自旋半经典朗之万方程一般形式的探讨

李德彰; 卢智伟; 赵宇军; 杨小宝

doi:10.7498/aps.72.20230106

摘要

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.

关键词:

Abstract

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:

作者及机构信息

1.
华南理工大学物理与光电学院, 广州　510640

2.
瑞典皇家理工学院工程科学院应用物理系, 斯德哥尔摩 SE-10691

通信作者: 杨小宝, scxbyang@scut.edu.cn

基金项目: 广东省基础与应用基础研究基金(批准号: 2021A1515010328)、广东省重点领域研发计划(批准号: 2020B010183001)和国家自然科学基金(批准号: 12074126)资助的课题.

Authors and contacts

1.
School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China

2.
Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of Technology, Stockholm SE-10691, Sweden

Corresponding author: Yang Xiao-Bao, scxbyang@scut.edu.cn

Funds: Project supported by the Guangdong Basic and Applied Basic Research Foundation, China (Grant No. 2021A1515010328), the Key-Area Research and Development Program of Guangdong Province, China (Grant No. 2020B010183001), and the National Natural Science Foundation of China (Grant No. 12074126).

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参考文献

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[2]	Antropov V P, Katsnelson M I, van Schilfgaarde M, Harmon B N 1995 Phys. Rev. Lett. 75 729 Google Scholar
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[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)
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[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
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[23]	Mayergoyz I D, Bertotti G, Serpico C 2009 Nonlinear Magnetization Dynamics in Nanosystems (Amsterdam: Elsevier)
[24]	Ma P-W, Dudarev S L, Semenov A A, Woo C H 2010 Phys. Rev. E 82 031111 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
[32]	Giordano S, Dusch Y, Tiercelin N, Pernod P, Preobrazhensky V 2013 Eur. Phys. J. B 86 249 Google Scholar
[33]	Aron C, Barci D G, Cugliandolo L F, Arenas Z G, Lozano G S 2014 J. Stat. Mech. :Theory Exp. 2014 P09008 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

施引文献

[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
[10]	Coffey W T, Kalmykov Y P 2012 J. Appl. Phys. 112 121301 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)
[24]	Ma P-W, Dudarev S L, Semenov A A, Woo C H 2010 Phys. Rev. E 82 031111 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
[32]	Giordano S, Dusch Y, Tiercelin N, Pernod P, Preobrazhensky V 2013 Eur. Phys. J. B 86 249 Google Scholar
[33]	Aron C, Barci D G, Cugliandolo L F, Arenas Z G, Lozano G S 2014 J. Stat. Mech. :Theory Exp. 2014 P09008 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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