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有限温度下自旋半经典系统的随机动力学行为通常由随机Landau-Lifshitz方程描述. 本文在朗之万随机微分方程的框架内, 推导出有效朗之万方程的一般形式, 及其对应的Fokker-Planck方程的表达式. 该有效朗之万方程能正确描述正则系综下自旋半经典系统的统计物理性质, 并且在阻尼项和随机项消失时能退化到自旋半经典运动方程, 因此是随机Landau-Lifshitz方程的一种推广. 在笛卡尔坐标系和球坐标系中, 分别给出有效朗之万方程的一般形式和对应的Fokker-Planck方程的显式表达式. 在球坐标系中, 讨论了朗之万方程中的纵场效应, 并从方程采取的形式中给出是否包含纵场效应的判断依据. 最后, 有效朗之万方程在一个单自旋、定值外磁场的体系中进行应用. 对方程采取特定的形式进行简便的求解, 并成功得到玻尔兹曼稳定分布, 该结果也检验了有效朗之万方程的准确性.
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关键词:
- 随机Landau-Lifshitz方程 /
- 朗之万方程 /
- Fokker-Planck方程 /
- 玻尔兹曼分布
The stochastic dynamics of spin semiclassical system at finite temperature is usually described by stochastic Landau-Lifshitz equation. In this work, the stochastic differential equation for spin semiclassical system is studied. The generalized formulation of effective Langevin equation and the corresponding Fokker-Planck equation are derived. The obtained effective Langevin equation offers an accurate description of the distribution in the canonical ensemble for spin semiclassical system. When the damping term and the stochastic term vanish, the effective Langevin equation reduces to the semiclassical equation of motion for spin system. Hence, the effective Langevin equation can be seen as a generalization of the stochastic Landau-Lifshitz equation. The explicit expressions for the effective Langevin equation and the corresponding Fokker-Planck equation are shown in both Cartesian coordinates and spherical coordinates. It is demonstrated that, the longitudinal effect can be easily illustrated from the expressions in spherical coordinates. The effective Langevin equation is applied to the simple system of a single spin in a constant magnetic field. Choosing an appropriate form, the Langevin equation can be easily solved and the stationary Boltzmann distribution can be obtained. The correctness of the Langevin approach for the spin semiclassical system is thus confirmed.-
Keywords:
- stochastic Landau-Lifshitz equation /
- Langevin equation /
- Fokker-Planck equation /
- Boltzmann distribution
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[1] Gilbert T L 2004 IEEE Trans. Magn. 40 3443
Google Scholar
[2] Antropov V P, Katsnelson M I, van Schilfgaarde M, Harmon B N 1995 Phys. Rev. Lett. 75 729
Google Scholar
[3] Antropov V P, Katsnelson M I, Harmon B N, van Schilfgaarde M, Kusnezov D 1996 Phys. Rev. B 54 1019
Google Scholar
[4] Ma P-W, Woo C H, Dudarev S L 2008 Phys. Rev. B 78 024434
Google Scholar
[5] Guo B, Ding S 2008 Landau-Lifshitz Equations (Singapore: World Scientific)
[6] Brown W F 1963 Phys. Rev. 130 1677
Google Scholar
[7] Kubo R, Hashitsume N 1970 Prog. Theor. Phys. Suppl. 46 210
Google Scholar
[8] García-Palacios J L, Lázaro F J 1998 Phys. Rev. B 58 14937
Google Scholar
[9] Ma P-W, Dudarev S L 2011 Phys. Rev. B 83 134418
Google Scholar
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Google Scholar
[11] Atxitia U, Hinzke D, Nowak U 2017 J. Phys. D:Appl. Phys. 50 033003
Google Scholar
[12] Landau L, Lifshitz E (edited by Pitaevski L P) 1992 Perspectives in Theoretical Physics (Amsterdam: Pergamon) p51
[13] Saslow W M 2009 J. Appl. Phys. 105 07D315
Google Scholar
[14] Lakshmanan M 2011 Philos. Trans. R. Soc. London, Ser. A 369 1280
Google Scholar
[15] Eriksson O, Bergman A, Bergqvist L, Hellsvik J 2017 Atomistic Spin Dynamics: Foundations and Applications (New York: Oxford University Press)
[16] Risken H 1989 The Fokker-Planck Equation: Methods of Solution and Applications (Berlin: Springer-Verlag)
[17] Zwanzig R 2001 Nonequilibrium statistical mechanics (New York: Oxford University Press)
[18] Kampen N G van 2009 Stochastic Processes in Physics and Chemistry (3rd Ed.) (Amsterdam: Elsevier)
[19] Klyatskin V I 2015 Stochastic Equations : Theory and Applications in Acoustics, Hydrodynamics, Magnetohydrodynamics, and Radiophysics, Volume 1, Understanding Complex Systems (Switzerland: Springer)
[20] Garanin D A, Ishchenko V V, Panina L V 1990 Theor. Math. Phys. 82 169
Google Scholar
[21] Garanin D A 1997 Phys. Rev. B 55 3050
Google Scholar
[22] Martínez E, López-Díaz L, Torres L, Alejos O 2004 Physica B 343 252
Google Scholar
[23] Mayergoyz I D, Bertotti G, Serpico C 2009 Nonlinear Magnetization Dynamics in Nanosystems (Amsterdam: Elsevier)
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Google Scholar
[25] Evans R F L, Hinzke D, Atxitia U, Nowak U, Chantrell R W, Chubykalo-Fesenko O 2012 Phys. Rev. B 85 014433
Google Scholar
[26] Coffey W T, Geoghegan L J 1996 J. Mol. Liq. 69 53
Google Scholar
[27] Fredkin D R 2001 Physica B 306 26
Google Scholar
[28] Cheng X Z, Jalil M B A, Lee H K, Okabe Y 2006 Phys. Rev. Lett. 96 067208
Google Scholar
[29] Denisov S I, Sakmann K, Talkner P, Hänggi P 2007 Phys. Rev. B 75 184432
Google Scholar
[30] Serpico C, Bertotti G, d'Aquino M, Ragusa C, Ansalone P, Mayergoyz I D 2008 IEEE Trans. Magn. 44 3157
Google Scholar
[31] Denisov S I, Polyakov A Y, Lyutyy T V 2011 Phys. Rev. B 84 174410
Google Scholar
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Google Scholar
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Google Scholar
[34] Titov S V, Coffey W T, Zarifakis M, Kalmykov Y P, Titov A S 2021 J. Magn. Magn. Mater. 539 168365
Google Scholar
[35] Ma P-W, Dudarev S L 2012 Phys. Rev. B 86 054416
Google Scholar
[36] Pan F, Chico J, Delin A, Bergman A, Bergqvist L 2017 Phys. Rev. B 95 184432
Google Scholar
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