谱截断Burgers-Hopf动力学的非平衡统计模型
A nonequilibrium statistical model of spectrally truncated Burgers-Hopf dynamics
摘要：
非受迫无粘Burgers-Hopf方程的精确谱截断是具有多个自由度的Hamilton系统，具有内禀随机性和相干标度行为。出于这个原因，最近的研究采用这些系统作为原型来测试随机模式减少策略。本文用一种新的统计模型降阶技术处理截断为n阶Fourier模的Burgers-Hopf动力学，导出了m阶最低模平均值的封闭演化方程组（m << n）。在简化模型中的m-模式宏态与n-模式微观态的相空间上的尝试概率密度相关联，并且引入成本泛函来量化这些密度的路径对刘维尔方程的拟合不足。通过最小化路径上的代价泛函得到最佳拟合的宏观动力学，然后从Hamilton-Jacobi理论导出控制闭合的方程。得到的约化方程具有分数阶扩散和修正的非线性相互作用，且两者的显式形式在单个封闭参数下确定。这种非平衡封闭的准确性和有效性范围进行评估，通过比较对直接数值模拟的统计合奏，预测的行为被发现是很好地代表了减少方程。
英文摘要：
Exact spectral truncations of the unforced, inviscid Burgers-Hopf equation are Hamiltonian systems with many degrees of freedom which exhibit intrinsic stochasticity and coherent scaling behavior. For this reason recent studies have employed these systems as prototypes to test stochastic mode reduction strategies. In the present paper the Burgers-Hopf dynamics truncated to n Fourier modes is treated by a new statistical model reduction technique, and a closed system of evolution equations for the mean values of the m lowest modes is derived for m << n. In the reduced model the m-mode macrostates are associated with trial probability densities on the phase space of the n-mode microstates, and a cost functional is introduced to quantify the lack of fit of paths of these densities to the Liouville equation. The best-fit macrodynamics is obtained by minimizing the cost functional over paths, and the equations governing the closure are then derived from Hamilton-Jacobi theory. The resulting reduced equations have a fractional diffusion and modified nonlinear interactions, and the explicit form of both are determined up to a single closure parameter. The accuracy and range of validity of this nonequilibrium closure is assessed by comparison against direct numerical simulations of statistical ensembles, and the predicted behaviour is found to be well represented by the reduced equations.
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