User:Egm6936.f11.qiandeng/topic

back to main page = Discussion on Atomistic Field Theory (AFT)=

Local density function
Atomistic Field Theory is a generalized continuum theory. As shown in the following picture, each material point is treated as an infinitesimal point in classical continuum mechanics; in the later developed micromorphic theory, the material point is not longer an infinitesimal rigid point but a deformable particle; AFT views a crystalline material as a continuous collection of lattice points, while embedded within each point is a group of discrete atoms.



The atomic position is decomposed into lattice position and the relative internal position:


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$$ \displaystyle
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{{\mathbf {R}}^{k\alpha }}={{\mathbf{R}}^{k}}+\Delta {{\mathbf{r}}^{k\alpha }}

$$     (1)
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where $${{\mathbf{R}}^{k\alpha }}$$ is the position of the $$\displaystyle \alpha -th$$ atom in the $$\displaystyle {k-th}$$ unit cell, $${{\mathbf{R}}^{k}}$$ is the position of the $$\displaystyle {k-th}$$ unit cell, and $$\Delta {{\mathbf{r}}^{k\alpha }}$$ is the relative position between the $$\displaystyle \alpha -th$$ atom and the centre of the $$\displaystyle {k-th}$$ unit cell. The correspondence of this decomposition in the physical space is:


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$$ \displaystyle
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\mathbf{z}=\mathbf{x}+\mathbf{y}

$$     (2)
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where $$\mathbf{z, x, y}$$ are the counterparts of $${{\mathbf{R}}^{k\alpha }}$$, $${{\mathbf{R}}^{k}}$$ and $$\Delta {{\mathbf{r}}^{k\alpha }}$$ , respectively.

The link between the local density function in physical space of any measurable dynamics function $$\mathbf{A}(\mathbf{r,p})$$ in phase space is established as


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$$ \displaystyle \begin{align} & \mathbf{a}(\mathbf{z},t)=\sum\limits_{k=1}^{N}{\sum\limits_{\xi =1}^{n} – A\left( \mathbf{r}(t)\mathbf{,p}(t) \right)\delta ({{\mathbf{R}}^{k\xi }}}-\mathbf{z}) \\ & \text{         }=\sum\limits_{k=1}^{N}{\sum\limits_{\xi =1}^{n} – A\left( \mathbf{r}(t)\mathbf{,p}(t) \right)\delta ({{\mathbf{R}}^{k}}+\Delta {{\mathbf{r}}^{k\xi }}}-\mathbf{x}-\mathbf{y}) \\ & \text{         }=\mathbf{a}(\mathbf{x},\mathbf{y},t) \\ \end{align}$$ (3)
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where N is the number of lattice cells and n is the number of atoms within a lattice cell; $$\mathbf{a}(\mathbf{x},\mathbf{y},t)$$ or $$\mathbf{a}(\mathbf{z},t)$$ is the local density function corresponds to $$\mathbf{A}(\mathbf{r,p})$$; $$\delta (\cdot )$$ is the localization function which can be a Dirac $$\alpha$$-function or a distribution of weighting function and has a unit of inverse volume. Note that the function $$\mathbf{a}(\mathbf{x},\mathbf{y},t)$$ is continuous for both x and y. The cell-averaged continuously distributed local density function is defined as:


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$$ \displaystyle \begin{align} & \bar{a}(\mathbf{x},t)=\int_{V(\mathbf{x})}{\sum\limits_{k=1}^{N}{\sum\limits_{\xi =1}^{n}{a(\mathbf{x},\mathbf{y},t)dy}}}/V(\mathbf{x}) \\ & \text{         }=\int_{V(\mathbf{x})}{\sum\limits_{k=1}^{N}{\sum\limits_{\xi =1}^{n}{A\left( \mathbf{r}(t)\mathbf{,p}(t) \right)\delta ({{\mathbf{R}}^{k}}+\Delta {{\mathbf{r}}^{k\xi }}-\mathbf{x}-\mathbf{y})}}d\mathbf{y}}/V(\mathbf{x}) \\ & \text{         }=\int_{V(\mathbf{x})}{\sum\limits_{k=1}^{N}{\sum\limits_{\xi =1}^{n}{A\left( \mathbf{r}(t)\mathbf{,p}(t) \right)\delta ({{\mathbf{R}}^{k}}+\Delta {{\mathbf{r}}^{k\xi }}-\mathbf{x}-\mathbf{y})}}d\mathbf{y}}\delta ({{\mathbf{R}}^{k}}-\mathbf{x}) \\ & \text{         }=\int_{V(\mathbf{x})}{d\mathbf{y}\sum\limits_{k=1}^{N}{\sum\limits_{\xi =1}^{n}{A\left( \mathbf{r}(t)\mathbf{,p}(t) \right)\delta ({{\mathbf{R}}^{k}}-\mathbf{x})\delta (\Delta {{\mathbf{r}}^{k\xi }}-\mathbf{y})}}} \\ \end{align}$$ (4)
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In the above equation, $$V(\mathbf{x})$$ is the volume of the lattice cell that is located at spatial place x. The condition $$1/V(\mathbf{x})=\sum\limits_{k=1}^{N}{\delta ({{\mathbf{R}}^{k}}-\mathbf{x})}$$ is derived from the conditions:


