Add Kokkos version of compute orientorder/atom

This commit is contained in:
Stan Moore
2020-02-24 13:27:40 -07:00
parent f0935feabe
commit 674781fe0e
322 changed files with 3229 additions and 2663 deletions

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@ -49,8 +49,8 @@ if(PKG_USER-PLUMED)
message(STATUS "PLUMED download requested - we will build our own") message(STATUS "PLUMED download requested - we will build our own")
include(ExternalProject) include(ExternalProject)
ExternalProject_Add(plumed_build ExternalProject_Add(plumed_build
URL https://github.com/plumed/plumed2/releases/download/v2.5.3/plumed-src-2.5.3.tgz URL https://github.com/plumed/plumed2/releases/download/v2.6.0/plumed-src-2.6.0.tgz
URL_MD5 de30d6e7c2dcc0973298e24a6da24286 URL_MD5 204d2edae58d9b10ba3ad460cad64191
BUILD_IN_SOURCE 1 BUILD_IN_SOURCE 1
CONFIGURE_COMMAND <SOURCE_DIR>/configure --prefix=<INSTALL_DIR> CONFIGURE_COMMAND <SOURCE_DIR>/configure --prefix=<INSTALL_DIR>
${CONFIGURE_REQUEST_PIC} ${CONFIGURE_REQUEST_PIC}

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@ -75,8 +75,11 @@ html: $(ANCHORCHECK)
@rm -rf html/_sources @rm -rf html/_sources
@rm -rf html/PDF @rm -rf html/PDF
@rm -rf html/USER @rm -rf html/USER
@rm -rf html/JPG
@cp -r src/PDF html/PDF @cp -r src/PDF html/PDF
@cp -r src/USER html/USER @cp -r src/USER html/USER
@mkdir -p html/JPG
@cp `grep -A2 '\.\. image::' src/*.rst | grep ':target:' | sed -e 's,.*:target: JPG/,src/JPG/,' | sort | uniq` html/JPG/
@rm -rf html/PDF/.[sg]* @rm -rf html/PDF/.[sg]*
@rm -rf html/USER/.[sg]* @rm -rf html/USER/.[sg]*
@rm -rf html/USER/*/.[sg]* @rm -rf html/USER/*/.[sg]*

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@ -1,4 +1,4 @@
.TH LAMMPS "4 February 2020" "2020-02-04" .TH LAMMPS "18 February 2020" "2020-02-18"
.SH NAME .SH NAME
.B LAMMPS .B LAMMPS
\- Molecular Dynamics Simulator. \- Molecular Dynamics Simulator.
@ -11,13 +11,18 @@ or
mpirun \-np 2 mpirun \-np 2
.B lmp .B lmp
<input file> [OPTIONS] ... \-in <input file> [OPTIONS] ...
or or
.B lmp .B lmp
\-r2data file.restart file.data \-r2data file.restart file.data
or
.B lmp
\-h
.SH DESCRIPTION .SH DESCRIPTION
.B LAMMPS .B LAMMPS
is a classical molecular dynamics code, and an acronym for \fBL\fRarge-scale is a classical molecular dynamics code, and an acronym for \fBL\fRarge-scale
@ -249,7 +254,7 @@ the chapter on errors in the
manual gives some additional information about error messages, if possible. manual gives some additional information about error messages, if possible.
.SH COPYRIGHT .SH COPYRIGHT
© 2003--2019 Sandia Corporation © 2003--2020 Sandia Corporation
This package is free software; you can redistribute it and/or modify This package is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License version 2 as it under the terms of the GNU General Public License version 2 as

