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从阻尼对历史加速度记忆的角度出发, 对阶数p (0, 2)的分数阶阻尼物理意义给出了统一的合理解释, 具体分析了不同阶数下的阻尼记忆特性, 在此基础上研究了空间对称势中分数阶单分子马达在无偏置双频简谐激励下的输运问题, 通过数值方法分析了输运速度与模型各参数的关系以及分数阶阻尼对输运现象的影响机理. 研究表明, 在不同阶数下历史加速度对当前时刻阻尼力的贡献与距当前时刻的时间长度呈单增或单减关系; 在适当参数下输运速度随空间势深和外力频率的增大均会出现广义共振现象, 特别地, 在存在输运且阻尼阶数较大的情况下输运速度随势深增大出现阶梯状变化而与外力频率呈正比例关系; 输运速度及方向对外力波形十分敏感, 在不同外力下阻尼力的记忆性会分别促进或阻碍粒子跃迁, 甚至引发与整数阶方向相反的定向流.Physical significance of fractional damping for order 0 p 2 is demonstrated from the perspective that it can be explained as the memory of acceleration. Based on Caputo's fractional derivatives, the transport phenomenon of fractional overdamped deterministic motors in spatial symmetric potentials driven by biharmonic forces is investigated numerically. Relationships between transport velocity and model parameters are analyzed. The effect of fractional order is discussed in detail. Research shows that the contribution of historical acceleration increases or decreases monotonously with the historical moment varying with different fractional orders. With certain parameters the transport velocity can show generalized resonance when the spatial potential depth or the external force frequency varies. Furthermore, for some large orders, the velocity varies in step with the variation of potential depth and is in a direct proportional to the frequency if there is transport. Effect of fractional damping is intimately linked with the shape of the force. The memory of damping force can promote or inhibit the particle transport under different conditions, thus triggering abundant transport behaviors.
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Keywords:
- deterministic motor /
- symmetric potential /
- biharmonic input /
- fractional
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[4] Mateos J L 2000 Phys. Rev. Lett. 84 258
[5] Tu Z C 2012 Chin. Phys. B 21 020513
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[13] Hanggi P, Marchesoni F 2009 Rev. Mod. Phys. 81 387
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[19] Zeng C H, Wang H, Wang H T 2011 Chin. Phys. B 20 050502
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[28] Hilfer R 2003 Applications of Fractional Calculus in Physics (Singapore: World Scientific)
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[31] Shen S J, Liu F W 2004 J. Xiamen Univ. (Nat. Sci.) 43 306 (in Chinese) [沈淑君, 刘发旺2004 厦门大学学报 (自然科学版) 43 306]
[32] del-Castillo-Negrete D, Gonchar V Y, Chechkin A V 2008 Physica A 387 6693
[33] Ai B Q, He Y F 2010 J. Chem. Phys. 132 094504
[34] Ai B Q, He Y F, Zhong W R 2010 Phys. Rev. E 82 061102
[35] Risau-Gusman S, Ibanez S, Bouzat S 2013 Phys. Rev. E 87 022105
[36] Ai B Q, Shao Z G, Zhong W R 2012 J. Chem. Phys. 137 174101
[37] Zhou X W, Lin L F, Ma H, Luo M K 2014 Acta Phys. Sin. 63 110501 (in Chinese) [周兴旺, 林丽烽, 马洪, 罗懋康 2014 63 110501]
[38] 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]
[39] Podlubny I 1999 Fractional Differential Equations (San Diegop: Academic Press)
[40] Zheng Z G 2004 Spatiotemporal Dynamics and Collective Behaviors in Coupled Nonlinear Systems (Beijing: Higher Education Press) (in Chinese) [郑志刚2004耦合非线性系统的时空动力学与合作行为(北京: 高等教育出版社)]
