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针对黏性介质引起的Brown粒子质量存在随机涨落以及阻尼力对历史速度具有记忆性等问题, 本文首次提出分数阶质量涨落谐振子模型, 以考察黏性介质中Brown粒子的动力学特性. 首先, 将Shapiro-Loginov 公式分数阶化, 使之适用于对含指数关联随机系数的分数阶随机微分方程的求解. 然后, 利用随机平均法和分数阶Shapiro-Loginov公式推导系统稳态响应振幅的解析表达式, 并据此研究系统的共振行为; 最后, 通过仿真实验验证理论结果的可靠性. 研究表明: 1)质量涨落噪声可诱导系统产生随机共振行为; 2)记忆性阻尼力可诱导系统产生参数诱导共振行为; 3)不同参数条件下, 系统表现出单峰共振、双峰共振等多样化的共振形式.When moving in viscous medium, the mass of a Brownian particle is fluctuant and its damping force depends on the past velocity history. Therefore, in order to investigate the characteristics of Brownian motion in viscous medium, fractional harmonic oscillator is proposed in this paper for the first time so for as we know. First, the Shapiro-Loginov formula is fractionized to solve fractional stochastic differential equation with exponential correlative stochastic coefficients. Then, by using stochastic averaging method and fractional Shapiro-Loginov formula, the analytical expression of a system’s steady response amplitude is presented and the system’s resonant behavior is discussed accordingly. Finally, the reliability of theoretical results is tested by simulation experiments. All the research shows that: (1) Stochastic resonant behavior can be induced by mass fluctuation noise. (2) Parameter-induced resonance can be induced by memory damping force. (3) Under different parameter conditions, the system’s resonant forms are diverse.
[1] Landau L D, Lifshitz E M (Translated by Li J F) 2007 Mechanics (5st Edn.) (Beijing: Higher Education Press) pp75–102 (in Chinese) [朗道 L. D., 栗弗席兹 E. M.著, 李俊峰译 2007 力学 (第五版) (北京: 高等教育出版社) 第75–102页]
[2] Li P, Nie L R, L X M, Zhang Q B 2011 Chin. Phys. B 20 100502
[3] Li P, Nie L R, Huang Q R, Sun X X 2012 Chin. Phys. B 21 050503
[4] Xue S H, Lin M, Meng Y 2012 Chin. Phys. B 21 090504
[5] Zhang L Y, Jin G X, Cao L, Wang Z Y 2012 Chin. Phys. B 21 120502
[6] Hu G 1994 Stochastic forces and nonlinear systems (Shanghai: Shanghai Scientific and Technological Education Press) p18 (in Chinese) [胡岗 1994 随机力与非线性系统 (上海: 上海科技教育出版社) 第18页]
[7] Bao J D 2012 Introduction to Anomalous Statistical Dynamics (Beijing: Science Press) pp67–70 (in Chinese) [包景东 2012 反常统计动力学导论 (北京: 科学出版社) 第67–70页]
[8] Mason T G, Weitz D A 1995 Phys. Rev. Lett. 74 1250
[9] Golding I, Cox E C 2006 Phys. Rev. Lett. 96 098102
[10] Banks D S, Fradin C 2005 Biophys. J. 89 2960
[11] Guigas G, Kalla C, Weiss M 2007 Biophys. J. 96 316
[12] Gitterman M, Klyatskin V I 2010 Phys. Rev. E 81 051139
[13] Gitterman M 2012 Physica A 391 3033
[14] Gitterman M 2012 Physic A 391 5343
[15] Soika E, Mankin R, Ainsaar A 2010 Phys. Rev. E 81 011141