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@ -880,6 +880,9 @@ USER-PLUMED package
Before building LAMMPS with this package, you must first build PLUMED. Before building LAMMPS with this package, you must first build PLUMED.
PLUMED can be built as part of the LAMMPS build or installed separately PLUMED can be built as part of the LAMMPS build or installed separately
from LAMMPS using the generic `plumed installation instructions <plumedinstall_>`_. from LAMMPS using the generic `plumed installation instructions <plumedinstall_>`_.
The USER-PLUMED package has been tested to work with Plumed versions
2.4.x, 2.5.x, and 2.6.x and will error out, when trying to run calculations
with a different version of the Plumed kernel.
PLUMED can be linked into MD codes in three different modes: static, PLUMED can be linked into MD codes in three different modes: static,
@ -1212,13 +1215,10 @@ USER-QMMM package
for a QM/MM simulation. You must also build Quantum ESPRESSO and for a QM/MM simulation. You must also build Quantum ESPRESSO and
create a new executable (pwqmmm.x) which links LAMMPS and Quantum create a new executable (pwqmmm.x) which links LAMMPS and Quantum
ESPRESSO together. These are steps 3 and 4 described in the ESPRESSO together. These are steps 3 and 4 described in the
lib/qmmm/README file. Unfortunately, the Quantum ESPRESSO developers lib/qmmm/README file. This requires a compatible Quantum espresso
have been breaking the interface that the QM/MM code in LAMMPS is using, and LAMMPS version. The current interface and makefiles have
so that currently (Summer 2018) using this feature requires either last been verified to work in February 2020 with Quantum Espresso
correcting the library interface feature in recent Quantum ESPRESSO versions 6.3 to 6.5.
releases, or using an outdated version of QE. The last version of
Quantum ESPRESSO known to work with this QM/MM interface was version
5.4.1 from 2016.
**CMake build**\ : **CMake build**\ :

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@ -17,9 +17,9 @@ General remarks
LAMMPS is developed and tested primarily on Linux machines. The vast LAMMPS is developed and tested primarily on Linux machines. The vast
majority of HPC clusters and supercomputers today runs on Linux as well. majority of HPC clusters and supercomputers today runs on Linux as well.
Thus portability to other platforms is desired, but not always achieved. While portability to other platforms is desired, it is not always achieved.
The LAMMPS developers strongly rely on LAMMPS users giving feedback and The LAMMPS developers strongly rely on LAMMPS users giving feedback and
providing assistance in resolving portability issues. This particularly providing assistance in resolving portability issues. This is particularly
true for compiling LAMMPS on Windows, since this platform has significant true for compiling LAMMPS on Windows, since this platform has significant
differences with some low-level functionality. differences with some low-level functionality.
@ -31,18 +31,20 @@ Running Linux on Windows
So before trying to build LAMMPS on Windows, please consider if using So before trying to build LAMMPS on Windows, please consider if using
the pre-compiled Windows binary packages are sufficient for your needs the pre-compiled Windows binary packages are sufficient for your needs
(as an aside, those packages themselves are build on a Linux machine (as an aside, those packages themselves are build on a Linux machine
using cross-compilers). If it is necessary for your to compile LAMMPS using cross-compilers). If it is necessary for you to compile LAMMPS
on a Windows machine (e.g. because it is your main desktop), please also on a Windows machine (e.g. because it is your main desktop), please also
consider using a virtual machine software and run a Linux virtual machine, consider using a virtual machine software and compile and run LAMMPS in
or - if have a recently updated Windows 10 installation - consider using a Linux virtual machine, or - if you have a recently updated Windows 10
the Windows subsystem for Linux, which allows to run a bash shell from installation - consider using the Windows subsystem for Linux. This
Ubuntu and from there on, you can pretty much use that shell like you optional Windows feature allows you to run the bash shell from Ubuntu
are running on an Ubuntu Linux machine (e.g. installing software via from within Windows and from there on, you can pretty much use that
apt-get). For more details on that, please see :doc:`this tutorial <Howto_bash>` shell like you are running on an Ubuntu Linux machine (e.g. installing
software via apt-get and more). For more details on that, please
see :doc:`this tutorial <Howto_bash>`
.. _gnu: .. _gnu:
Using GNU GCC ported to Windows Using a GNU GCC ported to Windows
----------------------------------------- -----------------------------------------
One option for compiling LAMMPS on Windows natively, that has been known One option for compiling LAMMPS on Windows natively, that has been known
@ -83,13 +85,13 @@ traditional build system, but CMake has also been successfully tested
using the mingw32-cmake and mingw64-cmake wrappers that are bundled using the mingw32-cmake and mingw64-cmake wrappers that are bundled
with the cross-compiler environment on Fedora machines. A CMake preset with the cross-compiler environment on Fedora machines. A CMake preset
selecting all packages compatible with this cross-compilation build selecting all packages compatible with this cross-compilation build
is provided. You likely need to disable the GPU package unless you is provided. You will likely need to disable the GPU package unless you
download and install the contents of the pre-compiled `OpenCL ICD loader library <https://download.lammps.org/thirdparty/opencl-win-devel.tar.gz>`_ download and install the contents of the pre-compiled `OpenCL ICD loader library <https://download.lammps.org/thirdparty/opencl-win-devel.tar.gz>`_
into your MinGW64 cross-compiler environment. The cross-compilation into your MinGW64 cross-compiler environment. The cross-compilation
currently will only produce non-MPI serial binaries. currently will only produce non-MPI serial binaries.
Please keep in mind, though, that this only applies to compiling LAMMPS. Please keep in mind, though, that this only applies to **compiling** LAMMPS.
Whether the resulting binaries do work correctly is no tested by the Whether the resulting binaries do work correctly is not tested by the
LAMMPS developers. We instead rely on the feedback of the users LAMMPS developers. We instead rely on the feedback of the users
of these pre-compiled LAMMPS packages for Windows. We will try to resolve of these pre-compiled LAMMPS packages for Windows. We will try to resolve
issues to the best of our abilities if we become aware of them. However issues to the best of our abilities if we become aware of them. However

