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arXiv 1410.6340cond-mat.stat-mech
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具有热涨落的非线性扩散方程的有限元离散

Finite element discretization of non-linear diffusion equations with thermal fluctuations

J. A. de la Torre, Pep Español, Aleksandar Donev

摘要:

我们提出了一个有限元离散的非线性扩散方程中使用的临界现象领域,最近,在动态密度泛函理论的背景下。离散化的方程保持了连续方程的结构。具体地说,它保持了粒子的总数,并满足了原始偏微分方程的H定理。在粗粒化理论的指导下,讨论了非线性扩散方程中热涨落的引入。这揭示了这样一个波动的流体力学方程的意义和所涉及的近似的局限性。在有限元方法中引入热涨落的方法是通用的,适用于任意尺寸的规则和不规则网格。在这里,我们专注于模拟的Ginzburg-Landau自由能功能,使用规则和不规则的一维网格。随着网格分辨率的增加,静、动结构因子的数值结果收敛。

英文摘要:

We present a finite element discretization of a non-linear diffusion equation used in the field of critical phenomena and, more recently, in the context of Dynamic Density Functional Theory. The discretized equation preserves the structure of the continuum equation. Specifically, it conserves the total number of particles and fulfills an H-theorem as the original partial differential equation. Guided by the Theory of Coarse-Graining, we discuss the inclusion of thermal fluctuations in the non-linear diffusion equation. This sheds light on the meaning of such a fluctuating hydrodynamics equation and to the limitations of the approximations involved. The methodology proposed for the introduction of thermal fluctuations in finite element methods is general and valid for both regular and irregular grids in arbitrary dimensions. We focus here on simulations of the Ginzburg-Landau free energy functional using both regular and irregular 1D grids. Convergence of the numerical results is obtained for the static and dynamic structure factors as the resolution of the grid is increased.

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