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I'm searching for an at least semi-rigorous Statistical Mechanics description/treatment of a (spatially resolved) chemical reaction process of a macroscopic portion of at least two different species along the lines of $$A_1+A_2\longrightarrow A_3,$$ (with $A_3=A_1A_2$, necessarily) or $$A_3\longrightarrow A_1+A_2.$$ Might be a description here or a reference.

I'm aware of the reaction kinetics description of combustion processes, where the idea is to modify diffusion equations with a reaction rate term

$$''\frac{\partial}{\partial t}[A_i]=D\ \Delta [A_i]+\Sigma_j\ R_{ij}(T,[A_k]),''$$

where the last term in the end is coming from a Boltzmann'esque theory, but I'm interested in "more modern" approaches or a self contained treatment from first principles, which might, for example, model the whole thing in therms of ensembles, obviously as linkage between thermodynamics and non-equilibrium physics.

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2 Answers 2

up vote 3 down vote accepted

Chapter 16 of Classical and Quantum Mechanics via Lie algebras contains a section on deriving stirred chemical reaction dynamics on a statistical basis. (It is silent about the space distributed case.)

Section 14G of the first edition of the Statistical Physics book by L.E. Reichl treats chemical reaction dynamics distributed in space. (The section seems to have been dropped in the second and third edition.) Unlike my claim in the original answer, Section 14G does not contain a derivation from statistical mechanics, but only one based on the general features of thermodynamically consistent fluid flow equations. However, a derivation from a generalized Boltzmann equation can be found e.g., in http://www.cmap.polytechnique.fr/~graille/papers/article1.pdf.

Form a more fundamental point of view, a reactive chemical system is modelled microscopically by a system of atoms whose Hamiltonian is given by the ground state of the corresponding electronic system in the Born-Oppenheimer approximation. At low energies, the atoms group themselves into molecules (defined by local minimum wells) and the Hamiltonian can be simplified to a classical force field describing the reachable energy landscape.

The dynamics of the mixed state is given by the standard Liouville equation for such a system. Some simplifications arise by approximating the mixed state by a Gibbs state of the form $e^{-S/k}$, where $k$ is the Boltzmann constant and the entropy $S$ is taken to have the form of a space integral of intensive fields multiplied by asymptotic one-particle field operators associated with the bound states (=molecules) of the system. Of course, this approximation leads to some neglect of microscopic degrees of freedom, which results in dissipative terms, producing to gether with the streaming terms (from the conservation laws) the traditional equations for distributed chemical reactions.

The dynamics for stirred reactions is obtained by assuming the intensive fields to be independent of space (hence only varying in time).

All this requires to work with a finite volume and typically periodic boundary conditions, letting the volume go to infinity in a thermodynamic limit applied at the very end of the caluclations.

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This is going to become a book someday; so I'd appreciate getting a list of typos and other things that should be improved! –  Arnold Neumaier Mar 2 '12 at 20:52
Okay, thanks again. If I'll find the time I try to read it. (What I can say by skimming over it is that one page $114$ there is an $=$ missing. Interestingly enough, it's under Section title $4.11$.) –  NikolajK Mar 4 '12 at 0:57
For the reactive systems without well stirred condition, one must need a spatial resolved description. As we all know that traditional Liouville equation is used to characterize Hamiltonian systems(many body particles without reactive interaction). So one need an extended Liouville equation and I think the research on reactive fluids using a modified Liouville equation are certainly fascinating but very complex. To my knowledge, one need an additional term in the modified Liouville equation for spatial resolved reactive systems,but I could not find the related context(modified Liouville equati –  Yasmine Lee Oct 27 '12 at 8:57
@YasmineLee: I added some details on the microscopic model underlying chemical reactions. –  Arnold Neumaier Oct 28 '12 at 12:53
@YasmineLee: I added an online reference to a derivation from a Boltzmann-like equation. –  Arnold Neumaier Oct 30 '12 at 10:25

The microscopic model or Boltzmann equation need more knowledge of details than traditional chemical rate equations(population/concentration of species is the only state variable!).

However, if one only limits in developing a general thermodynamical framework for chemical reaction that accommodates for momentum density transfer, that means the state variables would be populations of species and momentum density(if we only consider isothermal case). I am not clear if anyone has developed such affinity-flux relation(in way of thermodynamical description!) for this case(rate equations combined with momentum effect!).

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In answer,don't write thanks or names of any user. –  Curious Nov 29 '12 at 7:51

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