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Diffusion enhancement of the particle in disorder medium by biased force

Fan Li-Ming Bao Jing-Dong

Wen Lin, Liang Yi, Zhou Jing, Yu Peng, Xia Lei, Niu Lian-Bin, Zhang Xiao-Fei. Effects of linear Zeeman splitting on the dynamics of bright solitons in spin-orbit coupled Bose-Einstein condensates. Acta Phys. Sin., 2019, 68(8): 080301. doi: 10.7498/aps.68.20182013
Citation: Wen Lin, Liang Yi, Zhou Jing, Yu Peng, Xia Lei, Niu Lian-Bin, Zhang Xiao-Fei. Effects of linear Zeeman splitting on the dynamics of bright solitons in spin-orbit coupled Bose-Einstein condensates. Acta Phys. Sin., 2019, 68(8): 080301. doi: 10.7498/aps.68.20182013

Diffusion enhancement of the particle in disorder medium by biased force

Fan Li-Ming, Bao Jing-Dong
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  • The diffusion dynamics of a particle in the biased disorder medium is investigated in this paper. Based on the mean first passage time (MFPT) theory, the analytical approximate expression of effective diffusion coefficient of a particle in the biased disorder potential is obtained. The results show that the effective diffusion of a particle in the biased disorder potential is significantly enhanced. We explain the enhancement mechanism by using the wave packet broadening of probability density distribution function. In addition, we propose the concepts of effective kinetic temperature and effective friction, and further find that the effective diffusion behavior of a particle strongly depends on the biased force.
      PACS:
      03.75.Lm(Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations)
      03.75.Kk(Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow)
      03.75.Mn(Multicomponent condensates; spinor condensates)
      71.70.Ej(Spin-orbit coupling, Zeeman and Stark splitting, Jahn-Teller effect)
      Corresponding author: Bao Jing-Dong, jdbao@bnu.edu.cn
    • Funds: Project supported by the Key Program of the National Natural Science Foundation of China (Grant No. 11735005)

    孤子作为一种非线性波, 因其独特的传播性质及潜在的应用价值, 已成为非线性科学研究领域的重要研究课题之一. 孤子也是自然界中的一种普遍的非线性现象, 并广泛地存在于各种非线性介质中, 如水波、等离子体、粒子物理、分子生物学及光纤等[1]. 特别地, 随着玻色-爱因斯坦凝聚(BEC)和简并费米气体的实验实现, 大量的研究结果展示, 超冷原子气体中也存在物质波孤子现象, 实验上已经相继观察到了物质波亮孤子、暗孤子及涡旋孤子等非线性现象[2-13]. 由于物质波孤子在相干原子光学、原子干涉仪及原子输运等领域中存在着潜在的应用价值, 研究超冷原子气体中的孤子动力学性质也成为了近几十年的热点研究课题之一.

    近年来, 人造自旋-轨道耦合在超冷原子气体中的实验实现, 也为探索规范场中孤子的动力学性质提供了平台[14-17]. 一方面, 自旋-轨道耦合使得体系单粒子基态在有限动量处简并[18,19], 自旋-轨道耦合在BEC中将导致许多新奇的静态孤子, 例如条纹孤子和分数涡旋能隙孤子等[20-42]. 另一方面, 在多组分BEC中, 孤子可看作是具有赝自旋的粒子, 自旋-轨道耦合将孤子自旋与质心耦合在一起, 使得孤子的自旋旋进将影响其质心运动. 例如, 孤子自旋周期性翻转将提供一个周期性的力去驱使孤子质心做周期性振荡[43-46]. 由此可见, 自旋-轨道耦合为孤子的宏观量子调控提供了新的手段.

    实验上在超冷原子气体中产生人造自旋-轨道耦合时, 不同分量间的能量差及Raman激光的频率差将产生一个有效的线性塞曼劈裂[14-17], 它使得单粒子能谱的对称结构被破坏, 并导致许多新奇的量子态, 如BEC中的极化平面波态、超冷费米气体中的拓扑超流态和Majorana费米子等[47-51]. 除此以外, 甚至在没有自旋-轨道耦合的旋量BEC中, 塞曼劈裂也会导致一些新奇的拓扑孤子态[52]. 因此, 在具有自旋-轨道耦合的BEC中, 线性塞曼劈裂将对孤子的动力学性质产生明显的影响.

    本文以一维自旋-轨道耦合双组份BEC为研究对象, 深入研究线性塞曼劈裂对亮孤子动力学性质的影响. 假设体系的原子相互作用守恒SU(2)对称性, 并取双曲正弦函数作为亮孤子的试探波函数, 本文首先利用变分法导出了试探波函数中的变分参数(未知参数)随时间演化所满足的欧拉-拉格朗日方程; 然后求解欧拉-拉格朗日方程的固定点解, 在自旋-轨道耦合强度较弱时, 发现了两个近似的静态亮孤子解; 进一步对这两个静态孤子做线性稳定性分析, 获得了一个零能的Goldstone激发模和一个谐振激发模, 前者对应于平移对称性的破缺, 后者的振荡频率与线性塞曼劈裂的强度有关; 最后, 通过求解欧拉-拉格朗日方程, 获得了变分参数的含时精确解, 并发现线性塞曼劈裂明显地影响孤子的运动速度和振荡周期. 这些变分计算结果与基于Gross-Pitaevskii (GP)方程的直接数值模拟结果相吻合.