[41] Petras I 2011 Fractional-Order Nonlinear Systerms Modeling, Analysis and Simulation (1st Ed. ) (Beijing: Higher Education Press) p19
-
[1] Reimann P 2002 Phys. Rep. 361 57
[2] Jung P, Kissner J G, Hanggi P 1996 Phys. Rev. Lett. 76 3436
[3] Astumian R D 1997 Science 276 917
[4] Mateos J L 2000 Phys. Rev. Lett. 84 258
[5] Tu Z C 2012 Chin. Phys. B 21 020513
[6] Zhang H W, Wen S T, Zhang H T, Li Y X, Chen G R 2012 Chin. Phys. B 21 078701
[7] Falo F, Martinez P J, Mazo J J, Cilla S 1999 Europhys. Lett. 45 700
[8] Csahók Z, Family F, Vicsek T 1997 Phys. Rev. E 55 5179
[9] Ai B Q, He Y F, Zhong W R 2011 Phys. Rev. E 83 051106
[10] Gao T F, Liu F S, Chen J C 2012 Chin. Phys. B 21 020502
[11] Zheng Z G, Li X W 2001 Commun. Theor. Phys. 36 151
[12] Parrondo J M R, de Cisneros B J 2002 Appl. Phys. A 75 179
[13] Hanggi P, Marchesoni F 2009 Rev. Mod. Phys. 81 387
[14] Reguera D, Rubi J M 2001 Phys. Rev. E 64 061106
[15] Reguera D, Schmid G, Burada P S, Rubi J M, Reimann P, Hanggi P 2006 Phys. Rev. Lett. 96 130603
[16] Martens S, Schmid G, Schimansky-Geier L, Hanggi P 2011 Phys. Rev. E 83 051135
[17] Malgaretti P, Pagonabarraga I, Rubi J M 2013 J. Chem. Phys. 138 194906
[18] Liu J L, He J Z 2010 Chin. Phys. B 19 030504
[19] Zeng C H, Wang H, Wang H T 2011 Chin. Phys. B 20 050502
[20] Ai B Q, Wu J C 2013 J. Chem. Phys. 139 034114
[21] Flach S, Yevtushenko O, Zolotaryuk Y 2000 Phys. Rev. Lett. 84 11
[22] Quintero N R, Jose A, Cuesta J A, Alvarez-Nodarse R 2010 Phys. Rev. E 81 030102
[23] Savel'ev S, Marchesoni F, Hanggi P, Nori F 2004 Europhys. Lett. 67 179
[24] Borromeo M, Marchesoni F 2006 Phys. Rev. E 73 016142
[25] Machura L, Kostur M, Luczka J 2010 Chem. Phys. 375 445
[26] Brown M, Renzoni F 2008 Phys. Rev. A 77 033405
[27] Chen W, Sun H G, Li X C 2010 Fractional Derivative Modeling of Mechanical and Engineering Problems (Beijing: Science Press) p125 (in Chinese) [陈文, 孙洪广, 李西成 2010 机械和工程问题的分数阶导数模型 (北京: 科学出版社)第125页]
[28] Hilfer R 2003 Applications of Fractional Calculus in Physics (Singapore: World Scientific)
[29] Torvik P J, Bagley R L 1984 J. Appl. Mech. 51 294
[30] Gao S L, Zhong S C, Wei K, Ma H 2012 Acta Phys. Sin. 61 100502 (in Chinese) [高仕龙, 钟苏川, 韦鹍, 马洪 2012 61 100502]
[31] Shen S J, Liu F W 2004 J. Xiamen Univ. (Nat. Sci.) 43 306 (in Chinese) [沈淑君, 刘发旺2004 厦门大学学报 (自然科学版) 43 306]
[32] del-Castillo-Negrete D, Gonchar V Y, Chechkin A V 2008 Physica A 387 6693
[33] Ai B Q, He Y F 2010 J. Chem. Phys. 132 094504
[34] Ai B Q, He Y F, Zhong W R 2010 Phys. Rev. E 82 061102
[35] Risau-Gusman S, Ibanez S, Bouzat S 2013 Phys. Rev. E 87 022105
[36] Ai B Q, Shao Z G, Zhong W R 2012 J. Chem. Phys. 137 174101
[37] Zhou X W, Lin L F, Ma H, Luo M K 2014 Acta Phys. Sin. 63 110501 (in Chinese) [周兴旺, 林丽烽, 马洪, 罗懋康 2014 63 110501]
[38] 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]
[39] Podlubny I 1999 Fractional Differential Equations (San Diegop: Academic Press)
[40] Zheng Z G 2004 Spatiotemporal Dynamics and Collective Behaviors in Coupled Nonlinear Systems (Beijing: Higher Education Press) (in Chinese) [郑志刚2004耦合非线性系统的时空动力学与合作行为(北京: 高等教育出版社)]
[41] Petras I 2011 Fractional-Order Nonlinear Systerms Modeling, Analysis and Simulation (1st Ed. ) (Beijing: Higher Education Press) p19
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