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The directed transport of a Brownian particle in a spatially periodic symmetric field under a temporal asymmetric force is studied. Based on the Caputo’s fractional derivatives theory, we establish a differential aquation for an overdamped fractional Brownian motor as the system’s mathematic model, where the external force is zero-mean and the fractional order is used to describe the inhomogeneity of the real environment. Using the fractional differential algorithm, we analyze the relationships between transport velocity and model parameters. It is worth mentioning that the impact of fractional order is discussed in detail. According to the reflearch we find that a temporal asymmetric force can induce a net current without the application of a ratchet potential, even a noise. We also find that the velocity of the current increases monotonically with the increase in fractional order. Moreover with certain fractional orders, a generalized resonance phenomenon is reflealed since the velocity of the current varies non-monotonically with the system parameters, such as the height of the potential barrier and the noise strength etc. Research shows that the fractional system is a generalization of the traditional dynamic systems, which could probably give a more reasonable explanation of the directed transport as a consequence.
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Keywords:
- directed transport /
- Brownian motor /
- fractional Langevin equation /
- generalized stochastic resonance
[1] Reimann P 2002 Phys. Rep 361 57
[2] Charles R D, Werner H, Jason R 1994 Phys. Rev. Lett. 72 19
[3] Astumian R, Bier M 1994 Phys. Rev. Lett. 72 1766
[4] Magnasco M 1993 Phys. Rev. Lett. 71 1477
[5] Li F Z, Jiang L C 2010 Chin. Phys. B 19 02503
[6] Bouzat S 2014 Phys. Rev. E 89 062707
[7] Kula J, Czernik T, Luczka J 1998 Phys. Rev. Lett. 80 1377
[8] Astumian R D 1997 Science 277 917
[9] Fendrik A J, Romanelli L 2012 Phys. Rev. E 85 041149
[10] Zheng Z G 2004 Spatiotemporal Dynamics and Collective Behaviors in Coupled Nonlinear Systems (Beijing:Higher Education Presss) pp279-286 (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) 第279-286页]
[11] Hu G, Daffertshofer A, Haken H 1996 Phys. Rev. Lett. 76 26
[12] Vale R D 2003 Cell 112 467
[13] De Waele A, de Bruyn Ouboter R 1969 Physica A 41 225
[14] Mateos J L 2000 Phys. Rev. Lett. 20 364
[15] Xie T T, Zhang L, Wang F, Luo M K 2014 Acta Phys. Sin. 63 230503 (in Chinese) [谢天婷, 张路, 王飞, 罗懋康 2014 63 230503]
[16] Savel’ev S, Marchesoni F, Hannggi P, Nori F 2004 Euro. phys. Lett. 67 179
[17] Ai B Q, He Y F, Zhong W R 2011 Phys. Rev. E 83 051106
[18] Li C P, Han Y R, Zhan Y, Hu J J, Zhang L G, Qu J 2013 Acta. Phys. Sin. 62 230051 (in Chinese) [李晨璞, 韩英荣, 展永, 胡金江, 张礼刚, 曲蛟 2013 62 230051]
[19] Podlubny I 1998 Fractional Differential Equations (San Diego: Academic Press) pp78-81
[20] Ellis R J, Minton A P 2003 Nature 425 27
[21] Bhat D, Goalakrishnan M 2013 Phys. Rev. E 88 042702
[22] Yang J H, Liu X B 2011 Phys. Scr. 83 065008
[23] Bai W S M, Peng H, Tu Z, Ma H 2012 Acta Phys. Sin 61 210501 (in Chinese) [白文斯密, 彭浩, 屠浙, 马洪 2012 61 210501]
[24] Zheng Z G, Li X W 2001 Commun. Theor. Phys. 36 151
[25] Bao J D 2009 Stochastic Simulation Method of Classic and Quantum Dissipative System (Beijing:Science Press) p13 (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13页]
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[1] Reimann P 2002 Phys. Rep 361 57
[2] Charles R D, Werner H, Jason R 1994 Phys. Rev. Lett. 72 19
[3] Astumian R, Bier M 1994 Phys. Rev. Lett. 72 1766
[4] Magnasco M 1993 Phys. Rev. Lett. 71 1477
[5] Li F Z, Jiang L C 2010 Chin. Phys. B 19 02503
[6] Bouzat S 2014 Phys. Rev. E 89 062707
[7] Kula J, Czernik T, Luczka J 1998 Phys. Rev. Lett. 80 1377
[8] Astumian R D 1997 Science 277 917
[9] Fendrik A J, Romanelli L 2012 Phys. Rev. E 85 041149
[10] Zheng Z G 2004 Spatiotemporal Dynamics and Collective Behaviors in Coupled Nonlinear Systems (Beijing:Higher Education Presss) pp279-286 (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) 第279-286页]
[11] Hu G, Daffertshofer A, Haken H 1996 Phys. Rev. Lett. 76 26
[12] Vale R D 2003 Cell 112 467
[13] De Waele A, de Bruyn Ouboter R 1969 Physica A 41 225
[14] Mateos J L 2000 Phys. Rev. Lett. 20 364
[15] Xie T T, Zhang L, Wang F, Luo M K 2014 Acta Phys. Sin. 63 230503 (in Chinese) [谢天婷, 张路, 王飞, 罗懋康 2014 63 230503]
[16] Savel’ev S, Marchesoni F, Hannggi P, Nori F 2004 Euro. phys. Lett. 67 179
[17] Ai B Q, He Y F, Zhong W R 2011 Phys. Rev. E 83 051106
[18] Li C P, Han Y R, Zhan Y, Hu J J, Zhang L G, Qu J 2013 Acta. Phys. Sin. 62 230051 (in Chinese) [李晨璞, 韩英荣, 展永, 胡金江, 张礼刚, 曲蛟 2013 62 230051]
[19] Podlubny I 1998 Fractional Differential Equations (San Diego: Academic Press) pp78-81
[20] Ellis R J, Minton A P 2003 Nature 425 27
[21] Bhat D, Goalakrishnan M 2013 Phys. Rev. E 88 042702
[22] Yang J H, Liu X B 2011 Phys. Scr. 83 065008
[23] Bai W S M, Peng H, Tu Z, Ma H 2012 Acta Phys. Sin 61 210501 (in Chinese) [白文斯密, 彭浩, 屠浙, 马洪 2012 61 210501]
[24] Zheng Z G, Li X W 2001 Commun. Theor. Phys. 36 151
[25] Bao J D 2009 Stochastic Simulation Method of Classic and Quantum Dissipative System (Beijing:Science Press) p13 (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13页]
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