry). Given a Hamiltonian acting on kk particle species, there exists kk positive integer choices for ω n\omega_n which causes the following to vanish identically

    (4) i τTr(τ)ω i[n i(τ)m i(τ)]a n(τ)a m(τ)=0 \sum_i\sum_{\tau\in T} r(\tau)\omega_i[n_i(\tau) - m_i(\tau)]a^\dagger^{n(\tau)}a^{m(\tau)} = 0

    iff HH has a particle conservation symmetry.

As will soon be seen, this is precisely the case here. In other words, there exists ω 1\omega_1 and ω 2\omega_2 that take positive integer values which case the above quantity to vanish.

For our system to have a particle conservation symmetry, we must show that

(5)[N 1+2N 2,H]=[N 1,H]+2[N 2,H]=0 [N_1 + 2N_2, H] = [N_1, H] + 2[N_2,H] = 0

This vanishes since, from Lemma I, we calculate that

(6)[N 1,H]=2r 1a a b2r 2b aa [N_1, H] = 2r_1 a^\dagger a^\dagger b -2 r_2 b^\dagger a a
(7)[N 2,H]=r 1a a b+r 2b aa [N_2, H] = -r_1 a^\dagger a^\dagger b + r_2 b^\dagger a a

Here, Theorem II applies, are we are able to find two values, ω 1=1\omega_1=1 and ω 2=2\omega_2=2 that cause the commutator to vanish. The Hamiltonian therefore has a particle number conservation symmetry.

Master equation

Now for the master equation approach. Our Hamiltonian is given as

H=r 1(a a bb b)+r 2(b aaa a aa) H = r_1 (a^\dagger a^\dagger b - b^\dagger b) + r_2 (b^\dagger a a - a^\dagger a^\dagger a a)

and the evolution operator at time tt is given as

(8)V=e tH= k=0 H kk!t k V = e^{t H} = \sum_{k=0}^\infinity \frac{H^k}{k!}t^k

At each order in kk , we have a term that corresponds to the Hamiltonian HH acting kk times. One can think of these as alternative histories. In quantum mechanics, there is a spooky thing called coherence, where each of these histories seems to occur concurrently. In stochastic mechanics, each history only occurs with some probability. In terms of mathematical structure, the theories become closely related. While the semantic interpretation might differ, the syntactical form, given by a sum over histories unites quantum and stochastic mechanics. This enables us to e.g. apply tools from quantum mechanics to stochastic mechanics such as Feynman diagrams.

Supporting Material

Proof of Theorems I and II. Sometimes a long calculation can simplify matters. This is the case here, though we don’t want to muddy the water from what’s going on here, as this is just some algebra.

Here we will use the following notation.

a m(τ)a i =a i m i(τ)a i a m(τ) a^{m(\tau)}a^\dagger_i = a^{m_i(\tau)}_i a^\dagger_i a^{m'(\tau)}
a ia n(τ)=a n(τ)a ia i n i(τ) a_i a^\dagger^{n(\tau)} = a^\dagger^{n'(\tau)}a_i a_i^\dagger^{n_i(\tau)}

where the vector m(τ)m'(\tau) has its iith component zero which is why we are able to move term(s) a m(τ)a^{m'(\tau)} to the right. This enables us to express the more general commutation relations,

[a m(τ),a i ]=m i(τ)a i m i(τ)1a m(τ) [a^{m(\tau)},a^\dagger_i] = m_i(\tau)a_i^{m_i(\tau)-1}a^{m'(\tau)}
[a i,a n(τ)]=n i(τ)(a i ) n i(τ)1a n(τ) [a_i, a^\dagger^{n(\tau)}] = n_i(\tau) (a_i^\dagger)^{n_i(\tau)-1}a^\dagger^{n'(\tau)}

Using these relations, it follows that (is better notation possible?)

[H,a i ]= τTr(τ)(a n(τ)a m(τ))a i m i(τ)1a m(τ)m i(τ) [H, a_i^\dagger] = \sum_{\tau\in T} r(\tau)(a^\dagger^{n(\tau)} - a^\dagger^{m(\tau)})a_i^{m_i(\tau)-1} a^{m'(\tau)} m_i(\tau)

and also

[a i,H]= τTr(τ)[n i(τ)a n i(τ)1a n(τ)m i(τ)a m i(τ)1a m(τ)]a m(τ) [a_i, H] = \sum_{\tau\in T} r(\tau)[n_i(\tau)a^\dagger^{n_i(\tau)-1}a^\dagger^{n'(\tau)} - m_i(\tau)a^\dagger^{m_i(\tau)-1}a^\dagger^{m'(\tau)}]a^{m(\tau)}

These will simplify the calculation of the commutation of the Hamiltonian and the number operator.

[ ia i a i,H]= i[a i a i,H]= i(a i [a i,H][H,a i ]a i) [\sum_i a_i^\dagger a_i, H] = \sum_i [a_i^\dagger a_i, H] = \sum_i(a_i^\dagger [a_i, H] - [H, a_i^\dagger]a_i)

For this to vanish, the following quantity must vanish identically.

i(a i K 2K 1a i) \sum_i (a_i^\dagger K_2 - K_1 a_i)

where

a i K 2=a i [a i,H]= i τTr(τ)[n i(τ)a n(τ)m i(τ)a m(τ)]a m(τ) a_i^\dagger K_2 = a_i^\dagger[a_i, H] = \sum_i \sum_{\tau\in T} r(\tau)[n_i(\tau)a^\dagger^{n(\tau)} - m_i(\tau)a^\dagger^{m(\tau)}]a^{m(\tau)}
K 1a i=[H,a i ]a i= i τTr(τ)(a n(τ)a m(τ))a m(τ)m i(τ) K_1 a_i = [H, a_i^\dagger]a_i = \sum_i \sum_{\tau\in T} r(\tau)(a^\dagger^{n(\tau)} - a^\dagger^{m(\tau)})a^{m(\tau)}m_i(\tau)

Both of the theorems then follow from applications of the above.

Solution to exercises

Particle Conservation of the simple reversible reaction. In Network Theory Part 10 John considered the reversible reaction with Hamiltonian

H=(a b )(βbαa) H = (a^\dagger - b^\dagger)(\beta b - \alpha a)

We find its commutation relations with the creation (destruction) operators of both species to be

[H,a ]=α(b a ) [H, a^\dagger] = \alpha (b^\dagger - a^\dagger)
[H,a]=αaβb [H, a] = \alpha a - \beta b
[b ,H]=β(b a ) [b^\dagger, H] = \beta (b^\dagger - a^\dagger )
[b,H]=αaβb [b, H] = \alpha a - \beta b

Now we see that the particle number is conserved for this Hamiltonian by calculating

[N,H]=[a a,H]+[b b,H]=a [a,H]+[a ,H]a+b [b,H]+[b ,H]b=0 [N, H] = [a^\dagger a, H] + [b^\dagger b, H] = a^\dagger[a, H] + [a^\dagger, H]a + b^\dagger[b, H] + [b^\dagger, H]b = 0

This could have also been shown, using Theorem I directly.

New stuff

We’ve been working hard to understand the parallels and differences between quantum and stochastic mechanics. Last time we showed how the methods developed in prior posts can be used to model chemical reaction networks. This time, we are going to report the details of a battle.

This is the same quantum vs. stochastic battle we’ve talked about in prior posts, but this time we are going talk in detail about the odd nature of eigenstates in quantum mechanics, and how we can’t expect this structure in stochastic mechanics.