    考虑沿z方向的一维均匀自旋-轨道耦合双组分BEC[14-17]. 在由同一种原子的两种不同超精细态所形成的双组分BEC的实验中, 由于两个分量的种内原子相互作用强度和种间原子相互作用强度通常比较接近, 本文将假设两者相等, 即原子间的相互作用守恒SU(2)对称性. 因此, 在平均场近似下, 系统的动力学性质可用如下的无量纲化GP方程描述:

    iψt=12·2ψz2ikRψz+εψ+Ωψ+gnψ,
    (1a)
    iψt=12·2ψz2+ikRψzεψ+Ωψ+gnψ,
    (1b)

    方程中ψs(z,t)为描述两个分量的动力学性质的波函数(s=↑,代表两个不同的分量), 且满足归一化条件s=↑,+|ψs|2dz=1, zt分别为空间坐标和时间; n=|ψ|2+|ψ|2为总密度分布; ε为线性塞曼劈裂的强度; ΩkR分别表示Raman激光强度和自旋-轨道耦合强度; g<0代表原子间的吸引相互作用强度. 当自旋-轨道耦合存在时, GP方程((1a), (1b))不可积[25], 本文利用变分近似方法解析研究孤子的动力学性质[1]. 在变分近似方法中, 体系所对应的拉格朗日量为

    L(t)=+[i2s=↑,(ψsψstψsψst)+12ψ2ψz2+12ψ2ψz2+ikR(ψψzψψz)ε(|ψ|2|ψ|2)Ω(ψψ+ψψ)g2n4]dz,
    (2)

    其中代表复共轭.

    对于SU(2)自旋对称的均匀双组分BEC, 假设BEC凝聚在单个准动量为k的态上, 则可以利用双曲正弦函数作为亮孤子的试探波函数, 并取两个分量孤子的宽度η1、质心坐标z和波矢k相等, 即

    (ψψ)=η2(sinθei(kz+φ)cosθei(kz+φ))sech[η(zz)],
    (3)

    其中变分参数θ, η, z, kφs都是时间的函数; φs代表两个组分中亮孤子的相位; θ可描述两个亮孤子的振幅比. 将波函数(3)代入拉格朗日量(2)式, 并定义φ±=(φ±φ)/2β=kRk+ε, 则

    L(t)=zdkdt+cos(2θ)dφdtdφ+dt+βcos(2θ)Ωsin(2θ)cos(2φ)η(η+g)612k2.
    (4)

    然后利用欧拉-拉格朗日方程LAddt(L˙A)=0 (A为变分参数θ, k, z, φ±η, 且˙A=dA/dt), 可得变分参数随时间演化的运动方程

    η=g/2,
    (5a)
    dk/dt=0,
    (5b)
    dzdt=kkRcos(2θ),
    (5c)
    dθdt=Ωsin(2φ),
    (5d)
    dφdt=βΩcot(2θ)cos(2φ).
    (5e)

    方程(5a)和(5b)说明孤子的宽度仅由原子间的相互作用决定, 且两个孤子的动量守恒. 方程(5c)—(5e)表明, 自旋-轨道耦合将θzφ耦合在一起, 并且线性塞曼劈裂也影响这些参数的动力学演化, 这将导致孤子展现出有趣的动力学特征.

    通过设dθdt=dzdt=dφdt=0, 首先求解欧拉-拉格朗日方程(5a)—(5e)的固定点解(用符号“”作为上标标记), 它或许对应于静态孤子解[1,53]. 从方程(5c)和(5d)中可看出, ˜φ=nπ2 (n为整数)和~z=任意常数, 后者说明静态孤子具有平移对称性. 而固定点解˜θ˜k满足非线性方程组˜k=kRcos(2˜θ)0=k2Rcos(2˜θ)+ε+(1)nΩcot(2˜θ). 由于n取奇数和偶数时, ˜θ˜k的解仅仅相差一个负号, 因此本文考虑n=0的简单情形. 图1给出了固定点解˜θ˜k随线性塞曼劈裂强度的变化, 曲线颜色代表不同的解. 从图1(a)(b)中可看出, 当Ω/k2R<1时, 线性塞曼劈裂存在着一个临界值εc, 它随Raman耦合强度增加而单调递减(见图1(e)). 如果ε<εc, ˜θ˜k分别具有四个不同的解, 否则˜θ˜k分别具有两个不同的解. 而对于Ω/k2R>1, 不论线性塞曼劈裂强度取何值, ˜θ˜k都只具有两个不同的解(见图1(c)(d)). 特别的, 如图1(a)— (d)所示, 归咎于线性塞曼劈裂的存在, 这些固定点所对应的孤子总是具有非零的动量, 并且两个分量间的粒子数不相等, 即

