# Gillespie's stochastic framework valid for particles in aqueous solution?

Gillespie proposed a stochastic framework for simulating chemical reactions which is predicated on non-reactive elastic collisions serving to 'uniformize' particle position so that the assumption of well-mixedness is always satisfied (see page 409 in the linked version). This is formulated from kinetic theory.

A corollary to this is that a non-reactive collision between two molecules that are able to react does not induce local correlation, i.e., two particles able to react with each other that just collided, but didn't react are no more likely to react with each other in the next dt than any other particle pair in the volume.

Gillespie's algorithm is commonly used in biology where biochemical species are modeled in the aqueous environment of cells. Is this valid, and if so why? It seems the validity may depend on an assumption of Boltzmann-distributed velocities which may or may not be valid in aqueous phase. I recently asked about this (question here), however, the question was deemed a duplicate even though there was some disagreement among the answers.

-
I think that it is wrong, even the Boltzmann Stosszahlansatz only assumes no correlation before collision. Now as to the magnitude of this error, it is probably somewhat irrelevant in biochemistry, and is likely captured well by the effective reaction constants. I believe the lack of correlation is a much smaller sin than the lack of spatial heterogeneity (addressed by many methods, perhaps most notably by Green's Function Reaction Dynamics, gfrd.org), although one might argue that these are connected. – alarge May 22 '14 at 6:57
@amlrg I think they're definitely connected concepts since spatial heterogeneity begets spatial correlation. Can you weigh in on the validity of Boltzmann velocity distributions in liquid? That, to me, seems as or more important to the validity of Gillespie in modeling aqueous systems. – vector07 May 23 '14 at 18:05
From Liouville equation you get that the equilibrium phase space distribution Poisson bracketed with the Hamiltonian should vanish. Through repeated integration, then, you should be able to get the singlet reduced phase space distribution. I think this might in general be Maxwell-Boltzmann, but I'll get back to you if I get around to verifying this. – alarge May 25 '14 at 9:50
$\{\mathcal{H}, f^{[N]}\} = 0 \Rightarrow f^{[N]} = \frac{\mathcal{F}(\mathcal{H})}{\mathcal{Z}}$, where $\mathcal{Z} = \int \mathrm{d}x^N \mathcal{F}(\mathcal{H}(x^N))$ is the most general statement. Now $\mathcal{F}$ is going to depend on the ensemble, in the canonical one you have $\mathcal{F}(\mathcal{H}) = \frac{1}{N!h^{3N}}\frac{\exp(-\beta \mathcal{H})}{\mathcal{Z}}$, and so for the singlet $f^{(1)}(x) = N\int \mathrm{d}x^{(N-1)} f^{[N]}(x^N)$, with one more integration over the position coordinate yielding Maxwell--Boltzmann for basically anything. I think. – alarge May 26 '14 at 3:40