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@ -1,9 +0,0 @@
\documentclass[12pt]{article}
\begin{document}
$$
CS = \sum_{i = 1}^{N/2} | \vec{R}_i + \vec{R}_{i+N/2} |^2
$$
\end{document}

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\documentclass[12pt,article]{article}
\usepackage{indentfirst}
\usepackage{amsmath}
\begin{document}
\begin{eqnarray*}
r_{c}^{fcc} & = & \frac{1}{2} \left(\frac{\sqrt{2}}{2} + 1\right) \mathrm{a} \simeq 0.8536 \:\mathrm{a} \\
r_{c}^{bcc} & = & \frac{1}{2}(\sqrt{2} + 1) \mathrm{a} \simeq 1.207 \:\mathrm{a} \\
r_{c}^{hcp} & = & \frac{1}{2}\left(1+\sqrt{\frac{4+2x^{2}}{3}}\right) \mathrm{a}
\end{eqnarray*}
\end{document}

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@ -1,12 +0,0 @@
\documentclass[12pt,article]{article}
\usepackage{indentfirst}
\usepackage{amsmath}
\begin{document}
$$
Rc + Rs > 2*{\rm cutoff}
$$
\end{document}

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@ -1,14 +0,0 @@
\documentclass[12pt,article]{article}
\usepackage{indentfirst}
\usepackage{amsmath}
\begin{document}
\begin{eqnarray*}
r_{c}^{fcc} & = & \frac{1}{2} \left(\frac{\sqrt{2}}{2} + 1\right) \mathrm{a} \simeq 0.8536 \:\mathrm{a} \\
r_{c}^{bcc} & = & \frac{1}{2}(\sqrt{2} + 1) \mathrm{a} \simeq 1.207 \:\mathrm{a} \\
r_{c}^{hcp} & = & \frac{1}{2}\left(1+\sqrt{\frac{4+2x^{2}}{3}}\right) \mathrm{a}
\end{eqnarray*}
\end{document}

