# Chapter 0. Introduction

There have been very few rigorous mathematical results on Boltzmann equations arising in wave turbulence and quantum kinetic theories, despite their central roles in several aspects of sciences and engineering, due to their complexities in comparison with the classical Boltzmann equations. One of my research interests is to understand the rigorous mathematical properties of these Boltzmann equations.

1. Weak turbulence:

Wave turbulence is a branch of science studying the out-of-equilibrium statistical mechanics of random nonlinear waves of all kinds and scales. Despite the fact that wave fields in nature are enormous diverse; to describe the processes of random wave interactions, there is a common mathematical concept. The development of this mathematical concept and the applications is an established integral part of nonlinear science. Wave turbulence systems are important in a vast range of physical examples (cf. Zakharov, L’vov, Falkovich, Kolmogorov spectra of turbulence I: Wave turbulence, Springer 2012; Nazarenko, Wave turbulence, Springer 2011; Newell, Rumpf, World Scientific Series on Nonlinear Science Series, 2013.):

• water surface gravity and capillary waves;
•  inertial waves due to rotation and internal waves on density stratifications, which are important in planetary atmospheres and oceans;
• Alfvén Wave Turbulence in solar wind;
• planetary Rossby waves, which are important for the weather and climate evolutions;
• waves in Bose–Einstein condensates and in Nonlinear Optics;
• waves in plasmas of fusion devices;
• phonon interactions in anharmonic crystal lattices, solid state physics;
• waves on vibrating elastic plates, and many others.

Rigorous derivations and understanding properties of weak turbulence kinetic equations is a subject with a lot of interest for the recent years.

There are two types of wave turbulence kinetic equations.

The 4-wave turbulence kinetic equation:

$\partial_t f(t,p) = C_{4W}[f](t,p), f(0,k)=f_0(k),$

$C_{4W}[f](t,p)=\iiint_{\mathbb{R}^{3\times3}}|V_{p,p_1,p_2,p_3}|^2\delta(p+p_1-p_2-p_3)\delta(\omega(p)+\omega(p_1)-\omega(p_2)-\omega(p_3))[f_2f_3(f_1+f) -ff_1(f_2+f_3)dp_1dp_2dp_3.$

The most striking element of the theory of wave turbulence is the that the solutions of the 4- wave kinetic equation can describe the large time dynamic of high Sobolev norm $H^s$ of the solution $v$ of the cubic nonlinear Schr\”odinger equation on the torus

$-i\partial_t v + \frac{1}{2\pi}\Delta v = |v|^2v, v(t=0)= \epsilon v_0, x\in\mathbb{T}^d.$

The 3-wave turbulence kinetic equation:

$\partial_tf(t,p) =C_{3W}[f](t,p), f(0,p)=f_0(p),$

$C_{3W}[f](p) = \iint_{\mathbb{R}^{2d}} \Big[ R_{p,p_1,p_2}[f] - R_{p_1,p,p_2}[f] - R_{p_2,p,p_1}[f] \Big] dp_1dp_2$

$R_{p,p_1,p_2} [f]:= |V_{p,p_1,p_2}|^2\delta(p-p_1-p_2)\delta(\omega(p) -\omega({p_1})-\omega({p_2}))(f_1f_2-ff_1-ff_2),$

where $|V_{p,p_1,p_2}|^2$is the given transition probability, $\omega(p)$ is the dispersion relation and we have used the short-hand notation $f = f(t,p)$ and $f_j = f(t,p_j)$. Similar to the 4-wave kinetic equation, the 3-wave kinetic equation can be used to describe the long time dynamics of high Sobolev norms $H^s$ of solutions to the water wave equation (Zakharov’1965)

$\Delta \Phi(t,x,y,z) \ = \ 0, \mbox{ for } z<\zeta(t,x,y), (t,x,y,z)\in\mathbb{R}_+\times\mathbb{R}^3, \zeta_t(t,x,y)-\Phi_z(t,x,y,z) \ = \ -\zeta_x(t,x,y)\Phi_x(t,x,y,z) -\zeta_y(t,x,y)\Phi_y(t,x,y,z)\Big{|}_{z=\zeta}, \Phi_t(t,x,y,z) -\alpha (\zeta_{xx}(t,x,y)+\zeta_{yy}(t,x,y)) \ = \ \frac{|\nabla\Phi(t,x,y,z)|^2}{2}\Big{|}_{z=\zeta}, \Phi(t,x,y,z)\Big{|}_{z=-\infty}\ = \ 0,$

or several other equations (Lvov, Tabak’2001, Majda, Introduction to PDEs and waves for the atmosphere and ocean, 2003).