    图 1 (a),(b)$ \varOmega/k_{\rm R}^2 = 0.5 $时固定点解$ \tilde{\theta} $和$ \tilde{k} $随线性塞曼劈裂$ \varepsilon $的变化; (c),(d)$ \varOmega/k_{\rm R}^2 = 1.5 $时固定点解$ \tilde{\theta} $和$ \tilde{k} $随线性塞曼劈裂$ \varepsilon $的变化; (e)$ \varOmega/k_{\rm R}^2<1 $时临界值$ \varepsilon_{\rm c}$随$ \varOmega $的变化\r\nFig. 1. (a) and (b) show the $ \tilde{\theta} $ and $ \tilde{k} $ change with $ \varepsilon $ for $ \varOmega/k_{\rm R}^2 = 0.5 $; (c) and (d) display $ \tilde{\theta} $ and $ \tilde{k} $ change with $ \varepsilon $ for $ \varOmega/k_{\rm R}^2 = 1.5 $; (e) shows the critical value $ \varepsilon_{\rm c} $ versus $ \varOmega $ for $ \varOmega/k_{{\rm R}}^2<1 $.
    图 1  (a),(b)Ω/k2R=0.5时固定点解˜θ˜k随线性塞曼劈裂ε的变化; (c),(d)Ω/k2R=1.5时固定点解˜θ˜k随线性塞曼劈裂ε的变化; (e)Ω/k2R<1时临界值εcΩ的变化
    Fig. 1.  (a) and (b) show the ˜θ and ˜k change with ε for Ω/k2R=0.5; (c) and (d) display ˜θ and ˜k change with ε for Ω/k2R=1.5; (e) shows the critical value εc versus Ω for Ω/k2R<1.

    +(|ψ|2|ψ|2)dz=˜k/kR0.

    变分法仅仅是一种近似方法, 有必要将欧拉-朗格朗日方程(5a)(5e)的固定点解与GP方程的数值解作对比[1]. 一方面, 可利用虚时演化方法求解GP方程的静态孤子数值解, 其结果展示在图2(a)(b)中. 对于弱的自旋-轨道耦合(Ω/k2R1), 图2(a)展示欧拉-朗格朗日方程(5a)(5e)的固定点解与GP方程的静态孤子数值解相一致. 然而, 对于强自旋-轨道耦合(Ω/k2R1)的情况, 两者存在着明显的差别(见图2(b)), 图2(a), (b)中变分静态孤子解分别取自于图1(d)1(a)中的蓝色和红色曲线. 因此, 在弱自旋-轨道耦合情况下, 变分法能产生一个较好的静态孤子近似解. 另一方面, 也可以将欧拉-朗格朗日方程(5a)(5e)的固定点解作为初始条件去数值求解含时GP方程, 如果孤子在含时演化中能够保持其初始波形而不运动, 则欧拉-拉格朗日方程(5a)(5e)的固定点解可认作是GP方程的静态孤子近似解. 这些含时演化结果展示在图2(c)—(f)中, 从中可以看出, 对于弱自旋-轨道耦合的情况(Ω/k2R1), 尽管线性塞曼劈裂导致初始孤子具有一个有限的动量, 但是这些孤子总是能保持其初始的波形而静止在初始位置(见图2(c),(d)). 相反, 对于强自旋-轨道耦合的情形(Ω/k2R1), GP方程的含时数值演化结果表明, 孤子将偏离初始位置, 它不仅沿着z方向线性运动, 而且运动过程中还会出现振荡运动(见图2(e),(f)).

    图 2 (a)和(b)分别展示$ k_{\rm R} = 0.2\varOmega $和$ k_{\rm R} = 1.5\varOmega $时, 变分静态孤子解(圆圈)与GP方程(2)静态孤子的数值解(实线)的对比, 其他参数取值为$ \varepsilon = 0.3 $, $ \varOmega = 0.5 $及$ g = -10 $; (c)—(f)分别为(a)和(b)中的变分静态孤子解作为初始条件在含时GP方程中的动力学演化\r\nFig. 2. (a), (b) show the comparisons between the variationally predicted stationary soliton solutions (circles) and the numerical solutions (solid lines) of stationary solitons of GP equation (2) for $k_{\rm R}=0.2\varOmega$ and $k_{\rm R}=1.5\varOmega$ with $\varOmega=0.5$, respectively. The other parameters are $\varepsilon=0.3$ and $g=-10$;(c)−(f) are the dynamical evolutions of solitons in time-dependent GP simulations by using the variationally predicted stationary soliton solutions in (a) and (b) as initial wave functions, respectively
    图 2  (a)和(b)分别展示kR=0.2ΩkR=1.5Ω时, 变分静态孤子解(圆圈)与GP方程(2)静态孤子的数值解(实线)的对比, 其他参数取值为ε=0.3, Ω=0.5g=10; (c)—(f)分别为(a)和(b)中的变分静态孤子解作为初始条件在含时GP方程中的动力学演化
    Fig. 2.  (a), (b) show the comparisons between the variationally predicted stationary soliton solutions (circles) and the numerical solutions (solid lines) of stationary solitons of GP equation (2) for kR=0.2Ω and kR=1.5Ω with Ω=0.5, respectively. The other parameters are ε=0.3 and g=10;(c)−(f) are the dynamical evolutions of solitons in time-dependent GP simulations by using the variationally predicted stationary soliton solutions in (a) and (b) as initial wave functions, respectively

