| 57 |
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viscosity) in a fluid by imposing an artificial momentum flux between |
| 58 |
|
two thin parallel slabs of material that are spatially separated in |
| 59 |
|
the simulation cell.\cite{MullerPlathe:1997xw,Muller-Plathe:1999ek} The |
| 60 |
< |
artificial flux is typically created by periodically "swapping" either |
| 60 |
> |
artificial flux is typically created by periodically ``swapping'' either |
| 61 |
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the entire momentum vector $\vec{p}$ or single components of this |
| 62 |
|
vector ($p_x$) between molecules in each of the two slabs. If the two |
| 63 |
|
slabs are separated along the z coordinate, the imposed flux is either |
| 222 |
|
One may also define momentum flux (say along the x-direction) as: |
| 223 |
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\begin{equation} |
| 224 |
|
(1-x) P_c^x = j_z(p_x) |
| 225 |
+ |
\label{eq:fluxPlane} |
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\end{equation} |
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+ |
The above {\it flux equation} is essentially a plane which is |
| 228 |
+ |
perpendicular to the x-axis, with its position governed both by a |
| 229 |
+ |
targetted value, $j_z(p_x)$ as well as the instantaneous value of the |
| 230 |
+ |
momentum along the x-direction. |
| 231 |
+ |
|
| 232 |
+ |
Similarly, to satisfy a momentum flux as well as the conservation |
| 233 |
+ |
constraints, it is sufficient to determine the points ${x,y,z}$ which |
| 234 |
+ |
lie on both the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) |
| 235 |
+ |
and the flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of |
| 236 |
+ |
an ellipsoid and a plane in 3-dimensional space. |
| 237 |
+ |
|
| 238 |
+ |
To summarize, by solving respective equation sets, one can determine |
| 239 |
+ |
possible sets of scaling variables for cold slab. And corresponding |
| 240 |
+ |
sets of scaling variables for hot slab can be determine as well. |
| 241 |
+ |
|
| 242 |
+ |
The following problem will be choosing an optimal set of scaling |
| 243 |
+ |
variables among the possible sets. Although this method is inherently |
| 244 |
+ |
non-isotropic, the goal is still to maintain the system as isotropic |
| 245 |
+ |
as possible. Under this consideration, one would like the kinetic |
| 246 |
+ |
energies in different directions could become as close as each other |
| 247 |
+ |
after each scaling. Simultaneously, one would also like each scaling |
| 248 |
+ |
as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
| 249 |
+ |
large perturbation to the system. Therefore, one approach to obtain the |
| 250 |
+ |
scaling variables would be constructing an criteria function, with |
| 251 |
+ |
constraints as above equation sets, and solving the function's minimum |
| 252 |
+ |
by method like Lagrange multipliers. |
| 253 |
|
|
| 254 |
+ |
In order to save computation time, we have a different approach to a |
| 255 |
+ |
relatively good set of scaling variables with much less calculation |
| 256 |
+ |
than above. Here is the detail of our simplification of the problem. |
| 257 |
|
|
| 258 |
+ |
In the case of kinetic energy transfer, we impose another constraint |
| 259 |
+ |
${x = y}$, into the equation sets. Consequently, there are two |
| 260 |
+ |
variables left. And now one only needs to solve a set of two {\it |
| 261 |
+ |
ellipses equations}. This problem would be transformed into solving |
| 262 |
+ |
one quartic equation for one of the two variables. There are known |
| 263 |
+ |
generic methods that solve real roots of quartic equations. Then one |
| 264 |
+ |
can determine the other variable and obtain sets of scaling |
| 265 |
+ |
variables. Among these sets, one can apply the above criteria to |
| 266 |
+ |
choose the best set, while much faster with only a few sets to choose. |
| 267 |
|
|
| 268 |
+ |
In the case of momentum flux transfer, we impose another constraint to |
| 269 |
+ |
set the kinetic energy transfer as zero. In another word, we apply |
| 270 |
+ |
Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. After that, with one |
| 271 |
+ |
variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar set |
| 272 |
+ |
of equations on the above kinetic energy transfer problem. Therefore, |
| 273 |
+ |
an approach similar to the above would be sufficient for this as well. |
| 274 |
|
|
| 275 |
+ |
\section{Computational Details} |
| 276 |
|
|
| 277 |
|
\section{Acknowledgments} |
| 278 |
|
Support for this project was provided by the National Science |
