Chemical networks on a lattice and Boltzman distributions

Discussion in 'Chemistry' started by AlphaNumeric, Feb 3, 2012.

  1. AlphaNumeric Fully ionized Registered Senior Member

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    Chemical thermodynamics is not my strong point but hopefully someone else here knows about it.

    Suppose I have a set of chemical reactions with the various rates etc. The chemical network is zero deficiency and weakly reversible. I model them using mass action methods and the master equation and find the configurations which are stationary, which by the Zero Deficiency Theorem amounts to having a Poisson distribution for the number of the different particles. Suppose it's just one particle type doing say \(A \leftrightarrow 2B\). This ignores spatial position so suppose now I put one of these systems at each site in a large lattice, with L lattice sites. Obviously, even if you have diffusion between lattice sites, you get a stationary configuration for the system when ALL lattice sites have the same Poisson distribution of particles. So if the expected number of A at each site is \(\bar{a}\) then the system as a whole has \(L\bar{a}\) particles.

    Another way of looking at this is to start with \(L\bar{a}\) particles and assign each one with equal probability to particular lattice site. The expected number of particles at each lattice site is then \(\bar{a}\) and the distribution follows the same Poisson distribution.

    This second approach sounds a lot more like the Boltzman method to construct things like entropy of the system, because you're putting particles in boxes and looking at microstates. However, this method doesn't seem to have anything to do with the Boltzmann distribution. Suppose I make life easy for myself and say that the particles themselves have zero energy (ie ignore kinetic stuff), rather than the different \(E_{i}\) you see in partition functions etc. By chemical reaction methods the particles have chemical potential, which does appear in the Gibbs measure/partition function in the form of \(\beta \mu N\), but I don't see how to make any connection between these chemical network methods and the standard partition function approach. Chemical networks I have little familiarity with but obviously partition functions are more familiar to physicists.

    Can someone explain to me where, if at all, the Boltzman distribution comes into all of this? Does it play any role in chemical networks? They all talk about similar quantities and there's exponentials here and there but I can't quite get it. If this is a standard area (I would imagine so?) I'd greatly appreciate a textbook/paper/lecture note recommendation.
     
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  3. arfa brane call me arf Valued Senior Member

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    My suggestion is google "lattice boltzmann method" or "boltzmann algorithm". I've been looking at the connection between thermodynamics and algorithms, found a paper by Baez about algorithmic thermodynamics in which the number of particles is made conjugate to chemical potential divided by temperature, which might have some bearing on what you're looking for. But Baez' paper doesn't talk about lattices.
     
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  5. AlphaNumeric Fully ionized Registered Senior Member

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    I've read a lot about Baez's stuff, let's just say he leaves out a lot of critical information if you want to actually do anything more than the most trivial of trivial examples. I was somewhat pleased with myself that I deduced particular things myself which then turned out to be important things in the literature

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    Boltzmann lattice methods are a little different, I've seen them before in fluid mechanics simulation stuff.

    Having been staring blankly into space about this for the last week or so thinking about it I've reached the conclusion it's actually two different things. The distribution of particle number is not the same, in any limit or approximation, as the Boltzmann distribution because they are describing two different things. The confusion comes from the fact it's two close things for a single system type but there's no "Set \(E = \beta S\)* and take the limit of \(\langle N \rangle \to \infty\) and you get the Boltzmann distribution!" thing I was initially expecting there to be.

    * Warning, this equation makes no sense.
     
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  7. arfa brane call me arf Valued Senior Member

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    Hmm. The only "chemical networks" I know anything about are metabolic pathways.

    Since these are highly organised, I wouldn't expect Boltzmann distributions to have a whole lot of relevance. TBH I'm not that familiar with general methods or the master equation. I look at chemical and biochemical processes algorithmically, pretty much.

    Not sure if you will find it relevant, but I located a paper on arxiv that addresses convergence in networks using what looks like fixed point theory. Absolutely bristling with equations, too. http://arxiv.org/pdf/1112.3798.pdf
     
    Last edited: Feb 4, 2012
  8. AlphaNumeric Fully ionized Registered Senior Member

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    Got another vaguely related question (ie it's about chemistry and I don't feel another thread is necessary).

    According to here if you consider a single reversible reaction you can do lots of nice things like relate combinations of chemical potentials, \(\sum_{i}\nu_{i}\mu_{i}^{(0)}\) to ratios of activities at equilibrium, \(K_{eq} = \prod_{i}(a_{i})^{\nu_{i}}\), which in turn is expressible in terms of the rate coefficients \(K_{eq} = \frac{k_{-}}{k_{+}}\).

    How does this generalise if you're considering a system with multiple reactions, say \(A \leftrightarrow B \leftrightarrow C \leftrightarrow A\)? Would you just consider each reaction in the same way and everything should be consistent at the end or is it more complicated? For example, it all seems okay if you're doing a reversible reaction but what if each step wasn't reversible? What if you had a system which did \(A \to B \to C \to A\)? It could have a dynamic equilibrium but not in the same manner, as the \(K_{eq}\) values become infinite since \(k_{-} = 0\) for each reaction.

    It seems okay if the only way you can get an equilibrium is to reverse a reaction but if you can go through lengthy sequences to return to your initial configuration then I get the feeling things get very ugly very quickly.

    Another impression I get is that there's some way to work out the values \(\mu_{i}^{(0)}\) if you have lots of equilibria. Suppose you have a bunch of substances mixed together and reacting in some network. It reaches equilibria and you know the \(\{A\}\) etc values and you know the \(k_{\pm}\) rates (perhaps some zero) for all the individual reactions. Can you then compute the \(\mu_{i}^{(0)}\)? A single reversible reaction constraints their values via \(\Delta G = 0\) implying \(\sum_{i} \nu_{i}\mu_{i}^{(0)} = -RT \ln K_{eq}\), with \(K_{eq}\) expressible in terms of rate coefficients or activities. Does having more reactions (sufficiently many to uniquely define the potentials) allow you to determine the chemical potentials? Or is it done by some other method?
     

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