    利用线性稳定性理论可进一步分析这些静态孤子在微扰下的稳定性及激发模式[1,53]. 设变分参数A(t)=˜A+δeiωt, 其中δA代表在微扰下变分参数相对于其固定点解˜A的偏离, ω为激发的本征频率. 将A(t)=˜A+δAeiωt代入欧拉-拉格朗日方程, 并保留至δA的一阶项, 可得矩阵方程:

    (iω0001iω2kRsin(2˜θ)000iω2ΩkR02Ωsin2(2˜θ)iω)(δkδzδθδφ)=0.
    (6)

    解该矩阵方程, 可得到本征频率ω1=ω2=0ω±=±2Ωsin(2˜θ), 以及对应的本征矢量V1=(0,0,0,0)T, V2=(0,1,0,0)TV±=(0,kRΩsin3(2˜θ),±isin(2˜θ))T, 其中上标“T”表示转置. 根据线性稳定性分析理论可知[1], 所有4个本征频率都为实数, 说明静态孤子在微扰下是动力学稳定的. 与本征矢量V1相对应的零能模表明, 孤子在扰动下将保持不变. 而与本征矢量V2相对应的零能模就是所谓的Goldstone模. 由于本征矢量V2中的δz0, 而δk, δθδφ均为0, 说明孤子质心坐标在扰动下将偏离其平衡位置而以零频率振动, 即孤子在扰动下将以固定速度做线性运动, 其平移对称性被破缺[54]. 对于频率为ω±的谐振模, 孤子在扰动下将以频率ω±而振荡, 且ω±也与线性塞曼劈裂有关(见图3图3(a), (b)中的˜θ分别取图1(a)(c)中的数据).

    图 3 $ \varOmega/k_{\rm R}^2 = 0.5$ (a)和$ \varOmega/k_{\rm R}^2 = 1.5 $(b)时, 频率$ \omega_{\pm} = \pm 2\varOmega/\sin\left(2\tilde{\theta}\right) $随线性塞曼劈裂强度$ \varepsilon $的变化\r\nFig. 3. The frequency $\omega_{\pm}=\pm 2\varOmega/\sin\left(2\tilde{\theta}\right)$ changes with $\varepsilon$ for $\varOmega/k_{\rm R}^2=0.5$ and $\varOmega/k_{\rm R}^2=1.5$ in (a) and (b), respectively.
    图 3  Ω/k2R=0.5 (a)和Ω/k2R=1.5(b)时, 频率ω±=±2Ω/sin(2˜θ)随线性塞曼劈裂强度ε的变化
    Fig. 3.  The frequency ω±=±2Ω/sin(2˜θ) changes with ε for Ω/k2R=0.5 and Ω/k2R=1.5 in (a) and (b), respectively.

    通过求解含时欧拉-拉格朗日方程的解, 可研究线性塞曼劈裂对孤子的运动的影响. 由于自旋-轨道耦合将变分参数zθφ非线性地耦合在一起, 很难直接求解欧拉-拉格朗日方程的精确解. 为此, 首先引入满足条件s=↑,|χs|2=1的复值旋量χs=ψs/n, 并定义亮孤子自旋

    Sx=χχ+χχ=sin(2θ)cos(2φ),

    Sy=i(χχχχ)=sin(2θ)sin(2φ),

    Sz=|χ|2|χ|2=cos(2θ). 结合欧拉-拉格朗日方程, 可导出孤子自旋满足的运动方程

    dSxdt=2βSy,
    (7a)
    dSydt=2βSx2ΩSz,
    (7b)
    dSzdt=2ΩSy.
    (7c)

    该方程为常系数线性微分方程组, 其精确解为

    Sx=aβϖsin(2ϖtϕ)+Ωcϖ2,
    (8a)
    Sy=asin(2ϖt+ϕ),
    (8b)
    Sz=aΩϖsin(2ϖtϕ)+βcϖ2,
    (8c)

    其中

    ϖ=Ω2+β2,

    ϕ=arctan(b/Sy,0),

    a=b2+S2y,0,

    b=(βSx,0ΩSz,0)/ω,

    c=ΩSx,0+βSz,0, 下标“0”标记Sx,y,z的初始值, 它们由变分参数θφ的初始值θ0φ,0决定. 从解(8a)—(8c)式中可反解得出θφ的解, 即

    θ(t)=12arccos(Sz)φ=12arctan(SySx).

