In this post, we will see how a Lyapunov functional will help us to prove the explicit convergence to equilibrium for some chemical reactions.
We consider a single reversible reaction
with the forward and backward reaction rates are assumed to be . Denote by the concentrations of at time respectively. Using the law of mass action, we arrive at the following nonlinear system of ODEs:
We note that the system (1) possesses the mass conservation
where we call the initial mass.
The system (1) has a unique equilibrium which balances the reaction and satisfies the mass conservation:
We remark that, without satisfying the mass conservation, we would have many infinite stationary states to the system (1).
The main question is: do the concentrations tend to the equilibrium as .
Theorem. There exists an explicit constant depending on the initial mass such that,
The Theorem tells us that, we are not only able to show the convergence to equilibrium but also able to compute explicitly the rate of convergence. The idea of the proof is to use the free energy functional (or Boltzmann-type entropy functional).
Multiplying the equation of with and the equation of with we have
Summing up the two equations above we have
with and .
Remark. The functional is called free energy or entropy or Boltzmann-type entropy. In the latter case, we call entropy dissipation.
We note that iff , which combining with the mass conservation implies that,
This gives us a hope that the convergence actually happens!
To do that, we need two useful estimates relating to the relative entropy:
(i) ; and
Hints: The proof of (i) is easy with direct computations by using the mass conservation. The proof of (ii) is a little more tricky and is left as an exercise (for you, the reader, to check!)
The property (ii) tells us that, if we can prove the convergence of the entropy to as , the we will get the convergence of solutions.
To do that, we first observe
If we can prove that
then we have
which, by the help of Gronwall’s lemma, gives us exponentially as with the rate .
Therefore, our aim now is to prove the inequality (*). This inequality is called entropy-entropy dissipation estimate.
In order to do this, we use the idea of Bakry-Emery criterion: to investigate the second derivative of the relative entropy or equivalently the first derivative of the entropy dissipation .
By direct computations, we have
thanks to the positivity of solution (why do we have this? It would be nice if you could get the answer yourself). We thus get
Now, we integrate
from to and use and (one more why!!! Hints: is a Lyapunov functional), we have
We have proved (*) and thus complete the proof of Theorem.
(i) One could think of extending the method to the case of spatially imhomogeneous case with diffusion, that is the concentration does not only depend on time but also on spatial varibles. The system hence writes as
with homogeneous Neumann boundary condition. However, due to the appearance of spatial variables, the Bakry-Emery method leads to “nasty terms” which are (almost) impossible to control. There is a way to prove a corresponding version of (*), it was first given in a work of Desvillettes and Fellner.
(ii) By patient computations, one easily sees that the method would work for a more general reaction, which writes as
for any number and of substances.