# Chapter 5: Chemical reaction network and quantum kinetic theories. The key of a successful marriage.

This post is based on my joint work with Gheorghe Craciun.

In this chapter, I will try to answer the following conjecture: ”What is the key for the marriage of chemical reaction network and quantum kinetic theories to be successful?”

The behavior of reactor vessels used in chemical engineering are often based on systems of nonlinear ordinary differential equations (ODEs). Chemical reaction network theory (CRNT) has been developed over the last 45 years, to study such nonlinear ODEs, initially by the work of Horn, Jackson and Feinberg. The Global Attractor Conjecture (GAC), dated back to 1972, says that complex balanced mass-action network must have a global attractor within any compatibility class. Recently, the GAC has been solved by Gheorghe Craciun and a workshop has been organized to discuss the proof: ”The GAC is one of the oldest and best studied problems within CRNT. Since its formulation in the early 1970s, the conjecture has resulted in a flurry of research activity, dozens of papers, and a litany of false proofs. It is the intention of the workshop to go through the manuscript in detail, and see if this giant of CRNT has truly been slain.” Let me also quote the SIAM News paper: ”So, while Craciun’s proof may bookend decades of work on global dynamics in CRNT, the ideas of his proof will serve as the opening pages to new volumes of exploration.”

Now, let us consider the $C_{12}$ operator of Chapter 1, whose discrete form can be written as:

$\dot{F}_{k} =\sum_{k=k'+k''}\mathcal{K}^{12}_{k,k',k''}[F_{k'}F_{k''}-F_{k}- F_{k}F_{k'}-F_{k}F_{k''}]$

$+ 2\sum_{k+k'=k''}\mathcal{K}^{12}_{k,k',k''}[F_{k''}+F_{k}F_{k''}+F_{k'}F_{k''}-F_{k}F_{k'}] (1).$

To make the explication easier to understand, let us simplify (1) as follows:

$\dot{F}_{3} =F_{1}F_{2}-F_{3}- F_{3}F_{2}-F_{3}F_{1} (2).$

We will derive Equation (2) from the following chemical reaction network for species denoted $X_1, X_2, X_3$:

$X_2 + X_1 \leftrightarrow X_3 (3)$

$X_2 + X_3 \rightarrow 2X_2 + X_1 (4)$

$X_1 + X_3\rightarrow X_2 + 2X_2 (5).$

If we denote by $F_k$ the density function of the species $X_k$, $k=1,2,3$, we will show that, the differential equations satisfied by $F_3$ according the mass-action kinetics are the same as (2).

Consider the first reaction (3): In this reaction, $X_3$ is created from $X_1+X_2$ with the rate $F_1F_2$ and $X_3$ is decomposed into $X_1+X_2$ with the rate $-F_3$. The rate of change of the species $X_3$ is $F_1F_2 -F_3$.

For the second reaction (4): $X_3$ is lost with the rate $-F_2F_3$ to create $2X_2+X_1$. Therefore the rate of change of the species $X_3$ is $-F_2F_3$. By exchange the role of $X_2$ and $X_3$ in (4), we obtain the rate $-F_1F_3$ for (5).

As a result, the rate of change of $X_3$ in (3)-(5) is

$\dot{F}_{3} =F_{1}F_{2}-F_{3}- F_{3}F_{2}-F_{3}F_{1}.$

Based on the above observation, in this joint work with Gheorghe Craciun (Madison), we prove that discrete differential equations of quantum Boltzmann models are analogous to equations derived from chemical reaction networks, and the question about the convergence to equilibrium of quantum kinetic equations is analogous to the GAC for CRNT. Notice that chemical reaction networks associated to quantum kinetic equations are not complex balanced systems, as one can see in (3)-(5). We then employ a toric dynamical system approach, to study the convergence to equilibrium of several types of quantum kinetic equations. Our convergence results also apply to the quantum Boltzmann equation for phonon interactions in anharmonic crystals.

Our final conclusion for the success of the marriage of chemical reaction network and quantum kinetic theories is that they both share the same ODEs representation.

In the next chapter, we will discuss about derivations of quantum kinetic equations.

References for the picture:

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