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By choosing an appropriate damping kernel function of generalized Langevin equation, fractional Langevin equation (FLE) is derived in the case of overdamped condition. With the theory of anomalous diffusion and the memory of fractional derivatives, the physical meaning of FLE is discussed. Moreover, the internal mechanism of stochastic resonance about FLE is obtained. Finally, the numerical simulation shows that in a certain range of the order, stochastic resonance appears in FLE, and it is evident that the SNR gain in fractional Langevin equation is better than that of the integer-order situation.
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Keywords:
- stochastic resonance /
- fractional Langevin equations /
- bistable system /
- memory
[1] Bao J D 2009 Random Simulation Method of Classical and Quantum Dissipation System (Beijing: Science Press) p80 (in Chinese) [包景东 2009 经典和量子耗散系统的随机模拟方法 (北京:科学出版社) 第80页]
[2] Deng W H, Barkai E 2009 Phys. Rev. E 79 011112
[3] Benzi R, Sutera A, Vulpiana A 1981 J. Phys. A 14 L453
[4] Lin M, Fang L M 2009 Acta Phys. Sin. 58 2136 (in Chinese) [林敏, 方利民 2009 58 2136]
[5] Yang J H, Liu X B 2010 Chin. Phys. B 19 050504
[6] Gammaitoni L, Hanggi P, Jung P, Marchesoni F 1998 Rev. Modern Phys. 70 223
[7] Huang F, Liu F 2005 The ANZIAM J. 46 317
[8] Liu F, Turner I, Anh V 2003 J. Appl. Math. Comp. 13 233
[9] Wheatcraft S W, Benson D A, Meerschaert M M 2000 Water Resour. Res. 36 1403
[10] De Andrade M F, Lenzi E K, Evangelista L R, Mendes R S, Malacarne L C 2005 Phys. Lett. A 347 160
[11] Kenkre V M, Kus M, Dunlap D H, Parris P E 1998 Phys. Rev. E 58 99
[12] Dong X J 2009 Chin. Phys. B 18 70
[13] Zhang X Y, Xu W, Zhou B C 2011 Acta Phys. Sin. 60 060514 (in Chinese) [张晓燕, 徐伟, 周丙常 2011 60 060514]
[14] Podlubny I 1999 Fractional Differential Equations (San Diegop, CA: Academic Press)
[15] Samko S G, Kilbas A A, Marichev O I 1993 Marichev, Fractional Integrals and Derivatives Theory and Applications (New York, Gordon and Breach Science Publ.)
[16] Oldham K B, Spanier J 1974 The Fractional Calculus (New York: Academic Press)
[17] Kou S C, X Sunney X 2004 Phys. Rev. Lett. 93 180603
[18] Kou S C 2008 Ann. Appl. Statistics 2 501
[19] Hill T 1986 An Introduction to Statistical Thermodynamics (New York: Dover)
[20] Ahmed E, Elgazzar A S 2007 Physica A 379 607
[21] Tarasov V E 2009 J. Phys. A: Math. Theor. 42 465102
[22] Tarasov V E 2009 J. Math. Phys. 50 122703
[23] Goychuk I, Hanggi P 2003 Phys. Rev. Lett. 91 70601
[24] Fauve S, Hesolt F 1983 Phys. Lett. A 97 5
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[1] Bao J D 2009 Random Simulation Method of Classical and Quantum Dissipation System (Beijing: Science Press) p80 (in Chinese) [包景东 2009 经典和量子耗散系统的随机模拟方法 (北京:科学出版社) 第80页]
[2] Deng W H, Barkai E 2009 Phys. Rev. E 79 011112
[3] Benzi R, Sutera A, Vulpiana A 1981 J. Phys. A 14 L453
[4] Lin M, Fang L M 2009 Acta Phys. Sin. 58 2136 (in Chinese) [林敏, 方利民 2009 58 2136]
[5] Yang J H, Liu X B 2010 Chin. Phys. B 19 050504
[6] Gammaitoni L, Hanggi P, Jung P, Marchesoni F 1998 Rev. Modern Phys. 70 223
[7] Huang F, Liu F 2005 The ANZIAM J. 46 317
[8] Liu F, Turner I, Anh V 2003 J. Appl. Math. Comp. 13 233
[9] Wheatcraft S W, Benson D A, Meerschaert M M 2000 Water Resour. Res. 36 1403
[10] De Andrade M F, Lenzi E K, Evangelista L R, Mendes R S, Malacarne L C 2005 Phys. Lett. A 347 160
[11] Kenkre V M, Kus M, Dunlap D H, Parris P E 1998 Phys. Rev. E 58 99
[12] Dong X J 2009 Chin. Phys. B 18 70
[13] Zhang X Y, Xu W, Zhou B C 2011 Acta Phys. Sin. 60 060514 (in Chinese) [张晓燕, 徐伟, 周丙常 2011 60 060514]
[14] Podlubny I 1999 Fractional Differential Equations (San Diegop, CA: Academic Press)
[15] Samko S G, Kilbas A A, Marichev O I 1993 Marichev, Fractional Integrals and Derivatives Theory and Applications (New York, Gordon and Breach Science Publ.)
[16] Oldham K B, Spanier J 1974 The Fractional Calculus (New York: Academic Press)
[17] Kou S C, X Sunney X 2004 Phys. Rev. Lett. 93 180603
[18] Kou S C 2008 Ann. Appl. Statistics 2 501
[19] Hill T 1986 An Introduction to Statistical Thermodynamics (New York: Dover)
[20] Ahmed E, Elgazzar A S 2007 Physica A 379 607
[21] Tarasov V E 2009 J. Phys. A: Math. Theor. 42 465102
[22] Tarasov V E 2009 J. Math. Phys. 50 122703
[23] Goychuk I, Hanggi P 2003 Phys. Rev. Lett. 91 70601
[24] Fauve S, Hesolt F 1983 Phys. Lett. A 97 5
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