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# Experiments in 1-parameter family of equilibrium states

Created a page which illustrates our methods applied to a 1-parameter family of equilibrium states. In this example we illustrate that if the chemical rate equation vanishes, than so does the master equation. We arrive at this as a direct consequence of the former.

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edited September 2011

Nice!

We can why there's a 1-parameter family of solutions of the master equation in this case using Noether's theorem. We've got a reversible reaction

$$A + 2 B \leftrightarrow 3 A$$ and if we write

$$A + 2 B = 3 A$$ we can solve and get

$$B = A$$ so the overall effect of this reaction is to turn an $A$ into a $B$. This means that the number of $A$'s, plus the number of $B$'s, is conserved. So, we get a conserved quantity

$$O = N_A + N_B = a_A^\dagger a_A + a_B^\dagger a_B$$ and we therefore can apply any function of $O$ to an equilibrium solution $\Psi$ of the master equation and get another equilibrium solution:

$$H \Psi = 0 \implies H (f(O) \Psi) = 0$$ since Noether's theorem says

$$[H , O] = 0$$ so

$$[H, f(O)] = 0$$ so

$$H f(O) \Psi = f(O) H \Psi = 0$$ This kind of argument is fairly general. We just need to find a conserved quantity!

Comment Source:Nice! <img src = "http://math.ucr.edu/home/baez/emoticons/thumbsup.gif" alt = ""/> I've been a bit absorbed in blogging about this stuff and am just catching up with your work. We can why there's a 1-parameter family of solutions of the master equation in this case using Noether's theorem. We've got a reversible reaction $$A + 2 B \leftrightarrow 3 A$$ and if we write $$A + 2 B = 3 A$$ we can solve and get $$B = A$$ so the <i>overall effect</i> of this reaction is to turn an $A$ into a $B$. This means that the number of $A$'s, plus the number of $B$'s, is conserved. So, we get a conserved quantity $$O = N_A + N_B = a_A^\dagger a_A + a_B^\dagger a_B$$ and we therefore can apply any function of $O$ to an equilibrium solution $\Psi$ of the master equation and get another equilibrium solution: $$H \Psi = 0 \implies H (f(O) \Psi) = 0$$ since Noether's theorem says $$[H , O] = 0$$ so $$[H, f(O)] = 0$$ so $$H f(O) \Psi = f(O) H \Psi = 0$$ This kind of argument is fairly general. We just need to find a conserved quantity!