    对于孤子质心z, 由于欧拉-拉格朗日方程(5c)可写作dzdt=k+kRSz, 则将Sz的精确解代入, 并积分可得

    z(t)=kRΩ2ϖ2[2Sy,0sin2(ϖt)bsin(2ϖt)]+(kRβcϖ2+k)t,
    (9)

    其中已假设孤子质心的初始值为0. 从精确解中可以看出, 在自旋-轨道耦合、Raman耦合及线性塞曼劈裂的作用下, 孤子的质心运动是周期振荡及线性运动的叠加, 其振荡频率2ϖ及线性运动速度v=kRβc/ϖ2+k均与线性塞曼劈裂强度有关.

    对于给定的初始条件, 我们数值求解GP方程(1a), (1b), 并将GP方程的数值解与变分精确解做对比. 为了研究线性塞曼劈裂对孤子运动的影响, 本文考虑初始条件θ0=π4φ,0=0, 并假设BEC的凝聚动量k=0. 如果线性塞曼劈裂为零, 变分精确解为θ(t)=π4, φ=0z(t)=0, 说明孤子并不会运动, 与GP方程的数值模拟结果一致(见图4(a)—(c)). 图4(a)—(f)其他参数取值为g=10, η=g/2, kR=Ω/2Ω=0.5. 然而, 当线性塞曼劈裂不为零时, 孤子质心的变分精确解为

    图 4 (a)—(f)初始值为$ \theta_{0} = \dfrac{{\text{π}}}{4} $, $ \varphi_{-, 0} = 0 $及$ k = 0 $的孤子动力学演化 (a)—(c) $ \varepsilon = 0 $, (d)—(f) $ \varepsilon = 0.35 $; 孤子振荡周期$ T(g) $及速度$ v(h) $随线性塞曼劈裂的变化\r\nFig. 4. (a)−(f) show the dynamical evolutions of initially balanced solitons with $\theta_0=\dfrac{{\text{π}}}{4}$, $k=0$ and $\varphi_{-, 0}=0$ in GP simulations, $\varepsilon=0$ in (a)−(c), and $\varepsilon=0.35$ in (d)−(f); oscillation period $T(g)  $  and moving velocity $v(h) $ of solitons change with the linear Zeeman splitting.
    图 4  (a)—(f)初始值为θ0=π4, φ,0=0k=0的孤子动力学演化 (a)—(c) ε=0, (d)—(f) ε=0.35; 孤子振荡周期T(g)及速度v(h)随线性塞曼劈裂的变化
    Fig. 4.  (a)−(f) show the dynamical evolutions of initially balanced solitons with θ0=π4, k=0 and φ,0=0 in GP simulations, ε=0 in (a)−(c), and ε=0.35 in (d)−(f); oscillation period T(g) and moving velocity v(h) of solitons change with the linear Zeeman splitting.

    z(t)=kRΩε2(Ω2+ε2)3/2sin(2Ω2+ε2t)+kRΩεΩ2+ε2t,

    图4(d)—(f)所示, 孤子沿z方向线性运动的同时将做振荡运动, 振荡周期T及线性运动速度v与线性塞曼劈裂的强度有关(见图4(g),(h), 其中kR=Ω/2, Ω=0.5). 此外, 我们也选择了其他初始条件进行数值模拟, 并发现线性塞曼劈裂将影响孤子的运动速度及振荡频率, 且变分精确解与GP方程的数值模拟结果相吻合.

    本文研究了线性塞曼劈裂对一维自旋-轨道耦合双组份BEC中亮孤子的动力学性质的影响. 通过选择双曲正弦函数作为亮孤子的变分试探波函数, 可运用变分法导出变分参数随时间演化所满足的欧拉-拉格朗日方程. 求解不含时的欧拉-拉格朗日方程, 在弱自旋-轨道耦合情况下, 获得了两个近似的静态孤子解. 基于线性稳定性分析, 进一步发现了一个零能的Goldstone激发模和一个谐振激发模, 前者对应于在外界扰动下静态孤子的平移对称性破缺, 后者表明静态孤子在外界扰动下将做谐振运动, 其谐振频率也与线性塞曼劈裂有关. 最终, 通过求解含时欧拉-拉格朗日方程, 描述孤子质心运动的精确变分解被获得, 并发现线性塞曼劈裂将明显地影响孤子的运动速度和谐振周期. 所有这些变分计算结果都与GP方程的直接数值模拟相吻合.

