# Chapter 13: A reaction network approach to the theory of acoustic wave turbulence

This post is based on my paper

Let us consider the acoustic wave turbulence kinetic equation $\partial_tf \ = \ Q[f]$

where $Q[f]$ is the collision term, describing pure resonant three-wave interactions. The equation is a three-wave kinetic one, in which the collision operator is of the form $Q[f](k) \ = \ \iint_{\mathbb{R}^{2d}} \Big[ R_{k,k_1,k_2}[f] - R_{k_1,k,k_2}[f] - R_{k_2,k,k_1}[f] \Big] dk_1dk_2$

with $R_{k,k_1,k_2} [f]:= 4\pi |V_{k,k_1,k_2}|^2\delta(k-k_1-k_2)\delta(|k |-|k_1|-|k_2|)(f_1f_2-ff_1-ff_2).$

We have used the short-hand notation $f = f(t,k)$ and $f_j = f(t,k_j)$.

In this paper,  we discover, for the first time, the connection between the wave kinetic equation and chemical reaction networks. We prove that the discrete version of the wave kinetic equation can be associated with a chemical reaction network which takes the form $A_{k_2} + A_{k_3} -> A_{k_1},$ $A_{k_2} + A_{k_1} -> 2A_{k_2} + A_{k_3}.$

As a consequence, techniques that have been used to study the Global Attractor Conjecture in chemical reaction network theory can be applied to study the long time behavior of the wave kinetic equation  We prove that as time evolves, the solution of the discrete version of the wave kinetic equation converges to a steady state exponentially in time.