Chapter 4: A result on the convergence to equilibrium and some meditation

This post is based on my paper.

I googled ”relaxation to equilibrium” and found these pictures:

What is the meaning of “relaxation to equilibrium” in meditation? By meditation, we recreate an inner space that allows us to relax our mind from the external circumstances. That helps us develop the mental equilibrium.

But, are we discussing about meditation in this chapter? No.

However, similar to human beings, excited atoms also relax to equilibrium as time evolves i.e. the excited atoms density function   f(t,p)  converges to an equilibrium distribution   n_0(p) as   t tends to infinity. Let us recall the equation of   f(t,p) from Chapter 1:

\frac{df}{dt}=C_{12}[f], f(0,\cdot)=f_0, (1)

where the interaction operator is defined as

C_{12}[f]:=\int _{ \mathbb{R}^3}\int _{ \mathbb{R}^3}dp_1dp_2 [R(p, p_1, p_2)-R(p_1, p, p_2)-R(p_2, p_1, p) ],

R(p, p_1, p_2):= |\mathcal{M}(p, p_1, p_2)|^2 [\delta (\mathcal{E}_p-\mathcal{E}_{p_1}-\mathcal{E}_{p_2}) \delta (p-p_1-p_2)][ f(p_1)f(p_2)(1+f(p))-(1+f(p_1)(1+f(p_2))f(p)].

By the H-Theorem, Equation (1) has a family of equilibria:     n_0(p )=\frac {1} {e^{\frac{\mathcal{E}_p}{k_B T}}-1}, \beta >0, where  k_B is the Boltzmann’s constant and  T the temperature of the quasiparticles whose distribution is   n_0.   \mathcal{E}_p is approximated by the same form with Chapter 1.

Chapter 2 Pic 2

Figure: The Bose-Einstein Condensate (BEC) and the excited atoms.

In this work, we only study, instead, the relaxation process of the equation linearised around one equilibrium. Let us then write:  f(t, p)=n_0(p)+n_0(p)[1+n_0(p)]\Omega (t, p). Plugging this expression in the equation and keeping only the linear terms in  \Omega we obtain an equation of the form:

    n_0(p)[1+n_0(p )]\frac {\partial \Omega } {\partial t}(t, p)=\mathcal{L}(\Omega )(t, p)

  \mathcal{L}(\Omega )(t, p)=-M(p)\Omega (t, p)+\mathcal{T} (\Omega )(t, p)\mathcal{T} (\Omega )(t, p)=\int _{ \mathbb{R}^3 }\mathcal{U}(p, p') \Omega (t, p') dp',

then, we can prove that  ||\Omega (t)-\Theta ||_{L^2(\mathbb{R}^3, \frac {dp } {\mathrm{sinh}^2 |p|})} \le \frac{C}{(1+t)^{1/2 }}||\Omega _0-\Theta||_{L^2(\mathbb{R}^3, \frac {dp } {\mathrm{sinh}^{2} |p|} ) }, in which  \mathcal{L}(\Theta)(p)=0.

In the next Chapter, we will discuss the analog between chemical reaction networks and quantum kinetic equations.


References for the pictures:



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