    [1]

    Su Y, Ma X G, Lai P, Tong P 2017 Soft Matter 13 4773Google Scholar

    [2]

    Lindenberg K, Sancho J M, Khoury M, Lacasta A M 2012 Fluctuation Noise Lett. 11 1240004Google Scholar

    [3]

    Zwanzig R 1988 Proc. Natl. Acad. Sci. U.S.A. 85 2029Google Scholar

    [4]

    Lifson S, Jackson J L 1962 J. Chem. Phys. 36 2410Google Scholar

    [5]

    Longobardi L, Massarotti D, Rotoli G, Stornaiuolo D, Papari G, Kawakami A, Pepe G P, Barone A, Tafuri F 2011 Phys. Rev. B 84 184504Google Scholar

    [6]

    Fulde P, Pietronero L, Schneider W R, Strässler S 1975 Phys. Rev. Lett. 35 1776Google Scholar

    [7]

    Kurrer C, Schulten K 1995 Phys. Rev. E 51 6213Google Scholar

    [8]

    Reimann P 2002 Phys. Rep. 361 57Google Scholar

    [9]

    Lee S H, Grier D G 2006 Phys. Rev. Lett. 96 190601Google Scholar

    [10]

    Blickle V, Speck T, Seifert U, Bechinger C 2007 Phys. Rev. E 75 060101Google Scholar

    [11]

    Schiavoni M, Sanchez-Palencia L, Renzoni F, Grynberg G 2003 Phys. Rev. Lett. 90 094101Google Scholar

    [12]

    Lü K, Bao J D 2007 Phys. Rev. E 76 061119Google Scholar

    [13]

    Shi X Y, Bao J D 2019 Physica A 514 203Google Scholar

    [14]

    Gleeson J P, Sancho J M, Lacasta A M, Lindenberg K 2006 Phys. Rev. E 73 041102Google Scholar

    [15]

    Khoury M, Lacasta A M, Sancho J M, Lindenber K 2011 Phys. Rev. Lett. 106 090602Google Scholar

    [16]

    Slutsky M, Kardar M, Mirny L A 2004 Phys. Rev. E 69 061903Google Scholar

    [17]

    Bouchaud J P, Georges A 1990 Phys. Rep. 195 127Google Scholar

    [18]

    Hänggi P, Talkner P, Borkovec M 1990 Rev. Mod. Phys. 62 251Google Scholar

    [19]

    Hanes R D L, Dalle-Ferrier C, Schmiedeberg M, Jenkins M C, Egelhaaf S U 2012 Soft Matter 8 2714Google Scholar

    [20]

    Harris S J, Timmons A, Baker D R, Monroe C 2010 Chem. Phys. Lett. 485 265Google Scholar

    [21]

    Sieminskas L, Ferguson M, Zerda T W, Couch E 1997 J. Sol-Gel Sci. Technol. 8 1105Google Scholar

    [22]

    Kafri K, Lubensky D K, Nelson D R 2004 Biophys. J. 86 3373Google Scholar

    [23]

    Smith P, Morrison I E G, Wilson K M, Fernandez N, Cherry R J 1999 Biophys. J. 76 3331Google Scholar

    [24]

    Hyeon C, Thirumalai D 2003 Proc. Natl. Acad. Sci. U.S.A. 100 10249Google Scholar

    [25]

    Reimann P, Van den Broeck C, Linke H, Hänggi P, Rub J M, Perez-Madrid A 2001 Phys. Rev. Lett. 87 010602Google Scholar

    [26]

    Reimann P, Van den Broeck C, Linke H, Hänggi P, Rub J M, Perez-Madrid A 2002 Phys. Rev. E 65 031104Google Scholar

    [27]

    Evstigneev M, Zvyagolskaya O, Bleil S, Eichhorn R, Bechinger C, Reimann P 2008 Phys. Rev. E 77 041107Google Scholar

    [28]

    Lindner B 2010 New J. Phys. 12 063026Google Scholar

    [29]

    Simon M S, Sancho J M, Lacasta A M 2012 Fluctuation Noise Lett. 11 1250026Google Scholar

    [30]

    Simon M S, Sancho J M, Lindenberg K 2013 Phys. Rev. E 88 062105Google Scholar

    [31]

    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111页

    Bao J D 2009 Random Simulation Method of Classical and Quantum Dissipation System (Beijing: Science Press) p111 (in Chinese)

    [32]

    Hu M, Bao J D 2018 Phys. Rev. E 97 062143Google Scholar

    [33]

    Bao J D, Liu J 2013 Phys. Rev. E 88 022153Google Scholar

    [34]

    Reimann P, Eichhorn R 2008 Phys. Rev. Lett. 101 180601Google Scholar

    [35]

    Nixon G I, Slater G W 1996 Phys. Rev. E 53 4969Google Scholar

    [36]

    Yeh D C, Huntington H B 1984 Phys. Rev. Lett. 53 1469Google Scholar

    [37]

    Coh S, Gannett W, Zettl A, Cohen M L, Louie S G 2013 Phys. Rev. Lett. 110 185901Google Scholar

