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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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[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
-
[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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