PHYS 5120 - 计算能源材料和电子结构模拟 Lecture-3

Lecturer: Prof.PAN DING

1 radial distribution function RDF静态结构:

• 内容: This whiteboard serves as an excellent summary, pulling together all the key concepts we’ve discussed into a single, cohesive picture. Let’s connect everything on this slide to our detailed conversation.

1. RDF: The Static Structure RDF静态结构

On the top left, you see RDF (Radial Distribution Function).

• The Plots: The board shows the familiar $g(r)$ plot with its characteristic peaks for a liquid. Below it is a plot of the interatomic potential energy, $V(r)$. This addition is very insightful! It shows why the first peak in $g(r)$ exists: it corresponds to the minimum energy distance ($\sigma$) where particles are most stable and likely to be found. 白板展示了我们熟悉的$g(r)$图，它带有液体的特征峰。下方是原子间势能$V(r)$的图。这个补充非常有见地！它解释了为什么 $g(r)$ 中的第一个峰值存在：它对应于粒子最稳定且最有可能被发现的最小能量距离 ($\sigma$)。
• Connection: This section summarizes our first discussion. It’s the starting point for our analysis—a static snapshot of the material’s average atomic arrangement before we consider how the atoms move. 本节总结了我们的第一个讨论。这是我们分析的起点——在我们考虑原子如何运动之前，它是材料平均原子排列的静态快照。

2. MSD and The Einstein Relation: The Displacement Picture 均方位移 (MSD) 和爱因斯坦关系：位移图像

The board then moves to dynamics, presenting two methods to calculate the diffusion constant, D. The first is the Einstein relation. 两种计算扩散常数 D的方法。第一种是爱因斯坦关系。

• The Formula: It correctly states that the Mean Squared Displacement (MSD), $\langle r^2 \rangle$, is equal to $6Dt$ in three dimensions. It then rearranges this to solve for $D$: 它正确地指出了均方位移 (MSD)，$\langle r^2 \rangle$，在三维空间中等于 $6Dt$。然后重新排列该公式以求解 $D$： \(D = \lim_{t\to\infty} \frac{\langle |\vec{r}(t) - \vec{r}(0)|^2 \rangle}{6t}\)
• The Diagram: The central diagram beautifully illustrates the concept. It shows a particle in a simulation box (with “N=108” likely being the number of particles simulated) moving from an initial position $\vec{r}_i(0)$ to a final position $\vec{r}_i(t_j)$. The MSD is the average of the square of this displacement over all particles and many time origins. The graph labeled “MSD” shows how you would plot this data and find the slope (“fitting”) to calculate $D$. 中间的图表完美地阐释了这个概念。它展示了一个粒子在模拟框中（“N=108” 可能是模拟粒子的数量）从初始位置 $\vec{r}_i(0)$ 移动到最终位置 $\vec{r}_i(t_j)$。MSD 是该位移平方在所有粒子和多个时间原点上的平均值。标有“MSD”的图表显示了如何绘制这些数据并找到斜率（“拟合”）来计算 $D$。
• Connection: This is a perfect summary of the “Displacement Picture” we analyzed on the second whiteboard. It’s the most intuitive way to think about diffusion: how far particles spread out over time.这完美地总结了我们在第二个白板上分析的“位移图”。这是思考扩散最直观的方式：粒子随时间扩散的距离。

3. The Green-Kubo Relation: The Fluctuation Picture 格林-久保关系：涨落图

Finally, the board presents the more advanced but often more practical method: the Green-Kubo relation.

• The Equations: This section displays the two key equations from our last discussion:
  1. The MSD as the double integral of the Velocity Autocorrelation Function (VACF). 速度自相关函数 (VACF) 的二重积分的均方差 (MSD)。
  2. The crucial derivative step: $\frac{d\langle x^2(t)\rangle}{dt} = 2 \int_0^t dt’’ \langle V_x(t) V_x(t’’) \rangle$. 关键的导数步骤：$\frac{d\langle x^2(t)\rangle}{dt} = 2 \int_0^t dt’’ \langle V_x(t) V_x(t’’) \rangle$。
• The Diagram: The small diagram of a square with axes $t’$ and $t’’$ visually represents the two-dimensional domain of integration for the double integral. 一个带有轴 $t’$ 和 $t’’$ 的小正方形图直观地表示了二重积分的二维积分域。
• Connection: This summarizes the “Fluctuation Picture.” It shows the mathematical heart of the derivation that proves the Einstein and Green-Kubo methods are equivalent. As we concluded, this method is often numerically superior because it involves integrating a rapidly decaying function (the VACF) rather than finding the slope of a noisy, unbounded function (the MSD). 这概括了“涨落图”。它展示了证明爱因斯坦方法和格林-久保方法等价的推导过程的数学核心。正如我们总结的那样，这种方法通常在数值上更胜一筹，因为它涉及对快速衰减函数（VACF）进行积分，而不是求噪声无界函数（MSD）的斜率。