    [38]

    Berkovich R, Garcia-Manes S, Urbakh M, Klafter J, Fernandez J M 2010 Biophys. J. 98 2692Google Scholar

    [39]

    Smith S B, Cui Y J, Bustamante C 1996 Science 271 795Google Scholar

    [40]

    Liphardt J, Onoa B, Smith S B, Tinoco Jr. I, Bustamante C 2001 Science 292 733Google Scholar

  • 图 1  偏压OU-RCP及OU的空间导数RCP示意图. 参数选取为˜λ=0.5, ˜D=0.1, ˜F=0.8

    Figure 1.  The schematic diagram of the biased OU-RCP and the derivative of OU-RCP. The parameters used are ˜λ=0.5, ˜D=0.1, ˜F=0.8.

    图 2  偏压随机势中粒子的有效扩散系数Deff˜F的变化. 这里比较了OU-RCP和OU的导数RCP中的结果. 内图: 继续增大˜F, OU的导数RCP对应的绿色方块曲线的变化趋势. 参数选取为˜λ=0.5, ˜D=0.1

    Figure 2.  Dependence of the effective diffusion coefficient Deff on the biased force ˜F in ˜Vbr. Here, the results of OU-RCP and OU’s derivative RCP are compared. Illustration: The trend of the green square curve when continuing to increase ˜F. The parameters used are ˜λ=0.5, ˜D=0.1.

    图 3  分别叠加OU-RCP, OU的导数RCP的偏压随机势中粒子的概率密度分布函数. 内图: 叠加OU-RCP, OU的导数RCP的偏压随机势Vbr的示意图. 参数选取为˜λ=0.5, ˜D=0.1, ˜F=10.0

    Figure 3.  The PDF of a particle in Vbr, the OU-RCP and OU’s derivative RCP are considered. Illustration: the schematic diagram of Vbr. The parameters used are ˜λ=0.5, ˜D=0.1, ˜F=10.0

    图 4  3种势˜Vbr, ˜Vbpr˜Vbp中粒子的有效扩散系数Deff作为偏压力˜F的函数. 比较了解析和模拟结果. 参数选取为˜λ=0.5, ˜D=0.1

    Figure 4.  The effective diffusion coefficient Deff of a particle as a function of the biased force ˜F in ˜Vbr, ˜Vbpr and ˜Vbp. The analytical result and simulation result are compared. The parameters used are ˜λ=0.5, ˜D=0.1.

    图 5  ˜F=1.0时, ˜Vbp, ˜Vbr˜Vbpr中粒子的概率密度分布函数((a)(c)); (d) ˜F=1.7(图4的红线加三角形曲线的最大值对应的偏压力)时, ˜Vbpr中粒子的概率密度分布函数. 内图: ˜F=1.0时的˜Vbr, ˜Vbpr示意图. 参数选取为˜λ=0.5, ˜D=0.1

    Figure 5.  The PDF corresponding to ˜Vbp, ˜Vbr and ˜Vbpr for ˜F=1.0 ((a)–(c)); (d) the PDF of particle in ˜Vbpr for ˜F=1.7 (the optimal biased force for ˜Vbpr in Fig. 4). Illustration: the schematic diagram of ˜Vbr, ˜Vbpr for ˜F=1.0. The parameters used are ˜λ=0.5, ˜D=0.1.

    表 1  3种势结构下粒子的有效动力学温度kBT及有效阻尼γ随偏压力的变化

    Table 1.  The effective kinetic temperature kBT and effective friction γ of a particle under the three potential structures change with the biased force.

    ˜F=0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
    偏压周期势 kBTγ 0.2095.12 0.214.11 0.450.47 0.421.06 0.321.48 0.281.3 0.251.2 0.241.17 0.231.09
    偏压周期随机势 kBTγ 0.20360.32 0.21200.62 0.228.56 0.360.22 1.090.06 0.620.07 0.320.84 0.310.60 0.231.31
    偏压随机势 kBTγ 0.2077.43 0.2210.28 0.280.83 0.620.13 1.090.08 0.340.09 0.251.21 0.260.67 0.271.30
    DownLoad: CSV
  • [1]

    Su Y, Ma X G, Lai P, Tong P 2017 Soft Matter 13 4773Google Scholar

    [2]

    Lindenberg K, Sancho J M, Khoury M, Lacasta A M 2012 Fluctuation Noise Lett. 11 1240004Google Scholar

    [3]

    Zwanzig R 1988 Proc. Natl. Acad. Sci. U.S.A. 85 2029Google Scholar

    [4]

    Lifson S, Jackson J L 1962 J. Chem. Phys. 36 2410Google Scholar

    [5]

    Longobardi L, Massarotti D, Rotoli G, Stornaiuolo D, Papari G, Kawakami A, Pepe G P, Barone A, Tafuri F 2011 Phys. Rev. B 84 184504Google Scholar

    [6]