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\documentclass[12pt,article]{article}
\usepackage{indentfirst}
\usepackage{amsmath}
\begin{document}
$$
Rc + Rs > 2*{\rm cutoff}
$$
\end{document}

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@ -1,9 +0,0 @@
\documentclass[12pt]{article}
\begin{document}
$$
Q_{i} = \frac{1}{n_i}\sum_{j = 1}^{n_i} | \sum_{k = 1}^{n_{ij}} \vec{R}_{ik} + \vec{R}_{jk} |^2
$$
\end{document}

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@ -1,13 +0,0 @@
\documentstyle[12pt]{article}
\pagestyle{empty}
\begin{document}
\begin{eqnarray*}
U^{cond} = \displaystyle\sum_{i=1}^{N} u_{i}^{cond} \\
U^{mech} = \displaystyle\sum_{i=1}^{N} u_{i}^{mech} \\
U^{chem} = \displaystyle\sum_{i=1}^{N} u_{i}^{chem} \\
U = \displaystyle\sum_{i=1}^{N} (u_{i}^{cond} + u_{i}^{mech} + u_{i}^{chem}) \\
\theta_{avg} = (\frac{1}{N}\displaystyle\sum_{i=1}^{N} \frac{1}{\theta_{i}})^{-1} \\
\end{eqnarray*}
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
\[ \left< \frac{1}{1 + \exp\left[\left(U_1 - U_0 - \Delta_0^1A \right) /kT \right]} \right>_0 = \left< \frac{1}{1 + \exp\left[\left(U_0 - U_1 + \Delta_0^1A \right) /kT \right]} \right>_1 \]
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
\[ \Delta_0^1 A = \int_{\lambda=0}^{\lambda=1} \left( \frac{\partial
A(\lambda)}{\partial\lambda} \right)_\lambda \mathrm{d}\lambda
\approx \sum_{i=0}^{n-1} w_i \frac{A(\lambda_{i} + \delta) -
A(\lambda_i)}{\delta} \]
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
\[ \Delta_0^1 A = \sum_{i=0}^{n-1} \Delta_{\lambda_i}^{\lambda_{i+1}} A =
- kT \sum_{i=0}^{n-1} \ln \left< \exp \left( - \frac{U(\lambda_{i+1}) -
U(\lambda_i)}{kT} \right) \right>_{\lambda_i} \]
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
\begin{eqnarray*}
\lambda = 0 \quad\Rightarrow\quad U = U_{\mathrm{bg}} + U_0 \\
\lambda = 1 \quad\Rightarrow\quad U = U_{\mathrm{bg}} + U_1
\end{eqnarray*}
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
\[ \Delta_0^1 A = \int_{\lambda=0}^{\lambda=1} \left< \frac{\partial
U(\lambda)}{\partial\lambda} \right>_\lambda \mathrm{d}\lambda
\approx \sum_{i=0}^{n-1} w_i \left< \frac{U(\lambda_{i} + \delta) -
U(\lambda_i)}{\delta} \right>_{\lambda_i} \]
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
\[ U(\lambda) = U_{\mathrm{bg}} + U_1(\lambda) + U_0(\lambda) \]
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
\[ \Delta_0^1 A = - kT \sum_{i=0}^{n-1} \ln \frac{\left< V \exp \left( -
\frac{U(\lambda_{i+1}) - U(\lambda_i)}{kT} \right)
\right>_{\lambda_i}}{\left< V \right>_{\lambda_i}} \]
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
$$
{R_g}^2 = \frac{1}{M} \sum_i m_i (r_i - r_{cm})^2
$$
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
$$
NGP(t) = 3<(r(t)-r(0))^4>/(5<(r(t)-r(0))^2>^2) - 1
$$
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
$$
I=\frac{F^{*}F}{N}
$$
\end{document}

