fix bond/react, box/relax, ehex

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Axel Kohlmeyer 2020-02-11 16:14:00 +01:00
parent 6d5c001b61
commit e4a730d57a
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13 changed files with 32 additions and 65 deletions

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@ -1,9 +0,0 @@
\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[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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@ -374,8 +374,10 @@ reacting molecules.
The constraint of type 'arrhenius' imposes an additional reaction
probability according to the temperature-dependent Arrhenius equation:
.. image:: Eqs/fix_bond_react.jpg
:align: center
.. math::
k = AT^{n}e^{\frac{-E_{a}}{k_{B}T}}
The Arrhenius constraint has the following syntax:
@ -385,8 +387,8 @@ The Arrhenius constraint has the following syntax:
arrhenius *A* *n* *E_a* *seed*
where 'arrhenius' is the required keyword, *A* is the pre-exponential
factor, *n* is the exponent of the temperature dependence, *E\_a* is
the activation energy (:doc:`units <units>` of energy), and *seed* is a
factor, *n* is the exponent of the temperature dependence, :math:`E_a`
is the activation energy (:doc:`units <units>` of energy), and *seed* is a
random number seed. The temperature is defined as the instantaneous
temperature averaged over all atoms in the reaction site, and is
calculated in the same manner as for example

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@ -228,23 +228,28 @@ With this fix, the potential energy used by the minimizer is augmented
by an additional energy provided by the fix. The overall objective
function then is:
.. image:: Eqs/fix_box_relax1.jpg
:align: center
.. math::
where *U* is the system potential energy, *P*\ \_t is the desired
hydrostatic pressure, *V* and *V*\ \_0 are the system and reference
volumes, respectively. *E*\ \_\ *strain* is the strain energy expression
E = U + P_t \left(V-V_0 \right) + E_{strain}
where *U* is the system potential energy, :math:`P_t` is the desired
hydrostatic pressure, :math:`V` and :math:`V_0` are the system and reference
volumes, respectively. :math:`E_{strain}` is the strain energy expression
proposed by Parrinello and Rahman :ref:`(Parrinello1981) <Parrinello1981>`.
Taking derivatives of *E* w.r.t. the box dimensions, and setting these
to zero, we find that at the minimum of the objective function, the
global system stress tensor **P** will satisfy the relation:
.. image:: Eqs/fix_box_relax2.jpg
:align: center
.. math::
where **I** is the identity matrix, **h**\ \_0 is the box dimension tensor of
the reference cell, and **h**\ \_0\ *d* is the diagonal part of
**h**\ \_0. **S**\ \_\ *t* is a symmetric stress tensor that is chosen by LAMMPS
\mathbf P = P_t \mathbf I + {\mathbf S_t} \left( \mathbf h_0^{-1} \right)^t \mathbf h_{0d}
where **I** is the identity matrix, :math:`\mathbf{h_0}` is the box
dimension tensor of the reference cell, and ::math:`\mathbf{h_{0d}}`
is the diagonal part of :math:`\mathbf{h_0}`. :math:`\mathbf{S_t}`
is a symmetric stress tensor that is chosen by LAMMPS
so that the upper-triangular components of **P** equal the stress tensor
specified by the user.

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@ -78,13 +78,20 @@ additional thermostatting force to the equations of motion, such that
the time evolution of coordinates and momenta of particle :math:`i`
becomes :ref:`(Wirnsberger) <Wirnsberger>`
.. image:: Eqs/fix_ehex_eom.jpg
:align: center
.. math::
\dot{\mathbf r}_i &= \mathbf v_i, \\
\dot{\mathbf v}_i &= \frac{\mathbf f_i}{m_i} + \frac{\mathbf g_i}{m_i}.
The thermostatting force is given by
.. image:: Eqs/fix_ehex_f.jpg
:align: center
.. math::
\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}
where :math:`m_i` is the mass and :math:`k(\mathbf r_i)` maps the particle
position to the respective reservoir. The quantity