    Fulde P, Pietronero L, Schneider W R, Strässler S 1975 Phys. Rev. Lett. 35 1776Google Scholar

    [7]

    Kurrer C, Schulten K 1995 Phys. Rev. E 51 6213Google Scholar

    [8]

    Reimann P 2002 Phys. Rep. 361 57Google Scholar

    [9]

    Lee S H, Grier D G 2006 Phys. Rev. Lett. 96 190601Google Scholar

    [10]

    Blickle V, Speck T, Seifert U, Bechinger C 2007 Phys. Rev. E 75 060101Google Scholar

    [11]

    Schiavoni M, Sanchez-Palencia L, Renzoni F, Grynberg G 2003 Phys. Rev. Lett. 90 094101Google Scholar

    [12]

    Lü K, Bao J D 2007 Phys. Rev. E 76 061119Google Scholar

    [13]

    Shi X Y, Bao J D 2019 Physica A 514 203Google Scholar

    [14]

    Gleeson J P, Sancho J M, Lacasta A M, Lindenberg K 2006 Phys. Rev. E 73 041102Google Scholar

    [15]

    Khoury M, Lacasta A M, Sancho J M, Lindenber K 2011 Phys. Rev. Lett. 106 090602Google Scholar

    [16]

    Slutsky M, Kardar M, Mirny L A 2004 Phys. Rev. E 69 061903Google Scholar

    [17]

    Bouchaud J P, Georges A 1990 Phys. Rep. 195 127Google Scholar

    [18]

    Hänggi P, Talkner P, Borkovec M 1990 Rev. Mod. Phys. 62 251Google Scholar

    [19]

    Hanes R D L, Dalle-Ferrier C, Schmiedeberg M, Jenkins M C, Egelhaaf S U 2012 Soft Matter 8 2714Google Scholar

    [20]

    Harris S J, Timmons A, Baker D R, Monroe C 2010 Chem. Phys. Lett. 485 265Google Scholar

    [21]

    Sieminskas L, Ferguson M, Zerda T W, Couch E 1997 J. Sol-Gel Sci. Technol. 8 1105Google Scholar

    [22]

    Kafri K, Lubensky D K, Nelson D R 2004 Biophys. J. 86 3373Google Scholar

    [23]

    Smith P, Morrison I E G, Wilson K M, Fernandez N, Cherry R J 1999 Biophys. J. 76 3331Google Scholar

    [24]

    Hyeon C, Thirumalai D 2003 Proc. Natl. Acad. Sci. U.S.A. 100 10249Google Scholar

    [25]

    Reimann P, Van den Broeck C, Linke H, Hänggi P, Rub J M, Perez-Madrid A 2001 Phys. Rev. Lett. 87 010602Google Scholar

    [26]

    Reimann P, Van den Broeck C, Linke H, Hänggi P, Rub J M, Perez-Madrid A 2002 Phys. Rev. E 65 031104Google Scholar

    [27]

    Evstigneev M, Zvyagolskaya O, Bleil S, Eichhorn R, Bechinger C, Reimann P 2008 Phys. Rev. E 77 041107Google Scholar

    [28]

    Lindner B 2010 New J. Phys. 12 063026Google Scholar

    [29]

    Simon M S, Sancho J M, Lacasta A M 2012 Fluctuation Noise Lett. 11 1250026Google Scholar

    [30]

    Simon M S, Sancho J M, Lindenberg K 2013 Phys. Rev. E 88 062105Google Scholar

    [31]

    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111页

    Bao J D 2009 Random Simulation Method of Classical and Quantum Dissipation System (Beijing: Science Press) p111 (in Chinese)

    [32]

    Hu M, Bao J D 2018 Phys. Rev. E 97 062143Google Scholar

    [33]

    Bao J D, Liu J 2013 Phys. Rev. E 88 022153Google Scholar

    [34]

    Reimann P, Eichhorn R 2008 Phys. Rev. Lett. 101 180601Google Scholar

    [35]

    Nixon G I, Slater G W 1996 Phys. Rev. E 53 4969Google Scholar

    [36]

    Yeh D C, Huntington H B 1984 Phys. Rev. Lett. 53 1469Google Scholar

    [37]

    Coh S, Gannett W, Zettl A, Cohen M L, Louie S G 2013 Phys. Rev. Lett. 110 185901Google Scholar

    [38]

    Berkovich R, Garcia-Manes S, Urbakh M, Klafter J, Fernandez J M 2010 Biophys. J. 98 2692Google Scholar

    [39]

    Smith S B, Cui Y J, Bustamante C 1996 Science 271 795Google Scholar

    [40]

    Liphardt J, Onoa B, Smith S B, Tinoco Jr. I, Bustamante C 2001 Science 292 733Google Scholar

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Publishing process
  • Received Date:  06 December 2020
  • Accepted Date:  19 May 2021
  • Available Online:  16 September 2021
  • Published Online:  05 October 2021

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