过阻尼搓板势系统的随机共振

谢勇; 刘若男

doi:10.7498/aps.66.120501

摘要

研究在周期信号和高斯白噪声共同作用下过阻尼搓板势系统的随机共振.由于用直接模拟法研究随机系统所用时间较多，考虑用半解析的方法对系统的随机共振现象进行研究.在弱周期信号极限下，结合线性响应理论和扰动展开法提出一种计算系统线性响应的矩方法.在此基础上，利用Floquet理论和非扰动展开法将矩方法扩展到系统非线性响应的计算.通过直接数值模拟结果和矩方法所得结果的比较展示了矩方法的有效性并采用均方差作为量化指标给出其适用的参数范围.研究结果表明，以系统的功率谱放大因子作为量化指标，发现在适当的参数条件下，系统的共振曲线有一个单峰出现，说明过阻尼搓板势系统存在随机共振现象.而且在一定范围内调节偏置参数时，共振曲线的峰值随偏置参数的增大而增大；在调节驱动幅值时，随机共振效应随驱动幅值的增大而增强.

关键词:

Abstract

Brownian motion in a washboard potential has practical significance in investigating a lot of physical problems such as the electrical conductivity of super-ionic conductor, the fluctuation of super-current in Josephson junction, and the ad-atom motion on crystal surface. In this paper, we study the overdamped motion of a Brownian particle in a washboard potential driven jointly by a periodic signal and an additive Gaussian white noise. Since the direct simulation about stochastic system is always time-consuming, the purpose of this paper is to introduce a simple and useful technique to study the linear and nonlinear responses of overdamped washboard potential systems. In the limit of a weak periodic signal, combining the linear response theory and the perturbation expansion method, we propose the method of moments to calculate the linear response of the system. On this basis, by the Floquet theory and the non-perturbation expansion method, the method of moments is extended to calculating the nonlinear response of the system. The long time ensemble average and the spectral amplification factor of the first harmonic calculated from direct numerical simulation and from the method of moments demonstrate that they are in good agreement, which shows the validity of the method we proposed. Furthermore, the dependence of the spectral amplification factor at the first three harmonics on the noise intensity is investigated. It is observed that for appropriate parameters, the curve of the spectral amplification factor versus the noise intensity exhibits a peaking behavior which is a signature of stochastic resonance. Then we discuss the influences of the bias parameter and the amplitude of the periodic signal on the stochastic resonance. The results show that with the increase of the bias parameter in a certain range, the peak value of the resonance curve increases and the noise intensity corresponding to the resonance peak decreases. With the increase of the driven amplitude, comparing the changes of the resonance curves, we can conclude that the effect of stochastic resonance becomes more prominent. At the same time, by using the mean square error as the quantitative indicator to compare the difference between the results obtained from the method of moments and from the stochastic simulation under different signal amplitudes, we find that the method of moments is applicable when the amplitude of the periodic signal is lesser than 0.25.

Keywords:

作者及机构信息

谢勇¹,
刘若男²

1.
西安交通大学航天航空学院, 机械结构强度与振动国家重点实验室, 西安 710049;

2.
西安交通大学数学与统计学院, 西安 710049

通信作者: 谢勇, yxie@mail.xjtu.edu.cn

基金项目: 国家自然科学基金（批准号：11672219，11372233）资助的课题.

Authors and contacts

Xie Yong¹,
Liu Ruo-Nan²

1.
State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi'an Jiaotong University, Xi'an 710049, China;

2.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

Corresponding author: Xie Yong, yxie@mail.xjtu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11672219, 11372233).