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$$ \displaystyle \int\limits_{V}{\delta ({{\mathbf{R}}^{k}}-\mathbf{x})d\mathbf{x}}=1\text{  (}k =1, 2, 3, ...N\text{)} $$     (5)
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Discussion on $$\displaystyle\delta$$ -funciont
For $$\delta\,$$ -funciont:


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$$ \displaystyle \delta (x-y)=\left\{ \begin{matrix} 1\text{  if   }x=y  \\ 0\text{  if   }x\ne y  \\ \end{matrix} \right.$$ (6)
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Since:


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$$ \displaystyle a+b=c+d \mbox{ and  } a=c \qquad\Rightarrow\qquad a=c \mbox{  and  } b=d $$     (7)
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So we have:


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$$ \displaystyle \mathbf\delta [(a+b)-(c+d)]\delta (a-c)\qquad\Rightarrow\qquad \mathbf\delta (a-c)\delta (b-d) $$     (8)
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For this reason, we can write the term


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$$ \displaystyle \delta ({{\mathbf{R}}^{k}}+\Delta {{\mathbf{r}}^{k\xi }}-\mathbf{x}-\mathbf{y})\delta ({{\mathbf{R}}^{k}}-\mathbf{x}) $$     (9)
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into


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$$ \displaystyle \delta ({{\mathbf{R}}^{k}}-\mathbf{x})\delta (\Delta {{\mathbf{r}}^{k\xi }}-\mathbf{y})$$ (10)
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Density function
Professor Loc Vu-Quoc suggested the following way to derive the average density function $$a(\mathbf{z},t)\,$$ over an arbitrary volume.

For an arbitrary volume V, the mapping of a quantity $$A(\mathbf{r},t)\,$$ from $$(\mathbf{r},t)\,$$ space to $$(\mathbf{z},t)\,$$ space is:


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$$ \displaystyle A(\mathbf{z},t)=\int_{V}{A(\mathbf{r},t)w(\mathbf{r},z)d\mathbf{r}} $$     (11)
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where $$\int_{V}{w(\mathbf{r},\mathbf{z})d\mathbf{r}}=1$$

Then the average density of this quantity over the volume V is defined as:


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$$ \displaystyle a(\mathbf{z},t)=\frac{A(\mathbf{z},t)}=\frac{1}\int_{V}{A(\mathbf{r},t)w(\mathbf{r},\mathbf{z})d\mathbf{r}} $$     (12)
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Checking dimensions:


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$$ \displaystyle [w(\mathbf{z},\mathbf{r})]=\frac{1}{[V]}\Rightarrow [A(\mathbf{z},t)]=\frac{[A(\mathbf{r},t)][V]}{[V]}=[A(\mathbf{r},t)] $$     (13)
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$$ \displaystyle [a(z,t)]=\frac{[A(\mathbf{z},t)]}{[V]}=\frac{[A(\mathbf{r},t)]}{[V]} $$     (14)
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A useful special case of the above density is the unit cell average density:


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$$ \displaystyle\begin{align} & {{a}^{k}}(\mathbf{z},t)=\frac{A(\mathbf{z},t)}=\frac{1}\int_{A(\mathbf{r},t)w(\mathbf{r},\mathbf{z})d\mathbf{r}} \\ & \text{          }=\frac{1}\sum\limits_{\alpha =1}^{A({{\mathbf{r}}^{\alpha }},t)w({{\mathbf{r}}^{\alpha }},\mathbf{z})\Delta V_{k}^{\alpha }} \\ & \text{          }=\sum\limits_{\alpha =1}^{A({{\mathbf{r}}^{\alpha }},t)w({{\mathbf{r}}^{\alpha }},\mathbf{z})\frac{\Delta V_{k}^{\alpha }}} \\ \end{align} $$     (15)
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where $${{a}^{k}}(\mathbf{z},t)\,$$ is the cell average density of $$A(\mathbf{r},t)\,$$, $$ V^{k}\,$$ is the volume of an unit cell, $$ N_{}^\alpha\,$$ is the number of atoms in an unit cell, and $$\Delta V_{k}^{\alpha }\;$$ is the volume of the α-th atom in the k-th unit cell.

Obviously, the dimension of $${{a}^{k}}(\mathbf{z},t)\,$$ is $$\frac{[A(\mathbf{r},t)]}{[V]}\,$$