[16] Mankin R, Rekker A 2010 Phys. Rev. E 81 041122
[17] Zhong S C, Gao S L, Wei K, Ma H 2012 Acta Phys. Sin. 61 170501 (in Chinese) [钟苏川, 高仕龙, 韦鹍, 马洪 2012 61 170501]
[18] Tu Z, Wang F, Peng H, Ma H 2013 Acta Phys. Sin. 62 030502 (in Chinese) [屠浙, 彭皓, 王飞, 马洪 2013 62 030502]
[19] Yu T, Zhang Z, Luo M K 2013 Acta Phys. Sin. 62 120504 (in Chinese) [蔚涛, 张路, 罗懋康 2013 62 120504]
[20] Shapiro V E, Loginov V M 1978 Physica A 91 563
[21] Deng W H 2007 J. Comput. Phys. 227 1510
[22] Deng W H 2009 Phys. Rev. E 79 011112
[23] Deng W H, Li C 2012 Numerical Modelling (Rijeka: InTech) pp355–374 (in Chinese) [邓伟华, 李灿 2012 数值模拟 (里耶卡: InTech出版社) 第355–374页]
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[1] Landau L D, Lifshitz E M (Translated by Li J F) 2007 Mechanics (5st Edn.) (Beijing: Higher Education Press) pp75–102 (in Chinese) [朗道 L. D., 栗弗席兹 E. M.著, 李俊峰译 2007 力学 (第五版) (北京: 高等教育出版社) 第75–102页]
[2] Li P, Nie L R, L X M, Zhang Q B 2011 Chin. Phys. B 20 100502
[3] Li P, Nie L R, Huang Q R, Sun X X 2012 Chin. Phys. B 21 050503
[4] Xue S H, Lin M, Meng Y 2012 Chin. Phys. B 21 090504
[5] Zhang L Y, Jin G X, Cao L, Wang Z Y 2012 Chin. Phys. B 21 120502
[6] Hu G 1994 Stochastic forces and nonlinear systems (Shanghai: Shanghai Scientific and Technological Education Press) p18 (in Chinese) [胡岗 1994 随机力与非线性系统 (上海: 上海科技教育出版社) 第18页]
[7] Bao J D 2012 Introduction to Anomalous Statistical Dynamics (Beijing: Science Press) pp67–70 (in Chinese) [包景东 2012 反常统计动力学导论 (北京: 科学出版社) 第67–70页]
[8] Mason T G, Weitz D A 1995 Phys. Rev. Lett. 74 1250
[9] Golding I, Cox E C 2006 Phys. Rev. Lett. 96 098102
[10] Banks D S, Fradin C 2005 Biophys. J. 89 2960
[11] Guigas G, Kalla C, Weiss M 2007 Biophys. J. 96 316
[12] Gitterman M, Klyatskin V I 2010 Phys. Rev. E 81 051139
[13] Gitterman M 2012 Physica A 391 3033
[14] Gitterman M 2012 Physic A 391 5343
[15] Soika E, Mankin R, Ainsaar A 2010 Phys. Rev. E 81 011141
[16] Mankin R, Rekker A 2010 Phys. Rev. E 81 041122
[17] Zhong S C, Gao S L, Wei K, Ma H 2012 Acta Phys. Sin. 61 170501 (in Chinese) [钟苏川, 高仕龙, 韦鹍, 马洪 2012 61 170501]
[18] Tu Z, Wang F, Peng H, Ma H 2013 Acta Phys. Sin. 62 030502 (in Chinese) [屠浙, 彭皓, 王飞, 马洪 2013 62 030502]
[19] Yu T, Zhang Z, Luo M K 2013 Acta Phys. Sin. 62 120504 (in Chinese) [蔚涛, 张路, 罗懋康 2013 62 120504]
[20] Shapiro V E, Loginov V M 1978 Physica A 91 563
[21] Deng W H 2007 J. Comput. Phys. 227 1510
[22] Deng W H 2009 Phys. Rev. E 79 011112
[23] Deng W H, Li C 2012 Numerical Modelling (Rijeka: InTech) pp355–374 (in Chinese) [邓伟华, 李灿 2012 数值模拟 (里耶卡: InTech出版社) 第355–374页]
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