I have about the construction of master equation for chemical reaction i.e. I have to construct differential equations for the probability mass function for the number of particles A, B and C.
When I have this following reaction $$\emptyset\rightarrow^{k_A}{A}$$ $$\emptyset\rightarrow^{k_B}{B}$$ $$A\rightarrow^{k_A}{B}$$ $$B\rightarrow^{\gamma}{\emptyset}$$ I have no problem to construct the master equation because if i consider P(n,m,t), P(n,t), P(m,t) where n is the number of particles A and m the number of particles B at time t I should have the following equations with respect to each reaction (if I'm right): $$ P'(n,t)=P(n-1,t)k_{A}-P(n,t)k_A $$ $$ P'(m,t)=P(m-1,t)k_{B}-P(m,t)k_B $$ $$ P'(m,t)=P(m-1,t)nk_{A}-P(m,t)nk_A $$ $$ P'(m,t)=P(m+1,t)(m+1)\gamma-P(m,t)\gamma $$ But for example, when I have to consider more complicated reaction, in which way should I have to procede? For example, for the following reaction $$ A+B\rightarrow^{\gamma}\emptyset $$ is the following reaction correct? $$ P'(n,m,t)=P(n+1,m+1,t)(n+1)(m+1)\gamma-P(n,m,t)nm\gamma $$ And for other kind of reaction like these: $$ A+B\rightarrow^{\gamma_B}A $$ $$ A+A\rightarrow^{\gamma_{AA}}\emptyset $$ $$ A+B\rightarrow^{k_C}C $$ In which way should I procede? I will be very happy if someone could help me! Thank you very much in advance!