Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields.
Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields.
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空间不均匀转移概率作为统计均匀随机速度场中扩散的记忆效应。
DOI：
10.1103/physreve.81.056301
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
N. Suciu
中科院分区：
文献类型：
--
作者：
N. Suciu
每当人们使用平移不变均值格林函数来描述均值行为并估计随机速度场中扩散的色散系数时，就隐含地假设了传输过程的转移概率的空间均匀性。如果除了空间随机速度场的统计均匀性之外，还保证输运方程唯一经典解的存在，则可以在确定性初始条件下证明该性质。当唯一性条件失败并且无法假设均值格林函数的平移不变性时，如具有指数相关性的随机速度场的非光滑样本的情况，仍然可以使用 Itô 方程在替代方法中估计渐近色散系数。数值模拟证实了系数的预测渐近行为，但它们也显示了它们对早期初始条件的依赖性，这是非齐次转移概率的特征。这种记忆效应与随机初始条件更为相关，随机初始条件是相关速度场中扩散过程过去演化的结果，并且在幂律相关的情况下它们会无限期地持续存在。研究发现，只有存在长期正常扩散极限时，连续时间的转移概率才能在空间上均匀。此外，当确定性或随机初始状态的转移概率在空间上均匀时，它们可以明确地写为高斯分布。
Whenever one uses translation invariant mean Green's functions to describe the behavior in the mean and to estimate dispersion coefficients for diffusion in random velocity fields, the spatial homogeneity of the transition probability of the transport process is implicitly assumed. This property can be proved for deterministic initial conditions if, in addition to the statistical homogeneity of the space-random velocity field, the existence of unique classical solutions of the transport equations is ensured. When uniqueness condition fails and translation invariance of the mean Green's function cannot be assumed, as in the case of nonsmooth samples of random velocity fields with exponential correlations, asymptotic dispersion coefficients can still be estimated within an alternative approach using the Itô equation. Numerical simulations confirm the predicted asymptotic behavior of the coefficients, but they also show their dependence on initial conditions at early times, a signature of inhomogeneous transition probabilities. Such memory effects are even more relevant for random initial conditions, which are a result of the past evolution of the process of diffusion in correlated velocity fields, and they persist indefinitely in case of power law correlations. It was found that the transition probabilities for successive times can be spatially homogeneous only if a long-time normal diffusion limit exits. Moreover, when transition probabilities, for either deterministic or random initial states, are spatially homogeneous, they can be explicitly written as Gaussian distributions.