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The generalized Langevin equation with a power law memory kernel is derived via the gas/solid-surface model with fractional heat bath. Using Lapalce transformation, the dynamic evolution and long-time asymptotic behaviors of the gas particles occurring either in free or harmonic potentials are then investigated. In particular, the validity of three kinds of ergodic criteria is analyzed in detail, including the Khinchin criterion, Lee criterion, and the intrinsic and external behaviors. It is found that the Khinchin criterion holds for all ranges of diffusion and transport processes described by a generalized Langevin equation. Lee criterion is just applied to distinguish diffusion processes. Meanwhile, the intrinsic criterion and external behaviors can not only divide the nonergodicity into two classes but also reveal the underlying physical origins.
[1] Tamm M V, Nazarov L I, Gavrilov A A, Chertovich A V 2015 Phys. Rev. Lett. 114 178102
[2] Guo G, Bittig A, Uhrmacher A 2015 J. Comput. Phys. 288 167
[3] Hu W Y, Shao Y Z 2014 Acta Phys. Sin. 63 238202 (in Chinese) [胡文勇, 邵元智 2014 物理学报 63 238202]
[4] Lin F, Bao J D 2008 Acta Phys. Sin. 57 696 (in Chinese) [林方, 包景东 2008 物理学报 57 696]
[5] Montroll E, Weiss G 1965 J. Math. Phys. Lett. 6 167
[6] Viñales A D, Despósito M A 2006 Phys. Rev. E 73 016111
[7] Burov S, Barkai E 2008 Phys. Rev. Lett. 100 070601
[8] Amblard F, Maggs A C, Yurke B, Pargellis A N, Leibler S 1996 Phys. Rev. Lett. 77 4470
[9] Gu Q, Schiff E A, Grebner S, Wang F, Schwara R 1996 Phys. Rev. Lett. 76 3196
[10] Sciortion F, Tartaglia P 1997 Physcia A 236 140
[11] Stephenson J 1995 Physica A 222 234
[12] Lu H, Qin L, Bao J D 2009 Acta Phys. Sin. 58 8127 (in Chinese) [卢宏, 覃莉, 包景东 2009 物理学报 58 8127]
[13] Lu H, Bao J D 2013 Chin. Phys. Lett. 30 010502
[14] Brokmann X, Hermier J P, Messin G, Deebiolles P, Bouchard J P, Dahan M 2003 Phys. Rev. Lett. 90 120601
[15] Saubamea B, Leduc M, Cohen-Tannoudji C 1999 Phys. Rev. Lett. 83 3796
[16] Lutz E 2004 Phys. Rev. Lett. 93 190602
[17] Papoulis A 1965 Probability, Random Variables and Stochastic Processes (New York: McGraw-Hill)
[18] Lapas L C, Morgado R, Vainstein M H, Rubí J M, Oliveirs F A 2008 Phys. Rev. Lett. 101 230602
[19] Lee M H 2007 Phys. Rev. Lett. 98 110403
[20] Bai Z W, Bao J D, Song Y L 2005 Phys. Rev. E 72 061105
[21] Bao J D, Hänggi P, Zhuo Y Z 2005 Phys. Rev. E 72 061107
[22] Bao J D, Zhuo Y Z, Oliveira F A, Hänggi P 2006 Phys. Rev. E 74 061111
[23] Kupferman R 2004 J. Stat. Phys. 114 291
[24] Mainardia F, Goren R 2000 J. Comput. Appl. Math. 118 283
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[1] Tamm M V, Nazarov L I, Gavrilov A A, Chertovich A V 2015 Phys. Rev. Lett. 114 178102
[2] Guo G, Bittig A, Uhrmacher A 2015 J. Comput. Phys. 288 167
[3] Hu W Y, Shao Y Z 2014 Acta Phys. Sin. 63 238202 (in Chinese) [胡文勇, 邵元智 2014 物理学报 63 238202]
[4] Lin F, Bao J D 2008 Acta Phys. Sin. 57 696 (in Chinese) [林方, 包景东 2008 物理学报 57 696]
[5] Montroll E, Weiss G 1965 J. Math. Phys. Lett. 6 167
[6] Viñales A D, Despósito M A 2006 Phys. Rev. E 73 016111
[7] Burov S, Barkai E 2008 Phys. Rev. Lett. 100 070601
[8] Amblard F, Maggs A C, Yurke B, Pargellis A N, Leibler S 1996 Phys. Rev. Lett. 77 4470
[9] Gu Q, Schiff E A, Grebner S, Wang F, Schwara R 1996 Phys. Rev. Lett. 76 3196
[10] Sciortion F, Tartaglia P 1997 Physcia A 236 140
[11] Stephenson J 1995 Physica A 222 234
[12] Lu H, Qin L, Bao J D 2009 Acta Phys. Sin. 58 8127 (in Chinese) [卢宏, 覃莉, 包景东 2009 物理学报 58 8127]
[13] Lu H, Bao J D 2013 Chin. Phys. Lett. 30 010502
[14] Brokmann X, Hermier J P, Messin G, Deebiolles P, Bouchard J P, Dahan M 2003 Phys. Rev. Lett. 90 120601
[15] Saubamea B, Leduc M, Cohen-Tannoudji C 1999 Phys. Rev. Lett. 83 3796
[16] Lutz E 2004 Phys. Rev. Lett. 93 190602
[17] Papoulis A 1965 Probability, Random Variables and Stochastic Processes (New York: McGraw-Hill)
[18] Lapas L C, Morgado R, Vainstein M H, Rubí J M, Oliveirs F A 2008 Phys. Rev. Lett. 101 230602
[19] Lee M H 2007 Phys. Rev. Lett. 98 110403
[20] Bai Z W, Bao J D, Song Y L 2005 Phys. Rev. E 72 061105
[21] Bao J D, Hänggi P, Zhuo Y Z 2005 Phys. Rev. E 72 061107
[22] Bao J D, Zhuo Y Z, Oliveira F A, Hänggi P 2006 Phys. Rev. E 74 061111
[23] Kupferman R 2004 J. Stat. Phys. 114 291
[24] Mainardia F, Goren R 2000 J. Comput. Appl. Math. 118 283
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