随机非平衡系统中的跃迁：有效约化与分析
Transitions in Stochastic Non-equilibrium Systems: Efficient Reduction and Analysis
摘要：
物理学中的一个核心挑战是描述由随机性驱动的非平衡系统，例如随机增长的界面，或受随机波动影响的流体，例如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，标准形和中心流形理论已经显示出惊人的效率，通常完全表征线性不稳定性的开始如何转化为非线性模式的出现。然而，在存在随机波动的情况下，由于噪声引起的大偏移，这种减少过程受到严重挑战，并且需要重新审视该方法。我们提出了一个替代框架，以应付这些困难，利用随机不变流形和能量估计的逼近论测量的缺陷参数化的高模式。为了解决流体问题，这些误差估计是在假设高模带来的耗散效应的情况下得出的，高模适当地抵消了由于非线性项而导致的规律性损失。该方法使我们能够从简化方程预测大概率出现干草叉分叉的随机模拟，只要噪声强度和轻度不稳定模式的本征值大小相应地缩放即可。我们的参数化公式涉及非马尔可夫系数，这明确地取决于驱动SPDE动力学的噪声路径的历史，并且它们的记忆内容是自洽地由随机力的强度及其通过SPDE的非线性项的相互作用确定的。应用程序的随机瑞利-贝纳德问题的详细说明，其中的随机干草叉分叉（在大概率）发生的条件，澄清。
英文摘要：
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 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. However, in presence of random fluctuations, this reduction procedure is seriously challenged due to large excursions caused by the noise, and the approach needs to be revisited. We present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds and energy estimates measuring the defect of parameterization of the high-modes. To operate for fluid problems, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes that suitably counterbalance the loss of regularity due to the nonlinear terms. The approach enables us to predict, from the reduced equations, 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. Our parameterization formulas involve non-Markovian coefficients, which depend explicitly on the history of the noise path that drives the SPDE dynamics, 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-Benard problem are detailed, for which conditions for a stochastic pitchfork bifurcation (in large probability) to occur, are clarified.
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