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@ -1,9 +0,0 @@
\documentstyle[12pt]{article}
\begin{document}
$$
F(\mathbf{k})=\sum_{j=1}^{N}f_j(\theta)exp(2\pi i \mathbf{k}\cdot \mathbf{r}_j)
$$
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
$$
f_j\left ( \frac{sin(\theta)}{\lambda} \right )=\sum_{i}^{5}
a_i exp\left ( -b_i \frac{sin^{2}(\theta)}{\lambda^{2}} \right )
$$
\end{document}

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@ -1,13 +0,0 @@
\documentclass[12pt]{article}
\pagestyle{empty}
\begin{document}
\begin{eqnarray*}
c = l_z - 0.5(l_y+l_x) \\
b = l_y - l_x \\
k = \frac{3}{2} \frac{l_x^2+l_y^2+l_z^2}{(l_x+l_y+l_z)^2} - \frac{1}{2}
\end{eqnarray*}
\end{document}

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\documentclass[24pt]{article}
\pagestyle{empty}
\begin{document}
\begin{eqnarray*}
\theta_0 = {\tt rfac0} \frac{r-r_{min0}}{R_{ii'}-r_{min0}} \pi
\end{eqnarray*}
\end{document}

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\documentclass[24pt]{article}
\pagestyle{empty}
\begin{document}
\begin{eqnarray*}
u^j_{m,m'} = U^j_{m,m'}(0,0,0) + \sum_{r_{ii'} < R_{ii'}}{f_c(r_{ii'}) w_{i'} U^j_{m,m'}(\theta_0,\theta,\phi)}
\end{eqnarray*}
\end{document}

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\documentclass[24pt]{article}
\pagestyle{empty}
\begin{document}
\newcommand{\hcoeff}[9]{H\!\!{\tiny\begin{array}{l}#1 #2 #3 \\ #4 #5 #6 \\ #7 #8 #9 \end{array}}}
\begin{equation}
B_{j_1,j_2,j} = \\
\sum_{m_1,m'_1=-j_1}^{j_1}\sum_{m_2,m'_2=-j_2}^{j_2}\sum_{m,m'=-j}^{j} (u^j_{m,m'})^*
\hcoeff{j}{m}{m'}{j_1}{\!m_1}{\!m'_1}{j_2}{m_2}{m'_2}
u^{j_1}_{m_1,m'_1} u^{j_2}_{m_2,m'_2}
\end{equation}
\end{document}

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\documentclass[24pt]{article}
\pagestyle{empty}
\begin{document}
\begin{eqnarray*}
\label{eqn:f_c}
f_c(r) & = & \frac{1}{2}(\cos(\pi \frac{r-r_{min0}}{R_{ii'}-r_{min0}}) + 1), r \leq R_{ii'} \\
& = & 0, r > R_{ii'}
\end{eqnarray*}
\end{document}

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@ -1,12 +0,0 @@
\documentclass[24pt]{article}
\pagestyle{empty}
\begin{document}
\begin{equation}
- \sum_{i' \in I} \frac{\partial {B^{i'}_{j_1,j_2,j} }}{\partial {\bf r}_i}
\end{equation}
\end{document}

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\documentclass[24pt]{article}
\pagestyle{empty}
\begin{document}
\begin{eqnarray*}
- {\bf r}_i \otimes \sum_{i' \in I} \frac{\partial {B^{i'}_{j_1,j_2,j}}}{\partial {\bf r}_i}
\end{eqnarray*}
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
$$
I=Lp(\theta)\frac{F^{*}F}{N}
$$
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
$$
F(\mathbf{k})=\sum_{j=1}^{N}f_j(\theta)exp(2\pi i \mathbf{k}\cdot \mathbf{r}_j)
$$
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
$$
Lp(\theta)=\frac{1+cos^{2}(2\theta)}{cos(\theta)sin^{2}(\theta)}
$$
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
$$
\frac{sin(\theta)}{\lambda}=\frac{\left | \mathbf{k} \right |}{2}
$$
\end{document}

