On semi-classical limit of spatially homogeneous quantum Boltzmann equation: weak convergence. (English) Zbl 1469.82028
Summary: It is expected in physics that the homogeneous quantum Boltzmann equation with Fermi-Dirac or Bose-Einstein statistics and with Maxwell-Boltzmann operator (neglecting effect of the statistics) for the weak coupled gases will converge to the homogeneous Fokker-Planck-Landau equation as the Planck constant \(\hbar\) tends to zero. In this paper and the upcoming work (He et al. in On semi-classical limit of spatially homogeneous quantum Boltzmann equation: asymptotic expansion, preprint), we will provide a mathematical justification on this semi-classical limit. Key ingredients into the proofs are the new framework to catch the weak projection gradient, which is motivated by Villani (Arch Rational Mech Anal 143(3):273-307, 1998) to identify the \(H\)-solutions for Fokker-Planck-Landau equation, and the symmetric structure inside the cubic terms of the collision operators.
MSC:
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 82D05 | Statistical mechanics of gases |
| 35Q20 | Boltzmann equations |
| 35Q84 | Fokker-Planck equations |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 35R09 | Integro-partial differential equations |
| 45K05 | Integro-partial differential equations |
Keywords:
quantum Boltzmann equationReferences:
| [1] | Arkeryd, L., On the Boltzmann equation, Arch. Rational Mech. Anal., 45, 1-34 (1972) · Zbl 0245.76059 · doi:10.1007/BF00253392 |
| [2] | Arkeryd, L., A quantum Boltzmann equation for Haldane statistics and hard forces; the space-homogeneous initial value problem, Commun. Math. Phys., 298, 2, 573-583 (2010) · Zbl 1203.82008 · doi:10.1007/s00220-010-1046-3 |
| [3] | Balescu, R., Equilibrium and Nonequilibrium Statistical Mechanics (1975), New York: Wiley, New York · Zbl 0984.82500 |
| [4] | Benedetto, D.; Pulvirenti, M., The classical limit for the Uehling-Uhlenbeck operator, Bull. Inst. Math. Acad. Sin. (N.S.), 2, 4, 907-920 (2007) · Zbl 1214.82073 |
| [5] | Benedetto, D.; Pulvirenti, M.; Castella, F.; Esposito, R., On the weak-coupling limit for bosons and fermions, Math. Models Methods Appl. Sci., 15, 12, 1811-1843 (2005) · Zbl 1154.82314 · doi:10.1142/S0218202505000984 |
| [6] | Benedetto, D.; Castella, F.; Esposito, R.; Pulvirenti, M., Some considerations on the derivation of the nonlinear Quantum Boltzmann equation, J. Stat. Phys., 116, 1-4, 381-410 (2004) · Zbl 1134.82030 · doi:10.1023/B:JOSS.0000037205.09518.3f |
| [7] | Benedetto, D.; Castella, F.; Esposito, R.; Pulvirenti, M., From the N-body Schrödinger equation to the quantum Boltzmann equation: a term-by-term convergence result in the weak coupling regime, Commun. Math. Phys., 277, 1, 1-44 (2008) · Zbl 1148.82022 · doi:10.1007/s00220-007-0347-7 |
| [8] | Bhaduri, RK; Bhalero, RS; Murthy, MV, Haldane exclusion statistics and the Boltzmann equation, J. Stat. Phys., 82, 5-6, 1659-1668 (1996) · Zbl 1260.82075 · doi:10.1007/BF02183398 |
| [9] | Bobylev, A., Gamba, I., Potapenko, I.: On some properties of the Landau kinetic equation. J. Stat. Phys. 161(2015), no. 6, 1327-1338. Stat. Phys., 161(6), 1327-1338 (2015) · Zbl 1333.35151 |
| [10] | Bobylev, A.; Pulvirenti, M.; Saffirio, C., From particle systems to the Landau equation: a consistency result, Commun. Math. Phys., 319, 3, 683-702 (2013) · Zbl 1273.35281 · doi:10.1007/s00220-012-1633-6 |
