The Dean-Kawasaki equation - a strongly singular SPDE - is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of $N$ independent diffusing particles in the regime of large particle numbers $N\gg 1$. The singular nature of the Dean-Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions. In the present work, we give a rigorous and fully quantitative justification of the Dean-Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean-Kawasaki equation may approximate the density fluctuations of $N$ non-interacting diffusing particles to arbitrary order in $N^{-1}$ (in suitable weak metrics). In other words, the Dean-Kawasaki equation may be interpreted as a "recipe" for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.

中文翻译：

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Dean-Kawasaki 方程和扩散粒子系统中密度涨落的结构

Dean-Kawasaki 方程——一个强奇异的 SPDE——是波动流体动力学的基本方程；在物理文献中已经提出描述在大粒子数 $N\gg 1$ 范围内 $N$ 独立扩散粒子的密度波动。Dean-Kawasaki 方程的奇异性质对其分析和严格的数学论证提出了重大挑战。除了通过 Hairer 等人的规则结构理论不可重整化之外，最近还表明它甚至不承认非平凡的鞅解。在目前的工作中，我们通过考虑标准数值离散化提供的自然正则化，对 Dean-Kawasaki 方程给出了严格且完全定量的证明：我们表明，Dean-Kawasaki 方程的结构保持离散化可以将 $N$ 非相互作用扩散粒子的密度波动近似为 $N^{-1}$ 中的任意顺序（在合适的弱度量中）。换句话说，Dean-Kawasaki 方程可以解释为独立扩散粒子密度波动的精确和有效数值模拟的“配方”。