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Update pair_ylz.rst

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Corrected some typo in formulations
This commit is contained in:
Mehdi Baghaee
2022-10-19 18:52:31 +08:00
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commit 2590f9eea4
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doc/src/pair_ylz.rst
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@ -33,23 +33,21 @@ relative particle orientations via the formulas
.. math::
U ( \mathbf{r}_{ij}, \mathbf{n}_i, \mathbf{n}_j ) =\left\{\begin{matrix} \mathbf{u}_R(r)+\left [ 1-\phi (\mathbf{r\hat{}}_{ij}, \mathbf{n}_i, \mathbf{n}_j ) \right ]\epsilon, ~~ r<\mathbf{r}_{min} \\ \mathbf{u}_A(r)\phi (\mathbf{r\hat{}}_{ij}, \mathbf{n}_i, \mathbf{n}_j ),~~ \mathbf{r}_{min}<r<\mathbf{r}_{c} \\ \end{matrix}\right.\\\\ \phi (\mathbf{r\hat{}}_{ij}, \mathbf{n}_i, \mathbf{n}_j )=1+\mu (a(\mathbf{r\hat{}}_{ij}, \mathbf{n}_i, \mathbf{n}_j )-1) \\\\ a(\mathbf{r\hat{}}_{ij}, \mathbf{n}_i, \mathbf{n}_j )=(\mathbf{n}_i\times\mathbf{r\hat{}}_{ij} )\cdot (\mathbf{n}_j\times\mathbf{r\hat{}}_{ij} )+sin\mathbf{\theta}_0(\mathbf{n}_i-\mathbf{n}_j)\cdot \mathbf{r\hat{}}_{ij}\\\\ \mathbf{u}_R(r)=\epsilon \left [ \left ( \frac{{r}_{min}}{r} \right )^{4}-2\left ( \frac{{r}_{min}}{r}\right )^{2} \right ] \\\\ \mathbf{u}_A(r)=-\epsilon\;cos^{2\zeta }\left ( \frac{\pi}{2}\frac{\left ( {r}-{r}_{min} \right )}{\left ( {r}_{c}-{r}_{min} \right )} \right ) \\
U ( \mathbf{r}_{ij}, \mathbf{n}_i, \mathbf{n}_j ) =\left\{\begin{matrix} {u}_R(r)+\left [ 1-\phi (\mathbf{\hat{r}}_{ij}, \mathbf{n}_i, \mathbf{n}_j ) \right ]\epsilon, ~~ r<{r}_{min} \\ {u}_A(r)\phi (\mathbf{\hat{r}}_{ij}, \mathbf{n}_i, \mathbf{n}_j ),~~ {r}_{min}<r<{r}_{c} \\ \end{matrix}\right.\\\\ \phi (\mathbf{\hat{r}}_{ij}, \mathbf{n}_i, \mathbf{n}_j )=1+\left [ \mu (a(\mathbf{\hat{r}}_{ij}, \mathbf{n}_i, \mathbf{n}_j )-1) \right ] \\\\a(\mathbf{\hat{r}}_{ij}, \mathbf{n}_i, \mathbf{n}_j )=(\mathbf{n}_i\times\mathbf{\hat{r}}_{ij} )\cdot (\mathbf{n}_j\times\mathbf{\hat{r}}_{ij} )+{\beta}(\mathbf{n}_i-\mathbf{n}_j)\cdot \mathbf{\hat{r}}_{ij}-\beta^{2}\\\\ {u}_R(r)=\epsilon \left [ \left ( \frac{{r}_{min}}{r} \right )^{4}-2\left ( \frac{{r}_{min}}{r}\right )^{2} \right ] \\\\ {u}_A(r)=-\epsilon\;cos^{2\zeta }\left [ \frac{\pi}{2}\frac{\left ( {r}-{r}_{min} \right )}{\left ( {r}_{c}-{r}_{min} \right )} \right ]\\
where :math:`r_{i}` and :math:`r_{j}` are the center position vectors of
particles i and j, respectively, :math:`r_{ij}=r_{i}-r_{j}` is the
inter-particle distance vector, :math:`r=\left|r_{ij} \right|` and
:math:`{r\hat{}}_{ij}=\mathbf{r}_{ij}/r`. The unit vectors :math:`n_{i}`
and :math:`n_{j}` represent the axes of symmetry of particles i and j,
respectively. :math:`u_R` and :math:`u_A` are the repulsive and
attractive potentials. :math:`\phi` is an angular function which depends
on the relative orientation between pair particles. :math:`\mu` is the
parameter related to bending rigidity, :math:`\theta_{0}` is the
parameter related to the spontaneous curvature, and :math:`\epsilon` is
the energy unit, respectively. The :math:`\zeta` controls the slope of
the attractive branch and :math:`{r}_{c}`is the cutoff
radius. :math:`r_{min}` is the distance which minimizes the potential
energy :math:`u_{A}(r)`and :math:`r_{min}=2^{1/6}\sigma`, where
:math:`\sigma` is the length unit.
where :math:`\mathbf{r}_{i}` and :math:`\mathbf{r}_{j}` are the center position
vectors of particles i and j, respectively, :math:`\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}`
is the inter-particle distance vector, :math:`r=\left|\mathbf{r}_{ij} \right|` and
:math:`{\hat{\mathbf{r}}}_{ij}=\mathbf{r}_{ij}/r`. The unit vectors
:math:`\mathbf{n}_{i}` and :math:`\mathbf{n}_{j}` represent the axes of symmetry of
particles i and j, respectively, :math:`u_R` and :math:`u_A` are the
repulsive and attractive potentials, :math:`\phi` is an angular function which depends on
the relative orientation between pair particles, :math:`\mu` is the parameter
related to bending rigidity, :math:`\beta` is the parameter related to the spontaneous
curvature, and :math:`\epsilon` is the energy unit, respectively. The :math:`\zeta` controls
the slope of the attractive branch and :math:`{r}_{c}`is the cutoff radius. :math:`r_{min}`
is the distance which minimizes the potential energy :math:`u_{A}(r)`and
:math:`r_{min}=2^{1/6}\sigma`, where :math:`\sigma` is the length unit.
Use of this pair style requires the NVE, NVT, or NPT fixes with the *asphere* extension (e.g. :doc:`fix nve/asphere <fix_nve_asphere>`) in order to integrate particle rotation. Additionally, :doc:`atom_style ellipsoid <atom_style>` should be used since it defines the rotational state of each particle.
@ -59,7 +57,7 @@ The following coefficients must be defined for each pair of atoms types via the
* :math:`\sigma` = minimum effective particle radii (distance units)
* :math:`\zeta` = tune parameter for the slope of the attractive branch
* :math:`\mu` = parameter related to bending rigidity
* :math:`\sin(\theta _{0})` is the sine of parameter related to the spontaneous curvature
* :math:`\beta` = parameter related to the spontaneous curvature
* cutoff (distance units)
The last coefficient is optional. If not specified, the global