| [11] | Bressan, A.: Notes on the Boltzmann Equation Lecture notes for a summer course given at S.I.S.S.A. , Trieste (2005) |
| [12] | Briant, M.; Einav, A., On the Cauchy problem for the homogeneous Boltzmann-Nordheim equation for bosons: local existence, uniqueness and creation of moments, J. Stat. Phys., 163, 5, 1108-1156 (2016) · Zbl 1342.35203 · doi:10.1007/s10955-016-1517-9 |
| [13] | Cai, S.; Lu, X., The spatially homogeneous Boltzmann equation for Bose-Einstein particles: rate of strong convergence to equilibrium, J. Stat. Phys., 175, 2, 289-350 (2019) · Zbl 1444.82007 · doi:10.1007/s10955-019-02258-8 |
| [14] | Carlen, EA; Carvalho, MC; Lu, X., On strong convergence to equilibrium for the Boltzmann equation with soft potentials, J. Stat. Phys., 135, 4, 681-736 (2009) · Zbl 1191.76087 · doi:10.1007/s10955-009-9741-1 |
| [15] | Cercignani, C.; Illner, R.; Pulvirenti, M., The Mathematical Theory of Dilute Gases (1994), Berlin: Springer, Berlin · Zbl 0813.76001 · doi:10.1007/978-1-4419-8524-8 |
| [16] | Desvillettes, L., Entropy dissipation estimates for the Landau equation in the Coulomb case and applications, J. Funct. Anal., 269, 5, 1359-1403 (2015) · Zbl 1325.35223 · doi:10.1016/j.jfa.2015.05.009 |
| [17] | He, L.-B., Lu, X., Pulvirenti, M.: On semi-classical limit of spatially homogeneous quantum Boltzmann equation: asymptotic expansion, preprint |
| [18] | Landau, LD, Kinetic equation for the case of Coulomb interaction, Phys. Zs. Sov. Union, 10, 154-164 (1936) · Zbl 0015.38202 |
| [19] | Lifshitz, EM; Pitaevskii, LP, Course of Theoretical Physics (1981), Oxford: Pergamon, Oxford |
| [20] | Lu, X., Conservation of energy, entropy identity, and local stability for the spatially homogeneous Boltzmann equation, J. Stat. Phys., 96, 3-4, 765-796 (1999) · Zbl 0962.82062 · doi:10.1023/A:1004606525200 |
| [21] | Lu, X., On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles, J. Stat. Phys., 105, 1-2, 353-388 (2001) · Zbl 1156.82380 · doi:10.1023/A:1012282516668 |
| [22] | Lu, X., On isotropic distributional solutions of the Boltzmann equation for Bose-Einstein particles, J. Stat. Phys., 116, 5-6, 1597-1649 (2004) · Zbl 1097.82023 · doi:10.1023/B:JOSS.0000041750.11320.9c |
| [23] | Nordheim, LW, On the kinetic method in the new statistics and its application in the electron theory of conductivity, Proc. R. Soc. Lond. Ser. A, 119, 783, 689-698 (1928) · JFM 54.0988.05 · doi:10.1098/rspa.1928.0126 |
| [24] | Pulvirenti, M.: The weak-coupling limit of large classical and quantum systems. International Congress of Mathematicians Vol. III, 229-256, Eur. Math. Soc., Zurich (2006) · Zbl 1105.35129 |
| [25] | Rosenbluth, MN; MacDonald, WM; Judd, DL, Fokker-Planck equation for an inverse-square force, Phys. Rev., 107, 1-6 (1957) · Zbl 0077.44802 · doi:10.1103/PhysRev.107.1 |
| [26] | Rudin, W., Real and Complex Analysis (1974), New York: McGraw-Hill, New York · Zbl 0278.26001 |
| [27] | Uehling, EA; Uhlenbeck, GE, Transport phenomena in Einstein-Bose and Fermi-Dirac gases I, Phys. Rev., 43, 552-561 (1933) · Zbl 0006.33404 · doi:10.1103/PhysRev.43.552 |
| [28] | Villani, C., On a new class of weak solutions of the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143, 3, 273-307 (1998) · Zbl 0912.45011 · doi:10.1007/s002050050106 |
| [29] | Villani, C.: A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics, Vol. I, 71-305, North-Holland, Amsterdam (2002) · Zbl 1170.82369 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