Our recent contributions include:

i) Wave turbulence for capillary water wavesChapter 9, joint work with T. Nguyen

ii) Wave turbulence for internal waves in the ocean: Chapter 10, joint work with L. M. Smith and I. M. Gamba

iii) Optimal local well-posedness theory for the 4-wave equation: Chapter 11, joint work with Pierre Germain and Alexandru D. Ionescu

2. Quantum Boltzmann:

After the production of the first Bose-Einstein Condensates (BECs), that led to the 2001 Nobel Prize of Physics, there has been an explosion of physics research on the kinetic theory associated to BECs and their thermal clouds. The first attempt in this direction was the one by Theodore Kirkpatrick and Robert Dorfman (1985), based on the rich body of research carried out in the period 1940-67 by Bogoliubov, Lee and Yang, Beliaev, Pitaevskii, Hugenholtz and Pines, Hohenberg and Martin, Gavoret and Nozi`eres, Kane and Kadanoff and many others. Among the later works are the ones by Yves Pomeau et. al. (1999), Eugene Zaremba et. al. (1999), Hendricus Stoof et. al. (1999), Vladimir E. Zakharov and Sergey V. Nazarenko (2005), Hebert Spohn (2010), and several others. In 2001, Crispin Gardinier, Petter Zoller and collaborators derived a Master Quantum Kinetic Equation for BECs and introduced the terminology ”Quantum Kinetic Theory”. Quantum kinetic theory is both a genuine kinetic theory and a genuine quantum theory. In which, the kinetic part arises from the decorrelation between different momentum bands.

Cover of Scientific American

Note that the first proof of Bose-Einstein Condensation was given in 2002 by Elliott H. Lieb and Robert Seiringer.

Quantum kinetic theory is formally derived for trapped bose gases with short range interactions and the system is temperature dependent.  There are rigorous derivations for the excitation spectrum for  trapped bose gases with short range interactions and the system is temperature independent ( for instance, Seiringer ’11, Hepp, Rodnianski-Schlein, Grillakis-Margetis-Machedon ’10-’15, Ben Arous-Kirkpatrick-Schlein ’12, Bach-Breteaux-Chen-Fröhlich-Sigal ’16,  Deckert-Fröhlich-Pickl-Pizzo ’16 and references therein).

There are also two types of quantum Boltzmann collision operators, which are very similar to the wave turbulence collision operators.

The 4-wave quantum Boltzmann collision operators

$C_{22}[f](t,p) = \iiint_{\mathbb{R}^{3\times3}}|\mathcal{M}_{p,p_1,p_2,p_3}|^2\delta(p+p_1-p_2-p_3)\delta(\omega(p)+\omega(p_1)-\omega(p_2)-\omega(p_3))[f_2f_3(f_1+f+1) -ff_1(f_2+f_3+1)dp_1dp_2dp_3.$

The 3-wave quantum Boltzmann collision operators

$C_{12}[f](p) = \iint_{\mathbb{R}^{2d}} \Big[ \mathcal{R}_{p,p_1,p_2}[f] - \mathcal{R}_{p_1,p,p_2}[f] - \mathcal{R}_{p_2,p,p_1}[f] \Big] dp_1dp_2$

$\mathcal{R}_{p,p_1,p_2} [f]:= |\mathcal{M}_{p,p_1,p_2}|^2\delta(p-p_1-p_2)\delta(\omega(p) -\omega({p_1})-\omega({p_2}))(f_1f_2-ff_1-ff_2-f).$

This blog is devoted to the description of our recent research results on quantum kinetic theory, using several different points of view:

i) Kinetic equations approach:

– The Cauchy problem: Chapter 1, joint work with R. Alonso, I. M. Gamba

– Hydrodynamics limits: Chapter 8, joint work with S. Jin

– Convergence to equilibrium: Chapter 4, joint work with M. Escobedo

– Positivity: Chapter 2, joint work with T. Nguyen

ii) Dispersive equations approach:

– Scattering theory: Chapter 3, Chapter 7, joint work with A. Soffer

iii) Dynamical systems approach:

– The analog between the global attractor conjecture (GAC) in chemical reaction network and the convergence to equilibrium of quantum kinetic equations: Chapter 5, joint work with G. Craciun

iv) Quantum field theory approach:

– The Peletmiskii-Yatchenko derivation: Chapter 6, joint work with L. Reichl

3. Some aspects of my works on Scientific Computing-Computational Sciences and Uncertainty Quantification-Sensitivity Analysis:

i) Iterative Hybridized Discontinous Galerkind Algorithms: Extra Chapter 1 (simulations included), and Extra Chapter 2 joint work with Tan Bui-Thanh and Sriramkrishnan Muralikrishnan.

Our simulations for solutions of a hyperbolic system using our new iHDG method

References for the pictures:

[1] Sriramkrishnan Muralikrishnan, Minh-Binh Tran, Tan Bui-Thanh. An Iterative HDG Framework for Partial Differential Equations. SIAM Journal on Scientific Computing.

[2] http://bec.nist.gov/