【作者】 李勇；

【导师】 梅冬成；

【作者基本信息】 云南大学 ， 理论物理， 2018， 硕士

【摘要】 本学位论文我们主要研究非均匀环境中Lévy噪声诱导的次扩散及其遍历性破缺。首先,我们对布朗运动、反常扩散、Lévy飞行以及遍历性理论的研究现状做了综述,并给出本文研究的意义。其次,我们通过朗之万方程详细推导了过阻尼自由Lévy飞行的分数阶福克–普朗克方程。最后,我们借助分数阶福克普朗克方程通过理论计算及数值模拟的方法研究了非均匀环境中Lévy噪声诱导的次扩散及其遍历性破缺情况,其主要的结果如下:（i）本文通过引入非均匀环境（扩散系数D（x）=D₀|x|^α随空间分布,D₀是原始扩散系数强度）。在适当的参数范围内,幂率指数α和Lévy指数μ的竞争导致粒子概率密度函数P（x,t）～1/（|x|μ/p+1）在有限时间内迅速下降（其中μ/p>2,p=2/（2-α））,最后粒子方均位移的系综平均发散问题也得到了有效的解决。（ii）我们观察到次扩散现象存在—粒子方均位移的系综平均{x²（t）}∝t^κ,即κ<1。同时反常扩散指数κ随着幂率指数α增大而增大,随Lévy指数μ基本保持不变。（iii）我们还观察到弱的遍历性破缺—方均位移的时间平均δ2与系综平均不等价,我们通过遍历性破缺参数EB、EB和时间平均的幅度标度?（ξ）,其中（?）,来量化系统的遍历性行为。对于1≤μ<2（排除μ=2的情况）,遍历性破缺增强随着幂率指数α的增大,然而随着随Lévy指数μ和原始扩散系数强度D₀的增大而减弱,并且幅度标度?（ξ）的峰值主要集中在ξ=0.7附近。（iv）通过线性拟合得到遍历性破缺参数EB满足渐近线形式（?）。最后,从上面得出的结果来看,非均匀环境与Lévy噪声的竞争可以从本质上调控系统的动力学行为。更多还原

【Abstract】 In this dissertation,we mainly investigate the anomalous diffusion and er-godicity breaking of Brownian particles in heterogeneous environments subject to Lévy noise.Firstly,we summarized the research status about the Brown movement,anomalous diffusion,Lévy flight and the ergodic theory,and we also introduced the significance of the study in this dissertation.Secondly,we made a detailed derivation of the fractional Fokker Planck equation（FFPE）via the the Langevin equation with the damping free Lévy flight.Finally,the subdiffusion and ergodicity breaking of Brownian particles in heterogeneous environments（space-dependent diffusion coefficient D（x）=D₀|x|^α,D₀is the original inten-sity of diffusion）subject to Lévy noise are investigated,by both the analytically theory and the numerically simulating schemes,the main results are as follows:（i）Within the tailored parameter,the competition of the power-law exponentαand the Lévy indexμensures the long tail of probability density function with the form P（x）～1/（|x|μ/p+1）with p=2/（2-α）,falling rapidly whenμ/p>2.（ii）we observe the subdiffusive motion that is characterised by the scale of ensemble av-eraged mean-square displacement（MSD）,{x²（t）}∝t^κwithκ<1.Meanwhile,the anomalous exponentsκincreases as the increasing of power-law exponentα,but that remains unchanged with Lévy indexμ.（iii）we also observe the weak ergodicity breaking—the inequivalence between individual time averaged MSDδ² and the corresponding ensemble averaged MSD.We quantify the weak ergod-icity breaking behaviour in terms of the dimensionless parameters EB,EB and the amplitude scatter（?）（ξ）with（?）.For 1≤μ<2（except forμ=2）,the ergodicity breaking enhances as theαincreases,but weakens asμor D₀,and we have the peaks of the amplitude scatter（?）（ξ）close toξ=0.7.（iv）The EB follows the from,（?）,in the large time.Finally,from the above,we find that the competition of heterogeneous environments and Lévy noise can substantially modulate the dynamic behavior of systems.更多还原