In essence, this single whiteboard is a complete roadmap for analyzing diffusion in a molecular simulation. It shows how to first characterize the material’s structure ($g(r)$) and then how to compute its key dynamic property—the diffusion constant D—using two powerful, interconnected methods. 本质上，这块白板就是分子模拟中分析扩散的完整路线图。它展示了如何首先表征材料的结构（$g(r)$），然后如何使用两种强大且相互关联的方法计算其关键的动态特性——扩散常数 D。

This whiteboard beautifully concludes the derivation of the Green-Kubo relation, showing the final formulas and how they are used in practice. It provides the punchline to the mathematical story we’ve been following.

Let’s break down the details.

4. Finalizing the Derivation

The top lines of the board show the final step in connecting the Mean Squared Displacement (MSD) to the Velocity Autocorrelation Function (VACF).

\[\lim_{t\to\infty} \frac{d\langle x^2 \rangle}{dt} = 2 \int_0^\infty d\tau \langle V_x(0) V_x(\tau) \rangle\]

• The Left Side: As we know from the Einstein relation, the long-time limit of the derivative of the 1D MSD, $\lim_{t\to\infty} \frac{d\langle x^2 \rangle}{dt}$, is simply equal to $2D$.
• The Right Side: This is the result of the mathematical derivation from the previous slide. It shows that this same quantity is also equal to twice the total integral of the VACF.

By equating these two, we can solve for the diffusion coefficient, D.

5. The Velocity Autocorrelation Function (VACF)

The board explicitly names the key quantity here:

\[\Phi(\tau) = \langle V_x(0) V_x(\tau) \rangle\]

This is the “Velocity autocorrelation function” (abbreviated as VAF on the board), which we’ve denoted as VACF. The variable has been changed from t to τ (tau) to represent a “time lag” or interval, which is common notation.

• The Plot: The graph on the board shows a typical plot of the VACF, $\Phi(\tau)$, versus the time lag $\tau$.
  • It starts at a maximum positive value at $\tau=0$ (when the velocity is perfectly correlated with itself).
  • It rapidly decays towards zero as the particle undergoes collisions that randomize its velocity.
• The Integral: The shaded area under this curve represents the value of the integral $\int_0^\infty \Phi(\tau) d\tau$. The Green-Kubo formula states that the diffusion coefficient is directly proportional to this area.

6. The Green-Kubo Formulas for the Diffusion Coefficient

After canceling the factor of 2, the board presents the final, practical formulas for D.

• In 1 Dimension: \(D = \int_0^\infty d\tau \langle V_x(0) V_x(\tau) \rangle\)
• In 3 Dimensions: This is the more general and useful formula. \(D = \frac{1}{3} \int_0^\infty d\tau \langle \vec{v}(0) \cdot \vec{v}(\tau) \rangle\) There are two important changes for 3D:
  1. We use the full velocity vectors and their dot product, $\vec{v}(0) \cdot \vec{v}(\tau)$, to capture motion in all directions.
  2. We divide by 3 to get the average contribution to diffusion in any one direction (x, y, or z).

7. Practical Calculation in a Simulation

The last formula on the board shows how this is implemented in a computer simulation with a finite number of atoms.

\[D = \frac{1}{3N} \int_0^\infty d\tau \sum_{i=1}^{N} \langle \vec{v}_i(0) \cdot \vec{v}_i(\tau) \rangle\]

• $\sum_{i=1}^{N}$: This summation symbol indicates that you must compute the VACF for each individual atom (from atom i=1 to atom N).
• $\frac{1}{N}$: You then average the results over all N atoms in your simulation box.
• $\langle \dots \rangle$: The angle brackets here still imply an additional average over multiple different starting times (t=0) to get good statistics.

This formula is the practical recipe: to get the diffusion coefficient, you track the velocity of every atom, calculate each one’s VACF, average them together, and then integrate the result over time.