参考文献

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[2]	Gammaitoni L, Hanggi P, Hung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223
[3]	McNamara B, Wiesenfeld K, Roy R 1988 Phys. Rev. Lett. 60 2626
[4]	Paulsson J, Ehrenberg M 2000 Phys. Rev. Lett. 84 5447
[5]	Leonard D S, Reichl L E 1994 Phys. Rev. E 49 1734
[6]	Mao X M, Sun K, Ouyang Q 2002 Chin. Phys. 11 1106
[7]	Zhang G L, L X L, Kang Y M 2012 Acta Phys. Sin. 61 040501 (in Chinese) [张广丽, 吕希路, 康艳梅 2012 61 040501]
[8]	Jiao S B, Ren C, Huang W C, Liang Y M 2013 Acta Phys. Sin. 62 210501 (in Chinese) [焦尚彬, 任超, 黄伟超, 梁炎明 2013 62 210501]
[9]	Wallace R, Wallace D, Andrews H 1997 Environ. Plan. A 29 525
[10]	Asaklil A, Boughaleb Y, Mazroui M, Chhib M, Arroum L E 2003 Solid State Ion. 159 331
[11]	Falco A M 1976 Amer. J. Phys. 44 733
[12]	Hanggi P, Talkner P, Borkovec M 1990 Rev. Mod. Phys. 62 251
[13]	Kim Y W, Sung W 1998 Phys. Rev. E 57 R6237
[14]	Dan D, Mahato M C, Jayannavar A M 1999 Phys. Rev. E 60 6421
[15]	Tu Z, Lai L, Luo M K 2014 Acta Phys. Sin. 63 120503 (in Chinese) [屠浙, 赖莉, 罗懋康 2014 63 120503]
[16]	Fronzoni L, Mannela R 1993 J. Stat. Phys. 70 501
[17]	Marchesoni F 1997 Phys. Lett. A 231 61
[18]	Saikia S, Jayannavar A M, Mahato M C 2011 Phys. Rev. E 83 061121
[19]	Reenbohn W L, Pohlong S S, Mahato M C 2012 Phys. Rev. E 85 031144
[20]	Saikia S 2014 Physica A 416 411
[21]	Liu K H, Jin Y F 2013 Physica A 392 5283
[22]	Ma Z M, Jin Y F 2015 Acta Phys. Sin. 64 240502 (in Chinese) [马正木, 靳艳飞 2015 64 240502]
[23]	Risken H 1989 The Fokker Planck Equation (Berlin: Springer) pp287-289
[24]	Monnai T, Sugita A, Hirashima J, Nakamura K 2006 Physica D 219 177
[25]	Kang Y M, Jiang Y L 2008 Chin. Phys. Lett. 25 3578
[26]	Kang Y M, Jiang J, Xie Y 2011 J. Phys. A: Math. Theor. 44 035002
[27]	Evistigneev M, Pankov V, Prince R H 2001 J. Phys. A: Math. Gen. 34 2595
[28]	Fox R F, Gatland I R, Vemuri G, Roy R 1988 Phys. Rev. A 38 5938
[29]	Jung P 1993 Phys. Rep. 234 175
[30]	Asish K D 2015 Physica D 303 1
[31]	Qian M, Wang G X, Zhang X J 2000 Phys. Rev. E 62 6469

施引文献

[1]	Benzi R, Sutera A, Vulpiani A 1981 J. Phys. A 14 L453
[2]	Gammaitoni L, Hanggi P, Hung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223
[3]	McNamara B, Wiesenfeld K, Roy R 1988 Phys. Rev. Lett. 60 2626
[4]	Paulsson J, Ehrenberg M 2000 Phys. Rev. Lett. 84 5447
[5]	Leonard D S, Reichl L E 1994 Phys. Rev. E 49 1734
[6]	Mao X M, Sun K, Ouyang Q 2002 Chin. Phys. 11 1106
[7]	Zhang G L, L X L, Kang Y M 2012 Acta Phys. Sin. 61 040501 (in Chinese) [张广丽, 吕希路, 康艳梅 2012 61 040501]
[8]	Jiao S B, Ren C, Huang W C, Liang Y M 2013 Acta Phys. Sin. 62 210501 (in Chinese) [焦尚彬, 任超, 黄伟超, 梁炎明 2013 62 210501]
[9]	Wallace R, Wallace D, Andrews H 1997 Environ. Plan. A 29 525
[10]	Asaklil A, Boughaleb Y, Mazroui M, Chhib M, Arroum L E 2003 Solid State Ion. 159 331
[11]	Falco A M 1976 Amer. J. Phys. 44 733
[12]	Hanggi P, Talkner P, Borkovec M 1990 Rev. Mod. Phys. 62 251
[13]	Kim Y W, Sung W 1998 Phys. Rev. E 57 R6237
[14]	Dan D, Mahato M C, Jayannavar A M 1999 Phys. Rev. E 60 6421
[15]	Tu Z, Lai L, Luo M K 2014 Acta Phys. Sin. 63 120503 (in Chinese) [屠浙, 赖莉, 罗懋康 2014 63 120503]
[16]	Fronzoni L, Mannela R 1993 J. Stat. Phys. 70 501
[17]	Marchesoni F 1997 Phys. Lett. A 231 61
[18]	Saikia S, Jayannavar A M, Mahato M C 2011 Phys. Rev. E 83 061121
[19]	Reenbohn W L, Pohlong S S, Mahato M C 2012 Phys. Rev. E 85 031144
[20]	Saikia S 2014 Physica A 416 411
[21]	Liu K H, Jin Y F 2013 Physica A 392 5283
[22]	Ma Z M, Jin Y F 2015 Acta Phys. Sin. 64 240502 (in Chinese) [马正木, 靳艳飞 2015 64 240502]
[23]	Risken H 1989 The Fokker Planck Equation (Berlin: Springer) pp287-289
[24]	Monnai T, Sugita A, Hirashima J, Nakamura K 2006 Physica D 219 177
[25]	Kang Y M, Jiang Y L 2008 Chin. Phys. Lett. 25 3578
[26]	Kang Y M, Jiang J, Xie Y 2011 J. Phys. A: Math. Theor. 44 035002
[27]	Evistigneev M, Pankov V, Prince R H 2001 J. Phys. A: Math. Gen. 34 2595
[28]	Fox R F, Gatland I R, Vemuri G, Roy R 1988 Phys. Rev. A 38 5938
[29]	Jung P 1993 Phys. Rep. 234 175
[30]	Asish K D 2015 Physica D 303 1
[31]	Qian M, Wang G X, Zhang X J 2000 Phys. Rev. E 62 6469

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