forked from lijiext/lammps
fix bond/react, box/relax, ehex
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\documentstyle[12pt]{article}
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\pagestyle{empty}
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\begin{document}
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\begin{eqnarray*}
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k = AT^{n}e^{\frac{-E_{a}}{k_{B}T}}
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\end{eqnarray*}
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\end{document}
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\documentclass[12pt]{article}
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\begin{document}
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$$
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E = U + P_t \left(V-V_0 \right) + E_{strain}
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$$
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\end{document}
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\documentclass[12pt]{article}
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\begin{document}
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$$
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\mathbf P = P_t \mathbf I + {\mathbf S_t} \left( \mathbf h_0^{-1} \right)^t \mathbf h_{0d}
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$$
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\end{document}
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\documentclass[12pt]{article}
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\usepackage{amsmath}
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\begin{document}
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\begin{align*}
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\dot{\mathbf r}_i &= \mathbf v_i, \\
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\dot{\mathbf v}_i &= \frac{\mathbf f_i}{m_i} + \frac{\mathbf g_i}{m_i}.
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\end{align*}
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\end{document}
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\documentclass[12pt]{article}
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\usepackage{amsmath}
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\begin{document}
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\begin{equation*}
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\mathbf g_i =
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\begin{cases} \frac{m_i}{2} \frac{ F_{\Gamma_{k(\mathbf r_i)}}}{ K_{\Gamma_{k(\mathbf r_i)}}}
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\left(\mathbf v_i - \mathbf v_{\Gamma_{k(\mathbf r_i)}} \right) & \mbox{$k(\mathbf r_i)> 0$ (inside a reservoir),} \\
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0 & \mbox{otherwise, }
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\end{cases}
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\end{equation*}
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\end{document}
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@ -374,8 +374,10 @@ reacting molecules.
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The constraint of type 'arrhenius' imposes an additional reaction
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probability according to the temperature-dependent Arrhenius equation:
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.. image:: Eqs/fix_bond_react.jpg
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:align: center
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.. math::
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k = AT^{n}e^{\frac{-E_{a}}{k_{B}T}}
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The Arrhenius constraint has the following syntax:
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@ -385,8 +387,8 @@ The Arrhenius constraint has the following syntax:
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arrhenius *A* *n* *E_a* *seed*
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where 'arrhenius' is the required keyword, *A* is the pre-exponential
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factor, *n* is the exponent of the temperature dependence, *E\_a* is
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the activation energy (:doc:`units <units>` of energy), and *seed* is a
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factor, *n* is the exponent of the temperature dependence, :math:`E_a`
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is the activation energy (:doc:`units <units>` of energy), and *seed* is a
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random number seed. The temperature is defined as the instantaneous
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temperature averaged over all atoms in the reaction site, and is
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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
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by an additional energy provided by the fix. The overall objective
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function then is:
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.. image:: Eqs/fix_box_relax1.jpg
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:align: center
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.. math::
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where *U* is the system potential energy, *P*\ \_t is the desired
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hydrostatic pressure, *V* and *V*\ \_0 are the system and reference
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volumes, respectively. *E*\ \_\ *strain* is the strain energy expression
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E = U + P_t \left(V-V_0 \right) + E_{strain}
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where *U* is the system potential energy, :math:`P_t` is the desired
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hydrostatic pressure, :math:`V` and :math:`V_0` are the system and reference
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volumes, respectively. :math:`E_{strain}` is the strain energy expression
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proposed by Parrinello and Rahman :ref:`(Parrinello1981) <Parrinello1981>`.
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Taking derivatives of *E* w.r.t. the box dimensions, and setting these
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to zero, we find that at the minimum of the objective function, the
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global system stress tensor **P** will satisfy the relation:
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.. image:: Eqs/fix_box_relax2.jpg
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:align: center
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.. math::
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where **I** is the identity matrix, **h**\ \_0 is the box dimension tensor of
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the reference cell, and **h**\ \_0\ *d* is the diagonal part of
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**h**\ \_0. **S**\ \_\ *t* is a symmetric stress tensor that is chosen by LAMMPS
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\mathbf P = P_t \mathbf I + {\mathbf S_t} \left( \mathbf h_0^{-1} \right)^t \mathbf h_{0d}
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where **I** is the identity matrix, :math:`\mathbf{h_0}` is the box
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dimension tensor of the reference cell, and ::math:`\mathbf{h_{0d}}`
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is the diagonal part of :math:`\mathbf{h_0}`. :math:`\mathbf{S_t}`
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is a symmetric stress tensor that is chosen by LAMMPS
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so that the upper-triangular components of **P** equal the stress tensor
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specified by the user.
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@ -78,13 +78,20 @@ additional thermostatting force to the equations of motion, such that
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the time evolution of coordinates and momenta of particle :math:`i`
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becomes :ref:`(Wirnsberger) <Wirnsberger>`
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.. image:: Eqs/fix_ehex_eom.jpg
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:align: center
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.. math::
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\dot{\mathbf r}_i &= \mathbf v_i, \\
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\dot{\mathbf v}_i &= \frac{\mathbf f_i}{m_i} + \frac{\mathbf g_i}{m_i}.
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The thermostatting force is given by
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.. image:: Eqs/fix_ehex_f.jpg
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:align: center
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.. math::
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\mathbf g_i = \begin{cases}
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\frac{m_i}{2} \frac{ F_{\Gamma_{k(\mathbf r_i)}}}{ K_{\Gamma_{k(\mathbf r_i)}}}
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\left(\mathbf v_i - \mathbf v_{\Gamma_{k(\mathbf r_i)}} \right) & \mbox{$k(\mathbf r_i)> 0$ (inside a reservoir),} \\
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0 & \mbox{otherwise, }
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\end{cases}
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where :math:`m_i` is the mass and :math:`k(\mathbf r_i)` maps the particle
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position to the respective reservoir. The quantity
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