Transitions in stochastic non-equilibrium systems: Efficient reduction and analysis
Transitions in stochastic non-equilibrium systems: Efficient reduction and analysis
复制标题
DOI：
10.1016/j.jde.2022.11.025
复制
发表时间：
2022-02
影响因子：
2.4
通讯作者：
M. Chekroun;Honghu Liu;J. McWilliams;Shouhong Wang
中科院分区：
文献类型：
--
作者：
M. Chekroun;Honghu Liu;J. McWilliams;Shouhong Wang
物理学的一个核心挑战是描述由随机性驱动的非平衡系统，例如随机生长的界面，或受到随机波动影响的流体，例如随机波动。用于流体中与速度和温度梯度无关的局部应力和热通量。对于具有无限多个自由度的确定性系统，规范形式和中心流形理论已经显示出惊人的效率，通常可以完全描述线性不稳定的开始如何转化为与真正的物理状态相关的非线性模式的出现。然而，在存在随机波动的情况下，由于噪声引起的大偏移，中心流形的基本约简原理受到严重挑战，需要重新审视该方法。在本研究中，我们提出了一种替代框架来解决这些困难，一方面利用随机不变流形的逼近理论，另一方面利用能量估计测量高模参数化的缺陷。为了解决受随机搅拌力影响的流体问题，这些误差估计是在关于高模带来的耗散效应的假设下得出的，以便适当地平衡由于非线性项引起的规律性损失。因此，该方法使我们能够从随机流体问题的简化方程中预测，只要噪声的强度和轻度不稳定模态的特征值大小相应地发生，就会以很大的概率出现干草叉分叉的随机模拟。其轻度不稳定模式，我们的参数化公式表明噪声通过非马尔可夫系数传递到该模式，并且简化方程仅由后者随机驱动。这些系数明确取决于噪声路径的历史，并且它们的记忆内容由随机力的强度及其通过 SPDE 非线性项的相互作用自洽确定。详细介绍了随机瑞利-贝纳德问题的应用，阐明了随机干草叉分岔（大概率）发生的条件。
A central challenge in physics is to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes in the fluid which are not related to the velocity and temperature gradients. For deterministic systems with infinitely many degrees of freedom, normal form and center manifold theory have shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns, associated with genuine physical regimes. However, in presence of random fluctuations, the underlying reduction principle to the center manifold is seriously challenged due to large excursions caused by the noise, and the approach needs to be revisited.In this study, we present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds, on one hand, and energy estimates measuring the defect of parameterization of the high-modes, on the other. To operate for fluid problems subject to stochastic stirring forces, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes in order to suitably counterbalance the loss of regularity due to the nonlinear terms. As a result, the approach enables us to predict, from reduced equations of the stochastic fluid problem, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation, as long as the noise's intensity and the eigenvalue's magnitude of the mildly unstable mode scale accordingly.In the case of SPDEs forced by a multiplicative noise in the orthogonal subspace of e.g. its mildly unstable mode, our parameterization formulas show that the noise gets transmitted to this mode via non-Markovian coefficients, and that the reduced equation is only stochastically driven by the latter. These coefficients depend explicitly on the noise path's history, and their memory content is self-consistently determined by the intensity of the random force and its interaction through the SPDE's nonlinear terms. Applications to a stochastic Rayleigh-Bénard problem are detailed, for which conditions for a stochastic pitchfork bifurcation (in large probability) to occur, are clarified.