# American Institute of Mathematical Sciences

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 [3] and makes use of technicalities developed in [16]. 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:
 [1] 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. [2] 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. [3] 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. [4] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106, Springer-Verlag, New York, 1994. [5] 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. [6] 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. [7] K.-j. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [8] 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. [9] G. Gallavotti, Rigorous Theory Of The Boltzmann Equation In The Lorentz Gas, Nota interna Istituto di Fisica, Università di Roma, 358 (1973). [10] H. Grad, Principles of the kinetic theory of gases, Handbuch der Physik, 3 (1958), 205-294. [11] F. King, BBGKY Hierarchy for Positive Potentials, Ph. d. thesis, Department of Mathematics, Univ. California, Berkeley, 1975. [12] 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. [13] L. Landau, Kinetic equation in the case of Coulomb interaction. (in German), Phys. Zs. Sow. Union, 100 (1936), p154. [14] O. E. Lanford, Time evolution of large classical systems, in Dynamical Systems Theory and Applications, Lecture Notes in Physics, 38 (1975), 1-111. [15] J. Lebowitz and H. Spohn, Steady state self-diffusion at low density, J. Statist. Phys., 29 (1982), 39-55. doi: 10.1007/BF01008247. [16] 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. [17] 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. [18] 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. [19] H. Spohn, On the integrated form of the BBGKY hierarchy for hard spheres, arXiv: Math/0605068, 1-19. [20] K. Uchiyama, Derivation of the Boltzmann equation from particle dynamics, Hiroshima Mathematical Journal, 18 (1988), 245-297. [21] 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.

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##### References:
 [1] 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. [2] 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. [3] 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. [4] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106, Springer-Verlag, New York, 1994. [5] 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. [6] 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. [7] K.-j. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [8] 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. [9] G. Gallavotti, Rigorous Theory Of The Boltzmann Equation In The Lorentz Gas, Nota interna Istituto di Fisica, Università di Roma, 358 (1973). [10] H. Grad, Principles of the kinetic theory of gases, Handbuch der Physik, 3 (1958), 205-294. [11] F. King, BBGKY Hierarchy for Positive Potentials, Ph. d. thesis, Department of Mathematics, Univ. California, Berkeley, 1975. [12] 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. [13] L. Landau, Kinetic equation in the case of Coulomb interaction. (in German), Phys. Zs. Sow. Union, 100 (1936), p154. [14] O. E. Lanford, Time evolution of large classical systems, in Dynamical Systems Theory and Applications, Lecture Notes in Physics, 38 (1975), 1-111. [15] J. Lebowitz and H. Spohn, Steady state self-diffusion at low density, J. Statist. Phys., 29 (1982), 39-55. doi: 10.1007/BF01008247. [16] 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. [17] 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. [18] 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. [19] H. Spohn, On the integrated form of the BBGKY hierarchy for hard spheres, arXiv: Math/0605068, 1-19. [20] K. Uchiyama, Derivation of the Boltzmann equation from particle dynamics, Hiroshima Mathematical Journal, 18 (1988), 245-297. [21] 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 a three dimensional scattering
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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