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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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  • 1.
    edited September 2011

    Nice!

    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 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!
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