    June 2018, 11(3): 647-695. doi: 10.3934/krm.2018027

## The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit

 Dipartimento di Matematica, Università di Roma "La Sapienza", P.le A. Moro, 5, 00185 Roma, Italy

* Corresponding author: Nicolo' Catapano

Received  January 2017 Revised  September 2017 Published  March 2018

We consider a system of N particles interacting via a short-range smooth potential, in a weak-coupling regime. This means that the number of particles $N$ goes to infinity and the range of the potential $ε$ goes to zero in such a way that $Nε^{2} = α$, with $α$ diverging in a suitable way. We provide a rigorous derivation of the Linear Landau equation from this particle system. The strategy of the proof consists in showing the asymptotic equivalence between the one-particle marginal and the solution of the linear Boltzmann equation with vanishing mean free path. This point follows  and makes use of technicalities developed in . Then, following the ideas of Landau, we prove the asympotic equivalence between the solutions of the Boltzmann and Landau linear equation in the grazing collision limit.

Citation: Nicolo' Catapano. The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit. Kinetic & Related Models, 2018, 11 (3) : 647-695. doi: 10.3934/krm.2018027
##### References:
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##### References:
  G. Basile, A. Nota and M. Pulvirenti, A Diffusion Limit for a Test Particle in a Random Distribution of Scatterers, Journal of Statistical Physics, 155 (2014), 1087-1111. doi: 10.1007/s10955-014-0940-z.  A. V. Bobylev, M. Pulvirenti and C. Saffirio, From Particle Systems to the Landau Equation: A Consistency Result, Comm. Math. Phys., 319 (2013), 683-702. doi: 10.1007/s00220-012-1633-6.  T. Bodineau, I. Gallagher and L. Saint-raymond, The brownian motion as the limit of a deterministic system of hard-spheres, L. Invent. math., 203 (2016), 493-553. doi: 10.1007/s00222-015-0593-9.  C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106, Springer-Verlag, New York, 1994. M. Colangeli, F. Pezzotti and M. Pulvirenti, A Kac model for Fermions, M. Arch Rational Mech Anal, 216 (2015), 359-413. doi: 10.1007/s00205-014-0809-y.  L. Desvillettes and V. Ricci, A Rigorous Derivation of a Linear Kinetic Equation of Fokker-Planck Type in the Limit of Grazing Collisions, Journal of Statistical Physics, 104 (2001), 1173-1189. doi: 10.1023/A:1010461929872.  K.-j. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. I. Gallagher, L. Saint-raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, EMS Zurich Lectures in Advanced Mathematics, 2013. G. Gallavotti, Rigorous Theory Of The Boltzmann Equation In The Lorentz Gas, Nota interna Istituto di Fisica, Università di Roma, 358 (1973).  H. Grad, Principles of the kinetic theory of gases, Handbuch der Physik, 3 (1958), 205-294. F. King, BBGKY Hierarchy for Positive Potentials, Ph. d. thesis, Department of Mathematics, Univ. California, Berkeley, 1975. K. Kirkpatrick, Rigorous derivation of the landau equation in the weak coupling limit, Communications on Pure and Applied Analysis, 8 (2009), 1895-1916. doi: 10.3934/cpaa.2009.8.1895.  L. Landau, Kinetic equation in the case of Coulomb interaction. (in German), Phys. Zs. Sow. Union, 100 (1936), p154. O. E. Lanford, Time evolution of large classical systems, in Dynamical Systems Theory and Applications, Lecture Notes in Physics, 38 (1975), 1-111. J. Lebowitz and H. Spohn, Steady state self-diffusion at low density, J. Statist. Phys., 29 (1982), 39-55. doi: 10.1007/BF01008247.  M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Reviews in Mathematical Physics, 26 (2014), 1450001, 64 pp. M. Pulvirenti and S. Simonella, The Boltzmann-Grad limit of a hard sphere system: Analysis of the correlation error, Inventiones mathematicae, 207 (2017), 1135-1237, arXiv: 1405.4676 doi: 10.1007/s00222-016-0682-4. S. Simonella, Evolution of correlation functions in the hard sphere dynamics, Journal of Statistical Physics, 155 (2014), 1191-1221. doi: 10.1007/s10955-013-0905-7.  H. Spohn, On the integrated form of the BBGKY hierarchy for hard spheres, arXiv: Math/0605068, 1-19.  K. Uchiyama, Derivation of the Boltzmann equation from particle dynamics, Hiroshima Mathematical Journal, 18 (1988), 245-297. N. Ayi, From Newton's law to the linear Boltzmann equation without cut-offs, Commun. Math. Phys., 350 (2017), 1219-1274. doi: 10.1007/s00220-016-2821-6.   We denote with $\sigma\in S^{2}\left(\frac{v_{1}+v_{2}}{2}\right)$ the direction of $V^{'}$ and with $\theta$ the angle between $V$ and $V^{'}$ Here $\omega = \omega(\nu, V)$ is the unit vector bisecting the angle between $-V$ and $V'$, $\nu$ is the unit vector pointing from the particle with velocity $v_{1}$ to the particle with velocity $v_{2}$ when they are about to collide. We denote with $\beta$ the angle between $-V$ and $\omega$, with $\varphi$ the angle between $-V$ and $\nu$, with $\rho = \sin\varphi$ the impact parameter and with $\theta$ the deflection angle. It results that $\theta = \pi-2\beta$ A representation of the tree graph $(1, 1, 2)$. At the time $t_{1}$ we create the particle $2$ on the particle $1$. Then at time $t_{2}$ we create the particle $3$ on the particle $1$. Finally at time $t_{3}$ the particle $4$ is created on the particle $2$ We used a dashed line to evidence the virtual trajectory of the fourth particle The virtual trajectory of the praticles $i$ and $k$ and their backward history
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