From e4a730d57a2d344c66b3d87c07924ecaf3d262d1 Mon Sep 17 00:00:00 2001 From: Axel Kohlmeyer Date: Tue, 11 Feb 2020 16:14:00 +0100 Subject: [PATCH] fix bond/react, box/relax, ehex --- doc/src/Eqs/fix_bond_react.jpg | Bin 2427 -> 0 bytes doc/src/Eqs/fix_bond_react.tex | 9 --------- doc/src/Eqs/fix_box_relax1.jpg | Bin 3375 -> 0 bytes doc/src/Eqs/fix_box_relax1.tex | 9 --------- doc/src/Eqs/fix_box_relax2.jpg | Bin 3029 -> 0 bytes doc/src/Eqs/fix_box_relax2.tex | 9 --------- doc/src/Eqs/fix_ehex_eom.jpg | Bin 3272 -> 0 bytes doc/src/Eqs/fix_ehex_eom.tex | 8 -------- doc/src/Eqs/fix_ehex_f.jpg | Bin 10352 -> 0 bytes doc/src/Eqs/fix_ehex_f.tex | 12 ------------ doc/src/fix_bond_react.rst | 10 ++++++---- doc/src/fix_box_relax.rst | 25 +++++++++++++++---------- doc/src/fix_ehex.rst | 15 +++++++++++---- 13 files changed, 32 insertions(+), 65 deletions(-) delete mode 100644 doc/src/Eqs/fix_bond_react.jpg delete mode 100644 doc/src/Eqs/fix_bond_react.tex delete mode 100644 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zwe)*cdH)L56lu6`<1iciuFufs-*ozS|Ba&Qe19srtbrM3VnwO`yp$L`5BjDm9lbHvM{@yPqL7(=l-DEaZG>0O1!E0^It!K{XepN4|D$u D8v-&F diff --git a/doc/src/Eqs/fix_ehex_f.tex b/doc/src/Eqs/fix_ehex_f.tex deleted file mode 100644 index f6ffe1d720..0000000000 --- a/doc/src/Eqs/fix_ehex_f.tex +++ /dev/null @@ -1,12 +0,0 @@ -\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} - diff --git a/doc/src/fix_bond_react.rst b/doc/src/fix_bond_react.rst index ec431168b8..14d04009e7 100644 --- a/doc/src/fix_bond_react.rst +++ b/doc/src/fix_bond_react.rst @@ -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 ` of energy), and *seed* is a +factor, *n* is the exponent of the temperature dependence, :math:`E_a` +is the activation energy (:doc:`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 diff --git a/doc/src/fix_box_relax.rst b/doc/src/fix_box_relax.rst index d915d1a2a0..0c4cad7ad0 100644 --- a/doc/src/fix_box_relax.rst +++ b/doc/src/fix_box_relax.rst @@ -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) `. 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. diff --git a/doc/src/fix_ehex.rst b/doc/src/fix_ehex.rst index 34445fd284..9cc02261c7 100644 --- a/doc/src/fix_ehex.rst +++ b/doc/src/fix_ehex.rst @@ -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) ` -.. 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