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\documentstyle[12pt]{article}
\begin{document}
$$
f_j\left ( \frac{sin(\theta)}{\lambda} \right )=\sum_{i}^{4}
a_i exp\left ( -b_i \frac{sin^{2}(\theta)}{\lambda^{2}} \right )+c
$$
\end{document}

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\documentstyle[12pt]{article}
\pagestyle{empty}
\begin{document}
\begin{eqnarray*}
k = AT^{n}e^{\frac{-E_{a}}{k_{B}T}}
\end{eqnarray*}
\end{document}

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\documentclass[12pt]{article}
\begin{document}
$$
E = U + P_t \left(V-V_0 \right) + E_{strain}
$$
\end{document}

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\documentclass[12pt]{article}
\begin{document}
$$
\mathbf P = P_t \mathbf I + {\mathbf S_t} \left( \mathbf h_0^{-1} \right)^t \mathbf h_{0d}
$$
\end{document}

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\documentclass[24pt]{article}
\pagestyle{empty}
\Huge
\begin{document}
\begin{eqnarray*}
\frac{dc}{dt} &=&<EFBFBD> -\alpha (K_p e + K_i \int_0^t e \, dt + K_d \frac{de}{dt} ) \\
\end{eqnarray*}
\end{document}

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\documentclass[24pt]{article}
\pagestyle{empty}
\Huge
\begin{document}
\begin{eqnarray*}
c_n &=&<EFBFBD> c_{n-1} -\alpha (K_p \tau e_n + K_i \tau^2 \sum_{i=1}^n e_i + K_d (e_n - e_{n-1}) )
\end{eqnarray*}
\end{document}

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\documentclass[12pt]{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
\dot{\mathbf r}_i &= \mathbf v_i, \\
\dot{\mathbf v}_i &= \frac{\mathbf f_i}{m_i} + \frac{\mathbf g_i}{m_i}.
\end{align*}
\end{document}

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\documentclass[12pt]{article}
\usepackage{amsmath}
\begin{document}
\begin{equation*}
\mathbf g_i =
\begin{cases} \frac{m_i}{2} \frac{ F_{\Gamma_{k(\mathbf r_i)}}}{ K_{\Gamma_{k(\mathbf r_i)}}}
\left(\mathbf v_i - \mathbf v_{\Gamma_{k(\mathbf r_i)}} \right) & \mbox{$k(\mathbf r_i)> 0$ (inside a reservoir),} \\
0 & \mbox{otherwise, }
\end{cases}
\end{equation*}
\end{document}

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\documentstyle[12pt]{article}
\pagestyle{empty}
\begin{document}
$$
u_{i} = u^{mech}_{i} + u^{cond}_{i} = C_{V} \theta_{i}
$$
\end{document}

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\documentstyle[12pt]{article}
\pagestyle{empty}
\begin{document}
\begin{eqnarray*}
U_{i} = \displaystyle\sum_{j=1}^{m} c_{i,j}(u_{j} + \Delta H_{f,j}) + \frac{3k_{b}T}{2} + Nk_{b}T \\
\end{eqnarray*}
\end{document}

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\documentclass[12pt]{article}
\begin{document}
\begin{eqnarray*}
\mu &=&\mu^{id} + \mu^{ex}
\end{eqnarray*}
\end{document}

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\documentclass[12pt]{article}
\begin{document}
\begin{eqnarray*}
\mu^{id} &=& k T \ln{\rho \Lambda^3} \\
&=& k T \ln{\frac{\phi P \Lambda^3}{k T}}
\end{eqnarray*}
\end{document}

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\documentclass[12pt]{article}
\begin{document}
\begin{eqnarray*}
\Lambda &=& \sqrt{ \frac{h^2}{2 \pi m k T}}
\end{eqnarray*}
\end{document}

