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The paper "Perturbative approach to an $A + B \rightarrow C$ reaction-diffusion system", (Z. Phys. B 96, 137-144 (1994)), by Conrad and Trimper, applies the Fock Space formalism for the master equation developed by Doi and applies it to a $A+B \rightarrow C$ system.

The probability distribution $P (n,t)$ can be related to a state vector $|F(t)\rangle$ in a Fock space according to $P (n,t) = \langle n|F(t)\rangle$ with the basis vectors $|n \rangle = | {n_1 n_2 \dots n_i \dots}\rangle $. As a consequence, the master equation is is re-written as an equation in Fock space \begin{equation} \partial_t |F(t)\rangle = \hat{L} |F(t)\rangle \end{equation}.

Some other useful relations,

$$ |F(t)\rangle = \sum_{n_i} P(n,t) |n\rangle$$

$$\langle \hat{O} (t) \rangle = \langle s|O|F(t)\rangle$$

$$ \partial_t\langle \hat{O}(t) \rangle = \langle s|[\hat{O},\hat{L}]|F(t)\rangle $$

$$\langle s| = \sum\limits_{n_i} \langle n|$$

$$\langle s|F\rangle = 1 $$

$$\langle s|\hat{L} = 0$$

Now as far as I understand in the paper linked above they define bose annihilation operators $a_i$, $b_i$, $c_i$ at each lattice site, and similarly introduce bose creation operators $a^\dagger_i$, $b^\dagger_i$, $c^\dagger_i$. And number operators $A_i = a^\dagger_i a_i$ etc.

The reaction Liouvillean also makes sense to me derived as $$L_R = \eta \sum_i c^\dagger_ia_ib_i - a^\dagger_ia_ib^\dagger_i b_i$$

What I am having a hard time deriving is the time derivative of the expectation values, although the quoted result makes intuitive sense to me.

$$\partial_t \langle C_i \rangle = \eta \langle A_i B_i\rangle $$

The issue I am having is the time derivative given above should be the expectation value of the commutator, $[c^\dagger_i c_i,c^\dagger_ia_ib_i - a^\dagger_ia_ib^\dagger_i b_i] = c^\dagger_i a_i b_i \neq a^\dagger_ia_ib^\dagger_ib_i$ according to my computations. Any help would be appreciated

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  • $\begingroup$ This formalism is called Doi-Peliti approach. The definition of creation/annihilation operators is different in this method, this might be the reason for discrepancy (for calculating species c, you might have to use operator c). $\endgroup$ – Sunyam Jun 6 at 21:31
  • $\begingroup$ Hi!@Sunyam the authors seem to have defined $C_i = c^\dagger_i c_i$ in the standard way (text above equation 8 in the linked paper). $\endgroup$ – jcp Jun 6 at 21:35
  • $\begingroup$ Is $\hat L = L_{R} + L_{\text{Diffusion}}$. If so is $\langle s|L_{\text{Diffusion}}=0$, then $\langle s|L_{R}^{}=0 \Rightarrow \langle s| \eta \sum_i c_i^\dagger a_i b_i = \langle s| \eta \sum_i a_i^\dagger a_i b_i^\dagger b_i $? $\endgroup$ – Sunyam Jun 6 at 22:09
  • $\begingroup$ Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$ – Qmechanic Jun 6 at 22:54
  • $\begingroup$ @Sunyam. Yes that is true for $\hat{L}$. I am not sure about the other relations. I have only just begun studying this formalism. But I would assume in a process without diffusion, these relations should still hold? (I am not sure). $\endgroup$ – jcp Jun 6 at 23:35

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