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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.
[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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[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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