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\documentclass[12pt]{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
&{\bf F}_{j}(t) = {\bf F}^C_j(t)-\int \limits_{0}^{t} \Gamma_j(t-s) {\bf v}_j(s)~\text{d}s + {\bf F}^R_j(t) \\
&\Gamma_j(t-s) = \sum \limits_{k=1}^{N_k} \frac{c_k}{\tau_k} e^{-(t-s)/\tau_k} \\
&\langle{\bf F}^R_j(t),{\bf F}^R_j(s)\rangle = \text{k$_\text{B}$T} ~\Gamma_j(t-s)
\end{align*}
\end{document}

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\documentclass[12pt]{article}
\begin{document}
$$
T_{eff} = \lambda + \eta (H - H_0)
$$
\end{document}

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\documentclass[preview]{standalone}
\usepackage{varwidth}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath,amssymb,amsthm,bm,tikz}
\usetikzlibrary{automata,arrows,shapes,snakes}
\begin{document}
\begin{varwidth}{50in}
\begin{tikzpicture}
%Global
\node (v1) at (0,6.0) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] { $\bm{v} \leftarrow \bm{v}+L_v.\Delta t/2$ };
\node (s1) at (0,4.5) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] { $\bm{s} \leftarrow \bm{s}+L_s.\Delta t/2$ };
\node (r) at (0,3.0) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] { $\bm{r} \leftarrow \bm{r}+L_r.\Delta t$ };
\node (s2) at (0,1.5) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] { $\bm{s} \leftarrow \bm{s}+L_s.\Delta t/2$ };
\node (v2) at (0,0.0) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] { $\bm{v} \leftarrow \bm{v}+L_v.\Delta t/2$ };
\draw[line width=2pt, ->] (v1) -- (s1);
\draw[line width=2pt, ->] (s1) -- (r);
\draw[line width=2pt, ->] (r) -- (s2);
\draw[line width=2pt, ->] (s2) -- (v2);
%Spin
\node (s01) at (6,6.0) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] {$\bm{s}_0 \leftarrow \bm{s}_0+L_{s_0}.\Delta t/4$ };
\node (sN1) at (6,4.5) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] {$\bm{s}_{\rm N-1}\leftarrow\bm{s}_{\rm N-1}+L_{s_{\rm N-1}}.\Delta t/4$};
\node (sN) at (6,3.0) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] {$\bm{s}_{\rm N} \leftarrow \bm{s}_{\rm N}+L_{s_{\rm N}}.\Delta t/2$ };
\node (sN2) at (6,1.5) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] {$\bm{s}_{\rm N-1}\leftarrow\bm{s}_{\rm N-1}+L_{s_{\rm N-1}}.\Delta t/4$};
\node (s02) at (6,0.0) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] {$\bm{s}_0 \leftarrow \bm{s}_0+L_{s_0}.\Delta t/4$ };
\draw[line width=2pt,dashed, ->] (s01) -- (sN1);
\draw[line width=2pt, ->] (sN1) -- (sN);
\draw[line width=2pt, ->] (sN) -- (sN2);
\draw[line width=2pt,dashed, ->] (sN2) -- (s02);
%from Global to Spin
\draw[line width=2pt, dashed, ->] (s1) -- (s01.west);
\draw[line width=2pt, dashed, ->] (s1) -- (s02.west);
\end{tikzpicture}
\end{varwidth}
\end{document}

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\documentclass[12pt]{article}
\begin{document}
$$
\left(\partial_t + e_{i\alpha}\partial_{\alpha}\right)f_i = -\frac{1}{\tau}\left(f_i - f_i^{eq}\right) + W_i
$$
\end{document}

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\documentclass[12pt]{article}
\begin{document}
$$
{\bf F}_{j \alpha} = \gamma \left({\bf v}_n - {\bf u}_f \right) \zeta_{j\alpha}
$$
\end{document}

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